Exact solutions of the Dirac equation and induced representations of the Poincare group on

The electronic properties of grapheneA.H.Castro NetoDepartment of Physics,Boston University,590Commonwealth Avenue,Boston,Massachusetts02215,USAF.GuineaInstituto de Ciencia de Materiales de Madrid,CSIC,Cantoblanco,E-28049Madrid,SpainN.M.R.PeresCenter of Physics and Department of Physics,Universidade do Minho,P-4710-057,Braga,PortugalK.S.Novoselov and A.K.GeimDepartment of Physics and Astronomy,University of Manchester,Manchester,M139PL,United Kingdom͑Published14January2009͒This article reviews the basic theoretical aspects of graphene,a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations.The Dirac electrons can be controlled by application of external electric and magneticfields,or by altering sample geometry and/or topology.The Dirac electrons behave in unusual ways in tunneling,confinement,and the integer quantum Hall effect.The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers.Edge͑surface͒states in graphene depend on the edge termination͑zigzag or armchair͒and affect the physical properties of nanoribbons.Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties.The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.DOI:10.1103/RevModPhys.81.109PACS number͑s͒:81.05.Uw,73.20.Ϫr,03.65.Pm,82.45.MpCONTENTSI.Introduction110II.Elementary Electronic Properties of Graphene112A.Single layer:Tight-binding approach1121.Cyclotron mass1132.Density of states114B.Dirac fermions1141.Chiral tunneling and Klein paradox1152.Confinement and Zitterbewegung117C.Bilayer graphene:Tight-binding approach118D.Epitaxial graphene119E.Graphene stacks1201.Electronic structure of bulk graphite121F.Surface states in graphene122G.Surface states in graphene stacks124H.The spectrum of graphene nanoribbons1241.Zigzag nanoribbons1252.Armchair nanoribbons126I.Dirac fermions in a magneticfield126J.The anomalous integer quantum Hall effect128 K.Tight-binding model in a magneticfield128 ndau levels in graphene stacks130 M.Diamagnetism130 N.Spin-orbit coupling131 III.Flexural Phonons,Elasticity,and Crumpling132 IV.Disorder in Graphene134A.Ripples135B.Topological lattice defects136C.Impurity states137D.Localized states near edges,cracks,and voids137E.Self-doping138F.Vector potential and gaugefield disorder1391.Gaugefield induced by curvature1402.Elastic strain1403.Random gaugefields141G.Coupling to magnetic impurities141H.Weak and strong localization142I.Transport near the Dirac point143J.Boltzmann equation description of dc transport indoped graphene144 K.Magnetotransport and universal conductivity1451.The full self-consistent Born approximation͑FSBA͒146 V.Many-Body Effects148A.Electron-phonon interactions148B.Electron-electron interactions1501.Screening in graphene stacks152C.Short-range interactions1521.Bilayer graphene:Exchange1532.Bilayer graphene:Short-range interactions154D.Interactions in high magneticfields154VI.Conclusions154 Acknowledgments155 References155REVIEWS OF MODERN PHYSICS,VOLUME81,JANUARY–MARCH20090034-6861/2009/81͑1͒/109͑54͒©2009The American Physical Society109I.INTRODUCTIONCarbon is the materia prima for life and the basis of all organic chemistry.Because of the flexibility of its bond-ing,carbon-based systems show an unlimited number of different structures with an equally large variety of physical properties.These physical properties are,in great part,the result of the dimensionality of these structures.Among systems with only carbon atoms,graphene—a two-dimensional ͑2D ͒allotrope of carbon—plays an important role since it is the basis for the understanding of the electronic properties in other allotropes.Graphene is made out of carbon atoms ar-ranged on a honeycomb structure made out of hexagons ͑see Fig.1͒,and can be thought of as composed of ben-zene rings stripped out from their hydrogen atoms ͑Pauling,1972͒.Fullerenes ͑Andreoni,2000͒are mol-ecules where carbon atoms are arranged spherically,and hence,from the physical point of view,are zero-dimensional objects with discrete energy states.Fullerenes can be obtained from graphene with the in-troduction of pentagons ͑that create positive curvature defects ͒,and hence,fullerenes can be thought as wrapped-up graphene.Carbon nanotubes ͑Saito et al.,1998;Charlier et al.,2007͒are obtained by rolling graphene along a given direction and reconnecting the carbon bonds.Hence carbon nanotubes have only hexa-gons and can be thought of as one-dimensional ͑1D ͒ob-jects.Graphite,a three dimensional ͑3D ͒allotrope of carbon,became widely known after the invention of the pencil in 1564͑Petroski,1989͒,and its usefulness as an instrument for writing comes from the fact that graphite is made out of stacks of graphene layers that are weakly coupled by van der Waals forces.Hence,when one presses a pencil against a sheet of paper,one is actually producing graphene stacks and,somewhere among them,there could be individual graphene layers.Al-though graphene is the mother for all these different allotropes and has been presumably produced every time someone writes with a pencil,it was only isolated 440years after its invention ͑Novoselov et al.,2004͒.The reason is that,first,no one actually expected graphene to exist in the free state and,second,even with the ben-efit of hindsight,no experimental tools existed to search for one-atom-thick flakes among the pencil debris cov-ering macroscopic areas ͑Geim and MacDonald,2007͒.Graphene was eventually spotted due to the subtle op-tical effect it creates on top of a chosen SiO 2substrate ͑Novoselov et al.,2004͒that allows its observation with an ordinary optical microscope ͑Abergel et al.,2007;Blake et al.,2007;Casiraghi et al.,2007͒.Hence,graphene is relatively straightforward to make,but not so easy to find.The structural flexibility of graphene is reflected in its electronic properties.The sp 2hybridization between one s orbital and two p orbitals leads to a trigonal planar structure with a formation of a ␴bond between carbon atoms that are separated by 1.42Å.The ␴band is re-sponsible for the robustness of the lattice structure in all allotropes.Due to the Pauli principle,these bands have a filled shell and,hence,form a deep valence band.The unaffected p orbital,which is perpendicular to the pla-nar structure,can bind covalently with neighboring car-bon atoms,leading to the formation of a ␲band.Since each p orbital has one extra electron,the ␲band is half filled.Half-filled bands in transition elements have played an important role in the physics of strongly correlated systems since,due to their strong tight-binding charac-ter,the Coulomb energies are large,leading to strong collective effects,magnetism,and insulating behavior due to correlation gaps or Mottness ͑Phillips,2006͒.In fact,Linus Pauling proposed in the 1950s that,on the basis of the electronic properties of benzene,graphene should be a resonant valence bond ͑RVB ͒structure ͑Pauling,1972͒.RVB states have become popular in the literature of transition-metal oxides,and particularly in studies of cuprate-oxide superconductors ͑Maple,1998͒.This point of view should be contrasted with contempo-raneous band-structure studies of graphene ͑Wallace,1947͒that found it to be a semimetal with unusual lin-early dispersing electronic excitations called Dirac elec-trons.While most current experimental data in graphene support the band structure point of view,the role of electron-electron interactions in graphene is a subject of intense research.It was P .R.Wallace in 1946who first wrote on the band structure of graphene and showed the unusual semimetallic behavior in this material ͑Wallace,1947͒.At that time,the thought of a purely 2D structure was not reality and Wallace’s studies of graphene served him as a starting point to study graphite,an important mate-rial for nuclear reactors in the post–World War II era.During the following years,the study of graphite culmi-nated with the Slonczewski-Weiss-McClure ͑SWM ͒band structure of graphite,which provided a description of the electronic properties in this material ͑McClure,1957;Slonczewski and Weiss,1958͒and was successful in de-scribing the experimental data ͑Boyle and Nozières 1958;McClure,1958;Spry and Scherer,1960;Soule et al.,1964;Williamson et al.,1965;Dillon et al.,1977͒.From 1957to 1968,the assignment of the electron and hole states within the SWM model were oppositetoFIG.1.͑Color online ͒Graphene ͑top left ͒is a honeycomb lattice of carbon atoms.Graphite ͑top right ͒can be viewed as a stack of graphene layers.Carbon nanotubes are rolled-up cylinders of graphene ͑bottom left ͒.Fullerenes ͑C 60͒are mol-ecules consisting of wrapped graphene by the introduction of pentagons on the hexagonal lattice.From Castro Neto et al.,2006a .110Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009what is accepted today.In1968,Schroeder et al.͑Schroeder et al.,1968͒established the currently ac-cepted location of electron and hole pockets͑McClure, 1971͒.The SWM model has been revisited in recent years because of its inability to describe the van der Waals–like interactions between graphene planes,a problem that requires the understanding of many-body effects that go beyond the band-structure description ͑Rydberg et al.,2003͒.These issues,however,do not arise in the context of a single graphene crystal but they show up when graphene layers are stacked on top of each other,as in the case,for instance,of the bilayer graphene.Stacking can change the electronic properties considerably and the layering structure can be used in order to control the electronic properties.One of the most interesting aspects of the graphene problem is that its low-energy excitations are massless, chiral,Dirac fermions.In neutral graphene,the chemical potential crosses exactly the Dirac point.This particular dispersion,that is only valid at low energies,mimics the physics of quantum electrodynamics͑QED͒for massless fermions except for the fact that in graphene the Dirac fermions move with a speed v F,which is300times smaller than the speed of light c.Hence,many of the unusual properties of QED can show up in graphene but at much smaller speeds͑Castro Neto et al.,2006a; Katsnelson et al.,2006;Katsnelson and Novoselov, 2007͒.Dirac fermions behave in unusual ways when compared to ordinary electrons if subjected to magnetic fields,leading to new physical phenomena͑Gusynin and Sharapov,2005;Peres,Guinea,and Castro Neto,2006a͒such as the anomalous integer quantum Hall effect ͑IQHE͒measured experimentally͑Novoselov,Geim, Morozov,et al.,2005a;Zhang et al.,2005͒.Besides being qualitatively different from the IQHE observed in Si and GaAlAs͑heterostructures͒devices͑Stone,1992͒, the IQHE in graphene can be observed at room tem-perature because of the large cyclotron energies for “relativistic”electrons͑Novoselov et al.,2007͒.In fact, the anomalous IQHE is the trademark of Dirac fermion behavior.Another interesting feature of Dirac fermions is their insensitivity to external electrostatic potentials due to the so-called Klein paradox,that is,the fact that Dirac fermions can be transmitted with probability1through a classically forbidden region͑Calogeracos and Dombey, 1999;Itzykson and Zuber,2006͒.In fact,Dirac fermions behave in an unusual way in the presence of confining potentials,leading to the phenomenon of Zitter-bewegung,or jittery motion of the wave function͑Itzyk-son and Zuber,2006͒.In graphene,these electrostatic potentials can be easily generated by disorder.Since dis-order is unavoidable in any material,there has been a great deal of interest in trying to understand how disor-der affects the physics of electrons in graphene and its transport properties.In fact,under certain conditions, Dirac fermions are immune to localization effects ob-served in ordinary electrons͑Lee and Ramakrishnan, 1985͒and it has been established experimentally that electrons can propagate without scattering over large distances of the order of micrometers in graphene͑No-voselov et al.,2004͒.The sources of disorder in graphene are many and can vary from ordinary effects commonly found in semiconductors,such as ionized impurities in the Si substrate,to adatoms and various molecules ad-sorbed in the graphene surface,to more unusual defects such as ripples associated with the soft structure of graphene͑Meyer,Geim,Katsnelson,Novoselov,Booth, et al.,2007a͒.In fact,graphene is unique in the sense that it shares properties of soft membranes͑Nelson et al.,2004͒and at the same time it behaves in a metallic way,so that the Dirac fermions propagate on a locally curved space.Here analogies with problems of quantum gravity become apparent͑Fauser et al.,2007͒.The soft-ness of graphene is related with the fact that it has out-of-plane vibrational modes͑phonons͒that cannot be found in3D solids.Theseflexural modes,responsible for the bending properties of graphene,also account for the lack of long range structural order in soft mem-branes leading to the phenomenon of crumpling͑Nelson et al.,2004͒.Nevertheless,the presence of a substrate or scaffolds that hold graphene in place can stabilize a cer-tain degree of order in graphene but leaves behind the so-called ripples͑which can be viewed as frozenflexural modes͒.It was realized early on that graphene should also present unusual mesoscopic effects͑Peres,Castro Neto, and Guinea,2006a;Katsnelson,2007a͒.These effects have their origin in the boundary conditions required for the wave functions in mesoscopic samples with various types of edges graphene can have͑Nakada et al.,1996; Wakabayashi et al.,1999;Peres,Guinea,and Castro Neto,2006a;Akhmerov and Beenakker,2008͒.The most studied edges,zigzag and armchair,have drastically different electronic properties.Zigzag edges can sustain edge͑surface͒states and resonances that are not present in the armchair case.Moreover,when coupled to con-ducting leads,the boundary conditions for a graphene ribbon strongly affect its conductance,and the chiral Dirac nature of fermions in graphene can be used for applications where one can control the valleyflavor of the electrons besides its charge,the so-called valleytron-ics͑Rycerz et al.,2007͒.Furthermore,when supercon-ducting contacts are attached to graphene,they lead to the development of supercurrentflow and Andreev pro-cesses characteristic of the superconducting proximity effect͑Heersche et al.,2007͒.The fact that Cooper pairs can propagate so well in graphene attests to the robust electronic coherence in this material.In fact,quantum interference phenomena such as weak localization,uni-versal conductancefluctuations͑Morozov et al.,2006͒, and the Aharonov-Bohm effect in graphene rings have already been observed experimentally͑Recher et al., 2007;Russo,2007͒.The ballistic electronic propagation in graphene can be used forfield-effect devices such as p-n͑Cheianov and Fal’ko,2006;Cheianov,Fal’ko,and Altshuler,2007;Huard et al.,2007;Lemme et al.,2007; Tworzydlo et al.,2007;Williams et al.,2007;Fogler, Glazman,Novikov,et al.,2008;Zhang and Fogler,2008͒and p-n-p͑Ossipov et al.,2007͒junctions,and as“neu-111Castro Neto et al.:The electronic properties of graphene Rev.Mod.Phys.,V ol.81,No.1,January–March2009trino”billiards ͑Berry and Modragon,1987;Miao et al.,2007͒.It has also been suggested that Coulomb interac-tions are considerably enhanced in smaller geometries,such as graphene quantum dots ͑Milton Pereira et al.,2007͒,leading to unusual Coulomb blockade effects ͑Geim and Novoselov,2007͒and perhaps to magnetic phenomena such as the Kondo effect.The transport properties of graphene allow for their use in a plethora of applications ranging from single molecule detection ͑Schedin et al.,2007;Wehling et al.,2008͒to spin injec-tion ͑Cho et al.,2007;Hill et al.,2007;Ohishi et al.,2007;Tombros et al.,2007͒.Because of its unusual structural and electronic flex-ibility,graphene can be tailored chemically and/or struc-turally in many different ways:deposition of metal at-oms ͑Calandra and Mauri,2007;Uchoa et al.,2008͒or molecules ͑Schedin et al.,2007;Leenaerts et al.,2008;Wehling et al.,2008͒on top;intercalation ͓as done in graphite intercalated compounds ͑Dresselhaus et al.,1983;Tanuma and Kamimura,1985;Dresselhaus and Dresselhaus,2002͔͒;incorporation of nitrogen and/or boron in its structure ͑Martins et al.,2007;Peres,Klironomos,Tsai,et al.,2007͓͒in analogy with what has been done in nanotubes ͑Stephan et al.,1994͔͒;and using different substrates that modify the electronic structure ͑Calizo et al.,2007;Giovannetti et al.,2007;Varchon et al.,2007;Zhou et al.,2007;Das et al.,2008;Faugeras et al.,2008͒.The control of graphene properties can be extended in new directions allowing for the creation of graphene-based systems with magnetic and supercon-ducting properties ͑Uchoa and Castro Neto,2007͒that are unique in their 2D properties.Although the graphene field is still in its infancy,the scientific and technological possibilities of this new material seem to be unlimited.The understanding and control of this ma-terial’s properties can open doors for a new frontier in electronics.As the current status of the experiment and potential applications have recently been reviewed ͑Geim and Novoselov,2007͒,in this paper we concen-trate on the theory and more technical aspects of elec-tronic properties with this exciting new material.II.ELEMENTARY ELECTRONIC PROPERTIES OF GRAPHENEA.Single layer:Tight-binding approachGraphene is made out of carbon atoms arranged in hexagonal structure,as shown in Fig.2.The structure can be seen as a triangular lattice with a basis of two atoms per unit cell.The lattice vectors can be written asa 1=a 2͑3,ͱ3͒,a 2=a2͑3,−ͱ3͒,͑1͒where a Ϸ1.42Åis the carbon-carbon distance.Thereciprocal-lattice vectors are given byb 1=2␲3a͑1,ͱ3͒,b 2=2␲3a͑1,−ͱ3͒.͑2͒Of particular importance for the physics of graphene are the two points K and K Јat the corners of the graphene Brillouin zone ͑BZ ͒.These are named Dirac points for reasons that will become clear later.Their positions in momentum space are given byK =ͩ2␲3a ,2␲3ͱ3aͪ,K Ј=ͩ2␲3a ,−2␲3ͱ3aͪ.͑3͒The three nearest-neighbor vectors in real space are given by␦1=a 2͑1,ͱ3͒␦2=a 2͑1,−ͱ3͒␦3=−a ͑1,0͒͑4͒while the six second-nearest neighbors are located at ␦1Ј=±a 1,␦2Ј=±a 2,␦3Ј=±͑a 2−a 1͒.The tight-binding Hamiltonian for electrons in graphene considering that electrons can hop to both nearest-and next-nearest-neighbor atoms has the form ͑we use units such that ប=1͒H =−t͚͗i ,j ͘,␴͑a ␴,i †b ␴,j +H.c.͒−t Ј͚͗͗i ,j ͘͘,␴͑a ␴,i †a ␴,j +b ␴,i †b ␴,j +H.c.͒,͑5͒where a i ,␴͑a i ,␴†͒annihilates ͑creates ͒an electron with spin ␴͑␴=↑,↓͒on site R i on sublattice A ͑an equiva-lent definition is used for sublattice B ͒,t ͑Ϸ2.8eV ͒is the nearest-neighbor hopping energy ͑hopping between dif-ferent sublattices ͒,and t Јis the next nearest-neighbor hopping energy 1͑hopping in the same sublattice ͒.The energy bands derived from this Hamiltonian have the form ͑Wallace,1947͒E ±͑k ͒=±t ͱ3+f ͑k ͒−t Јf ͑k ͒,1The value of t Јis not well known but ab initio calculations ͑Reich et al.,2002͒find 0.02t Շt ЈՇ0.2t depending on the tight-binding parametrization.These calculations also include the effect of a third-nearest-neighbors hopping,which has a value of around 0.07eV.A tight-binding fit to cyclotron resonance experiments ͑Deacon et al.,2007͒finds t ЈϷ0.1eV.FIG.2.͑Color online ͒Honeycomb lattice and its Brillouin zone.Left:lattice structure of graphene,made out of two in-terpenetrating triangular lattices ͑a 1and a 2are the lattice unit vectors,and ␦i ,i =1,2,3are the nearest-neighbor vectors ͒.Right:corresponding Brillouin zone.The Dirac cones are lo-cated at the K and K Јpoints.112Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009f ͑k ͒=2cos ͑ͱ3k y a ͒+4cosͩͱ32k y a ͪcosͩ32k x a ͪ,͑6͒where the plus sign applies to the upper ͑␲*͒and the minus sign the lower ͑␲͒band.It is clear from Eq.͑6͒that the spectrum is symmetric around zero energy if t Ј=0.For finite values of t Ј,the electron-hole symmetry is broken and the ␲and ␲*bands become asymmetric.In Fig.3,we show the full band structure of graphene with both t and t Ј.In the same figure,we also show a zoom in of the band structure close to one of the Dirac points ͑at the K or K Јpoint in the BZ ͒.This dispersion can be obtained by expanding the full band structure,Eq.͑6͒,close to the K ͑or K Ј͒vector,Eq.͑3͒,as k =K +q ,with ͉q ͉Ӷ͉K ͉͑Wallace,1947͒,E ±͑q ͒Ϸ±vF ͉q ͉+O ͓͑q /K ͒2͔,͑7͒where q is the momentum measured relatively to the Dirac points and v F is the Fermi velocity,given by v F =3ta /2,with a value v F Ӎ1ϫ106m/s.This result was first obtained by Wallace ͑1947͒.The most striking difference between this result and the usual case,⑀͑q ͒=q 2/͑2m ͒,where m is the electron mass,is that the Fermi velocity in Eq.͑7͒does not de-pend on the energy or momentum:in the usual case we have v =k /m =ͱ2E /m and hence the velocity changes substantially with energy.The expansion of the spectrum around the Dirac point including t Јup to second order in q /K is given byE ±͑q ͒Ӎ3t Ј±vF ͉q ͉−ͩ9t Јa 24±3ta 28sin ͑3␪q ͉͒ͪq ͉2,͑8͒where␪q =arctanͩq x q yͪ͑9͒is the angle in momentum space.Hence,the presence of t Јshifts in energy the position of the Dirac point and breaks electron-hole symmetry.Note that up to order ͑q /K ͒2the dispersion depends on the direction in mo-mentum space and has a threefold symmetry.This is the so-called trigonal warping of the electronic spectrum ͑Ando et al.,1998,Dresselhaus and Dresselhaus,2002͒.1.Cyclotron massThe energy dispersion ͑7͒resembles the energy of ul-trarelativistic particles;these particles are quantum me-chanically described by the massless Dirac equation ͑see Sec.II.B for more on this analogy ͒.An immediate con-sequence of this massless Dirac-like dispersion is a cy-clotron mass that depends on the electronic density as its square root ͑Novoselov,Geim,Morozov,et al.,2005;Zhang et al.,2005͒.The cyclotron mass is defined,within the semiclassical approximation ͑Ashcroft and Mermin,1976͒,asm *=12␲ͫץA ͑E ͒ץEͬE =E F,͑10͒with A ͑E ͒the area in k space enclosed by the orbit andgiven byA ͑E ͒=␲q ͑E ͒2=␲E 2v F2.͑11͒Using Eq.͑11͒in Eq.͑10͒,one obtainsm *=E Fv F2=k Fv F.͑12͒The electronic density n is related to the Fermi momen-tum k F as k F2/␲=n ͑with contributions from the two Dirac points K and K Јand spin included ͒,which leads tom *=ͱ␲v Fͱn .͑13͒Fitting Eq.͑13͒to the experimental data ͑see Fig.4͒provides an estimation for the Fermi velocity andtheFIG.3.͑Color online ͒Electronic dispersion in the honeycomb lattice.Left:energy spectrum ͑in units of t ͒for finite values of t and t Ј,with t =2.7eV and t Ј=−0.2t .Right:zoom in of the energy bands close to one of the Diracpoints.FIG.4.͑Color online ͒Cyclotron mass of charge carriers in graphene as a function of their concentration n .Positive and negative n correspond to electrons and holes,respectively.Symbols are the experimental data extracted from the tem-perature dependence of the SdH oscillations;solid curves are the best fit by Eq.͑13͒.m 0is the free-electron mass.Adapted from Novoselov,Geim,Morozov,et al.,2005.113Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009hopping parameter as v F Ϸ106ms −1and t Ϸ3eV,respec-tively.Experimental observation of the ͱn dependence on the cyclotron mass provides evidence for the exis-tence of massless Dirac quasiparticles in graphene ͑No-voselov,Geim,Morozov,et al.,2005;Zhang et al.,2005;Deacon et al.,2007;Jiang,Henriksen,Tung,et al.,2007͒—the usual parabolic ͑Schrödinger ͒dispersion im-plies a constant cyclotron mass.2.Density of statesThe density of states per unit cell,derived from Eq.͑5͒,is given in Fig.5for both t Ј=0and t Ј 0,showing in both cases semimetallic behavior ͑Wallace,1947;Bena and Kivelson,2005͒.For t Ј=0,it is possible to derive an analytical expression for the density of states per unit cell,which has the form ͑Hobson and Nierenberg,1953͒␳͑E ͒=4␲2͉E ͉t 21ͱZ 0F ͩ␲2,ͱZ 1Z 0ͪ,Z 0=Άͩ1+ͯE t ͯͪ2−͓͑E /t ͒2−1͔24,−t ഛE ഛt4ͯE t ͯ,−3t ഛE ഛ−t ∨t ഛE ഛ3t ,Z 1=Ά4ͯE t ͯ,−t ഛE ഛtͩ1+ͯE tͯͪ2−͓͑E /t ͒2−1͔24,−3t ഛE ഛ−t ∨t ഛE ഛ3t ,͑14͒where F ͑␲/2,x ͒is the complete elliptic integral of thefirst kind.Close to the Dirac point,the dispersion is ap-proximated by Eq.͑7͒and the density of states per unit cell is given by ͑with a degeneracy of 4included ͒␳͑E ͒=2A c ␲͉E ͉v F2,͑15͒where A c is the unit cell area given by A c =3ͱ3a 2/2.It is worth noting that the density of states for graphene is different from the density of states of carbon nanotubes ͑Saito et al.,1992a ,1992b ͒.The latter shows 1/ͱE singu-larities due to the 1D nature of their electronic spec-trum,which occurs due to the quantization of the mo-mentum in the direction perpendicular to the tube axis.From this perspective,graphene nanoribbons,which also have momentum quantization perpendicular to the ribbon length,have properties similar to carbon nano-tubes.B.Dirac fermionsWe consider the Hamiltonian ͑5͒with t Ј=0and theFourier transform of the electron operators,a n =1ͱN c͚ke −i k ·R na ͑k ͒,͑16͒where N c is the number of unit ing this transfor-mation,we write the field a n as a sum of two terms,coming from expanding the Fourier sum around K Јand K .This produces an approximation for the representa-tion of the field a n as a sum of two new fields,written asa n Ӎe −i K ·R n a 1,n +e −i K Ј·R n a 2,n ,b n Ӎe −i K ·R n b 1,n +e −i K Ј·R n b 2,n ,͑17͒ρ(ε)ε/tρ(ε)ε/tFIG.5.Density of states per unit cell as a function of energy ͑in units of t ͒computed from the energy dispersion ͑5͒,t Ј=0.2t ͑top ͒and t Ј=0͑bottom ͒.Also shown is a zoom-in of the density of states close to the neutrality point of one electron per site.For the case t Ј=0,the electron-hole nature of the spectrum is apparent and the density of states close to the neutrality point can be approximated by ␳͑⑀͒ϰ͉⑀͉.114Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009where the index i =1͑i =2͒refers to the K ͑K Ј͒point.These new fields,a i ,n and b i ,n ,are assumed to vary slowly over the unit cell.The procedure for deriving a theory that is valid close to the Dirac point con-sists in using this representation in the tight-binding Hamiltonian and expanding the opera-tors up to a linear order in ␦.In the derivation,one uses the fact that ͚␦e ±i K ·␦=͚␦e ±i K Ј·␦=0.After some straightforward algebra,we arrive at ͑Semenoff,1984͒H Ӎ−t͵dxdy ⌿ˆ1†͑r ͒ͫͩ3a ͑1−i ͱ3͒/4−3a ͑1+i ͱ3͒/4ͪץx +ͩ3a ͑−i −ͱ3͒/4−3a ͑i −ͱ3͒/4ͪץy ͬ⌿ˆ1͑r ͒+⌿ˆ2†͑r ͒ͫͩ3a ͑1+i ͱ3͒/4−3a ͑1−i ͱ3͒/4ͪץx +ͩ3a ͑i −ͱ3͒/4−3a ͑−i −ͱ3͒/4ͪץy ͬ⌿ˆ2͑r ͒=−i v F͵dxdy ͓⌿ˆ1†͑r ͒␴·ٌ⌿ˆ1͑r ͒+⌿ˆ2†͑r ͒␴*·ٌ⌿ˆ2͑r ͔͒,͑18͒with Pauli matrices ␴=͑␴x ,␴y ͒,␴*=͑␴x ,−␴y ͒,and ⌿ˆi†=͑a i †,b i †͒͑i =1,2͒.It is clear that the effective Hamil-tonian ͑18͒is made of two copies of the massless Dirac-like Hamiltonian,one holding for p around K and the other for p around K Ј.Note that,in first quantized lan-guage,the two-component electron wave function ␺͑r ͒,close to the K point,obeys the 2D Dirac equation,−i v F ␴·ٌ␺͑r ͒=E ␺͑r ͒.͑19͒The wave function,in momentum space,for the mo-mentum around K has the form␺±,K ͑k ͒=1ͱ2ͩe −i ␪k /2±e i ␪k /2ͪ͑20͒for H K =v F ␴·k ,where the Ϯsigns correspond to the eigenenergies E =±v F k ,that is,for the ␲*and ␲bands,respectively,and ␪k is given by Eq.͑9͒.The wave func-tion for the momentum around K Јhas the form␺±,K Ј͑k ͒=1ͱ2ͩe i ␪k /2±e −i ␪k /2ͪ͑21͒for H K Ј=v F ␴*·k .Note that the wave functions at K and K Јare related by time-reversal symmetry:if we set the origin of coordinates in momentum space in the M point of the BZ ͑see Fig.2͒,time reversal becomes equivalent to a reflection along the k x axis,that is,͑k x ,k y ͒→͑k x ,−k y ͒.Also note that if the phase ␪is rotated by 2␲,the wave function changes sign indicating a phase of ␲͑in the literature this is commonly called a Berry’s phase ͒.This change of phase by ␲under rotation is char-acteristic of spinors.In fact,the wave function is a two-component spinor.A relevant quantity used to characterize the eigen-functions is their helicity defined as the projection of the momentum operator along the ͑pseudo ͒spin direction.The quantum-mechanical operator for the helicity has the formhˆ=12␴·p ͉p ͉.͑22͒It is clear from the definition of h ˆthat the states ␺K͑r ͒and ␺K Ј͑r ͒are also eigenstates of h ˆ,h ˆ␺K ͑r ͒=±12␺K͑r ͒,͑23͒and an equivalent equation for ␺K Ј͑r ͒with inverted sign.Therefore,electrons ͑holes ͒have a positive ͑negative ͒helicity.Equation ͑23͒implies that ␴has its two eigen-values either in the direction of ͑⇑͒or against ͑⇓͒the momentum p .This property says that the states of the system close to the Dirac point have well defined chiral-ity or helicity.Note that chirality is not defined in regard to the real spin of the electron ͑that has not yet ap-peared in the problem ͒but to a pseudospin variable as-sociated with the two components of the wave function.The helicity values are good quantum numbers as long as the Hamiltonian ͑18͒is valid.Therefore,the existence of helicity quantum numbers holds only as an asymptotic property,which is well defined close to the Dirac points K and K Ј.Either at larger energies or due to the presence of a finite t Ј,the helicity stops being a good quantum number.1.Chiral tunneling and Klein paradoxIn this section,we address the scattering of chiral elec-trons in two dimensions by a square barrier ͑Katsnelson et al.,2006;Katsnelson,2007b ͒.The one-dimensional scattering of chiral electrons was discussed earlier in the context on nanotubes ͑Ando et al.,1998;McEuen et al.,1999͒.We start by noting that by a gauge transformation the wave function ͑20͒can be written as115Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009。

Exact solutions of extended Boussinesq equations S.Hamdi(samir.hamdi@utoronto.ca)The Fields Institute for Research in Mathematical Sciences,222College Street,Toronto,Ontario,Canada,M5T3J1W.H.Enright(enright@)Department of Computer Science,University of Toronto,10King’s College Road,Toronto,Canada,M5S3G4Y.Ouellet(yvon.ouellet@gci.ulaval.ca)Département de Génie Civil,UniversitéLaval,Québec,Canada G1K7P4W.E.Schiesser(wes1@)Mathematics and Engineering,Lehigh University,Bethlehem,PA18015USASubmitted January2003Abstract.The problem addressed in this paper is the verification of numerical solutions of nonlinear dispersive wave equations such as Boussinesq-like system of equations.A practical verification tool is to compare the numer-ical solution to an exact solution if available.In this work,we derive some exact solitary wave solutions and several invariants of motion for a wide range of Boussinesq-like equations using Maple software.The exact solitary wave solutions can be used to specify initial data for the incident waves in the Boussinesq numerical model and for the verification of the associated computed solution.When exact solutions are not available,four invariants of motions can be used as verification tools for the conservation properties of the numerical model.Keywords:verification tools,exact solutions,Bousinesq equations,Invariants of motion1.IntroductionIn recent years,efforts have been made by a number of researchers to extend the range of appli-cability of Boussinesq equations to deeper water by improving their dispersion characteristics (see Madsen et al.[5],Nwogu[6],and Beji et al.[1]).Although most extended Boussinesq systems of equations have equivalent linearised dis-persion characteristics,similar shoaling properties and formally the same accuracy,the ex-tended Boussinesq equations proposed by Nwogu[6]have recently generated the most interest because they are easier to solve numerically in the case of variable depth[8].Several numerical schemes have been proposed to solve the Nwogu’s extended Boussinesq equations.Recently, Wei and Kirby[9]developed a numerical code,that is fourth-order accurate in time and space, for solving the Nwogu’s extended Boussinesq equations.The numerical solutions of Wei and Kirby[9]are more accurate than those obtained by Nwogu[6]since their solution technique is based on a higher orderfinite difference discretisation scheme coupled with a high-order2Hamdi,Enright,Ouellet and Schiesserpredictor-corrector time integration method.More recently,Walkley and Berzins[8]imple-mented a high order accurate method of lines solution using a Galerkinfinite element spatial discretisation technique coupled with the adaptive time integration package SPRINT[8] Both,Wei and Kirby[9]and Walkley and Berzins[8]investigated solitary wave propaga-tion over a long andflat bottom in order to assess the accuracy,the stability and the conservation properties of their numerical schemes.To apply this important test problem,Wei and Kirby[9] derived approximate analytical solitary wave solutions following a procedure described in Schember[7].The approximate analytical solutions are used to specify initial data for the incident waves in their numerical models and also to assess the accuracy of the numerical solution.The numerical results of Wei and Kirby[9]indicate that a slightly higher amplitude solitary wave is formed together with a small dispersive tail lagging behind,compared to the approximate analytical solution[7,9].The wave profiles show that the amplitude of the tail and the initial deviation in solitary-wave height both increase with increasing initial wave height. They also observed that the numerically predicted phase speed is somewhat smaller than the analytically predicted one,and that the difference increases with increasing wave height.Such discrepancies are explained by the fact that the analytical solution is only an approximation and does not correspond exactly to a solitary waveform as predicted by the model.Walkley and Berzins[8]reported very similar results to those of Wei and Kirby[9]for the same solitary wave test problem.They have also observed that there is an identical slight phase error in the numerical results and a small dispersive tail.They concluded that the approximate analytical solution is only an approximation and therefore exact agreement is unlikely.In this paper,we derive exact solitary wave solutions for the extended Nwogu Boussi-nesq equations.The exact solutions are obtained following an approach devised recently by Chen[2],which is more general than the procedure used in[7,9],that leads to approximate solutions only.To overcome the problems reported by Wei et al.[9]and Walkley et al.[8],it is recommended to use the exact solitary wave solutions that we propose in this study,instead of using approximate solutions which are not accurate enough for testing high-order accurate schemes such as those given in[8,9].New analytical expressions of four invariants of motions (mass,momentum,energy and Hamiltonian)are also derived.These constants of motion can be used to assess the accuracy and the conservative properties of numerical schemes for Nwogu’s Boussinesq models.2.The one-dimensional extended Bousssinesq equationsIn the case of wave propagation in the one-dimensional(1D)horizontal direction with constant depth,the extended Boussinesq equations derived by Nwogu[6]and used by Wei et al.[9]in their numerical code,reduce to the following:ηt hu xηu xα13h3u xxx0(1a)u t gηx uu xαh2u txx0(1b)withα1h2zαExact solutions of extended Boussinesq equations3 whereηsurface elevation;h local water depth;u u x t horizontal velocity at an ar-bitrary depth zα;and g the gravitational acceleration.These equations are statements of conservation of mass and momentum,respectively.Two important length scales are the characteristic water depth h0in the vertical direc-tion and a typical wavelength l in the horizontal direction.The following independent,non-dimensional variables can be defined:x ˜xhtl˜t(3)The tildes are used to connote dimensional variables as in the set of equations(1a)and(1b).For effects related to the motion of the free water surface,the typical wave amplitude a0is alsoimportant.The following dependent,non-dimensional variables can also be defined:u hgh0˜uη˜ηh0(4)Using the transformation(3)and(4),the Nwogu’s set of equations(1a)and(1b)are rewritten in dimensionless form as followsηt u xδηu xµ2α13u xxx0(5a)u tηxδuu xµ2αu txx0(5b) The dimensionless parametersδa0h0andµh0l are measures of nonlinearity andfrequency dispersion,respectively,and are assumed to be small.The parameterαreduces toα14Hamdi,Enright,Ouellet and SchiesserSubstituting into(5a)and(5b),the functionsηξand uξsatisfy the third order nonlinear system of ordinary differential equations(ODEs)Cηuδηuµ2α13u0(9a)Cuηδuuµ2αC u0(9b) in which the derivatives are performed with respect to the coordinateξ.The solitary wave solutions are localized in space,i.e.,the solution and its derivatives at large distance from the pulse are extremely small and vanish asymptoticallyηnξu nξ0asξ∞(10) Integrating once,with zero boundary conditions at infinity,Cδuηuµ2α13u0(11a)Cuη1(15a)α13α13CA2Exact solutions of extended Boussinesq equations5 Note that A2is well defined sinceαα13is always strictly positive for all values of interest of the parameterα.Substituting(14a)into(13a),we readilyfind that the functionηξsatisfies a second order nonlinear ODE,21A2ηµ2α13A2ηδA2η2(16) To allow another integration,wefirst multiply byη.Then each term can be integrated sepa-rately to obtain,1A2η213δA2η3(17)which can be written as the separable ODE.dη133αA(23)2δη033δη03(24)α13αα13(25)boussinesq.tex;6/01/2003;12:38;p.56Hamdi,Enright,Ouellet and Schiesser the maximum wave amplitudeη031A249α12µ2α13A2(27)(28) 12η0δ69α13u12Cδu2C21u0(32)The coefficients of this ODE depend on the unknown C.To solve this equation,we adopt a technique similar to that used by Kichennassamy and Olver[4].This technique is easier than solving uξdirectly from(32)and also more general than the procedure used in[7,9],which leads to approximate solutions only.First,we assume that the function uξcan be expressed as the solution of a singlefirst order ordinary differential equationφu u2(33) Solutions can also be reconstructed using functionsφu in different forms(see for example Yang et al.[10]).From equation(33),for u0,we haveu1Exact solutions of extended Boussinesq equations7 where u0is the maximum velocity amplitude,then the functionφdefined by equation(33)has to be the cubic polynomial:φu4κ2u21u(37)Now substituting the expression(36)forφinto(34),we can write the second derivative u as a second order polynomial in u.uλu33λ1uδCµ2αCλC32δ33ρu2δ12µ2αCρu30(39)It follows from(39)that all the coefficients of this polynomial must be zero in order to obtain a nontrivial solution u.Setting the coefficients to zero yields a nonlinear algebraic system of three equations for the three unknowns C,λ,andρ:δ12µ2αCρ0δCµ2αCλC32δ33ρ0C Cµ2αCλµ2α129α138Hamdi,Enright,Ouellet and SchiesserTherefore,from(37)we deduce the analytical expressions for the wave numberκ12α13µ2α(44) and the peak amplitude of the wave velocityu029α1αδ3α1(45)which yields the exact solitary wave solution for u x tu x t u0sech2κx x0Ct(46) The exact solution for the surface elevationηcan be obtained by substituting the solution(46) into(11b).After simplifications and collecting all the terms in the right-hand side,we obtain the exact expression for the solitary wave solutionηx t.ηx tη0sech2κx x0Ct(47) where the maximum wave amplitudeη0is given byη01αδ(48)It is worth confirming that the exact solutions u andηare proportionalu x t u0sech2κx x0Ct Aη0sech2κx x0Ct Aηx t(49) where the constant A is given byAu0αExact solutions of extended Boussinesq equations9 Using(20)forηx t,we obtainI12η06αµ2α139α1(53) Another,obvious invariant of motion is the integralI 2∞∞u x t d x(54)Substituting the expression(21)of the horizontal velocity in the above integral we obtainI2AI12η0κδη03(56)26µ2α139α13η20αδ23α1(61)Because dissipation is ignored in the derivation of Nwogu Boussinesq model,we can define a Hamiltonian form for the system(5a)and(5b)I 4Hηu∞∞η2x u2xη2u2u2ηd x(62)The Hamiltonian is a constant of motion,which to say that the functional H satisfiesHηx t u x t Hηx0u x0(63)boussinesq.tex;6/01/2003;12:38;p.910Hamdi,Enright,Ouellet and SchiesserSubstituting(20)and(21)forηx t and u x t respectively in(62),we obtainI44κ(64)which can be written in explicit formI4209α1µ29α12。

a rXiv:g r-qc/3124v56Mar28An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry F.Finster,N.Kamran ∗,J.Smoller †,and S.-T.Yau ‡June 2004Abstract We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon.We derive an integral representation which expresses the solution as a superposition of solutions of the radial and angular ODEs which arise in the separation of variables.In particular,we prove completeness of the solutions of the separated ODEs.This integral representation is a suitable starting point for a detailed analysis of the long-time dynamics of scalar waves in the Kerr geometry.Contents 1Introduction 22Preliminaries 73Spectral Properties of the Hamiltonian in a Finite Box 114Resolvent Estimates 165Separation of the Resolvent216WKB Estimates257Contour Deformations32References 401IntroductionIn a recent paper[8],the long-term behavior of Dirac spinorfields in the Kerr-Newman geometry,which describes a charged rotating black hole in equilibrium,was investigated. It was shown that solutions of the Dirac equation for Cauchy data in L2outside the event horizon and bounded near the event horizon,decay in L∞loc as t→∞.In this paper,we turn our attention to the scalar wave equation in the Kerr geometry.Our main result is to derive an integral representation for the propagator,similar to the one obtained for the Dirac equation in[8].In our next paper[9],we will use this integral representation to analyze the long-time dynamics and the decay of solutions in L∞loc.The analysis of the wave equation is quite different from that for the Dirac equation. The main difficulty is that,in contrast to the Dirac equation,there is no conserved density for the scalar wave equation which is positive everywhere outside the event horizon.This is due to the fact that the charge density,which was positive for the Dirac equation, is not positive for the wave equation.The other conserved density,the energy density, is non-positive either:it is in general negative inside the ergosphere,a region outside the event horizon in which the Killing vector corresponding to time translations becomes space-like.For these reasons,it is not possible to introduce a positive scalar product which is conserved in time.In more technical terms,we are faced with the difficulty that it is impossible to represent the Hamiltonian(i.e.the operator generating time translations) as a selfadjoint operator on a Hilbert space.We remark that the existence of the ergosphere is a direct consequence of the fact that the Kerr black hole has angular momentum[4].Thus the ergosphere vanishes in the spherically symmetric limit.This simplifies the analysis considerably.A number of important contributions have been made to the rigorous study of the scalar wave equation in black hole geometries.The current last word on the stability of spherical black holes under scalar wave perturbations is the paper by Kay and Wald[14], who proved using energy estimates together with a reflection argument that all solutions of the wave equation in the Schwarzschild geometry are bounded in L∞.More recently, Klainerman,Machedon,and Stalker[15]proved decay in L∞loc of spherically symmetric solutions.These papers use the spherical symmetry of the Schwarzschild metric in an essential way.Whiting[21]proved the absence of exponentially growing modes for the Teukolsky equation with general spin s=0,1selfadjoint,and has a spectral decomposition involving afinite set of complex spectral points,which appear in complex conjugate pairs,together with a discrete spectrum of real eigenvalues.We write the projectors onto the invariant subspaces as contour integrals of the resolvent.In order to obtain estimates for the resolvent,it is useful to consider the Hamiltonian as a non-selfadjoint operator on a Hilbert space.This procedure also works in the original infinite volume setting,and we derive operator estimates which compare the resolvent infinite volume to that in infinite ing these estimates,we can represent the spectral projector corresponding to the non-real spectrum as integrals over contours which are not closed and lie inside a region of of the form|Imω|<c(1+|Reω|)−1 around the real axis.At this point,we make use of the fact that the scalar wave equation in the Kerr geometry is separable into ordinary differential equations for the radial and angular parts[4].For the angular equation,we rely on the results of[10],where a spectral representation is obtained for the angular operator,and estimates for the eigenvalues and spectral projectors are derived.For the radial equation,we here derive rigorous estimates which are based on the semi-classical WKB ing these estimates,we can express the resolvent in terms of solutions of the ing furthermore Whiting’s result that the ODEs admit no normalizable solutions for complexω,we can deform the contours onto the real line.Thisfinally gives an integral representation for the propagator in terms of the solutions of the ODEs withωreal.To be more precise,recall that in Boyer-Lindquist coordinates(t,r,ϑ,ϕ)with r>0, 0≤ϑ≤π,0≤ϕ<2π,the Kerr metric takes the form[4,12]ds2=g jk dx j x k=∆∆+dϑ2 −sin2ϑM2−a2and r1=M+∂t,g ijξiξj=g tt=∆−a2sin2ϑU.(1.2)This shows thatξis space-like in the open region of space-time wherer2−2Mr+a2cos2ϑ<0,(1.3) the so-called ergosphere.It is a bounded region of space outside the event horizon,and intersects the event horizon at the polesϑ=0,π.The scalar wave equation in the Kerr geometry isΦ:=g ij∇i∇jΦ=1−g∂−g g ij∂∂r ∆∂∆ (r2+a2)∂∂ϕ2−∂∂cosϑ−1∂t+∂∂r∆∂∆((r2+a2)ω+ak)2(1.8) Aω,k=−∂∂cosϑ+1the wave equation(1.5)takes the form i ∂t Ψ=H Ψ,(1.12)where H is the HamiltonianH = 01αβ .(1.13)Here αand βare the differential operatorsα= (r 2+a 2)2∆−1∆−a 2sin 2ϑ −1 r 2+a 22πi k ∈Z e −ikϕ n ∈I N lim εց0 C ε−2π 2π0e ikϕΨ0(r,ϑ,ϕ)dϕ.We consider ωin the lower complex half plane {Im ω<0},and C εis a contour which joins the points ω=−∞with ω=∞and stays in an ε-neighborhood of the real line.A typical example is C ε={x −iεe −x 2:x ∈R }.2πi Ce −iωt (A −ω)−1dω,where A is a finite-dimensional matrix and C a contour which encloses the whole spectrum of A .For given ωand k ,the wave operator is a sum of a radial operator R ω,k and an angular operator A ω,k .As shown in [10],the angular operator has for ωnear the real line a purely discrete spectrum consisting of eigenvalues (λn )n ∈I N (see Lemma 2.1).The spectralU (dt −a sin 2ϑdϕ),(1.18)where q denotes the charge of the black hole,and the parameters M,a,q satisfy the inequality M 2>a 2+q 2.2PreliminariesIn this section we briefly recall the variational formulation of the wave equation and the separation of variables.Furthermore,we bring the equation into afirst-order Hamilto-nian form.Finally,we introduce and discuss scalar products which are needed for the construction of the propagator.The wave equation(1.5)is the Euler-Lagrange equation corresponding to the action S= ∞−∞dt ∞r1dr 1−1d(cosϑ) π0dϕL(Φ,∇Φ),(2.1) where the Lagrangian L is given byL=−∆|∂rΦ|2+1sin2ϑ (a sin2ϑ∂t+∂ϕ)Φ 2.(2.2) According to Noether’s theorem,symmetries of the Lagrangian give rise to conserved quantities.The symmetry under local gauge transformations yields that the vectorfieldJ k=−Im(2πQ,where Q is the charge densityQ=i∂L∆r2+a2 −a2sin2ϑa sin2ϑ .Moreover,since the Kerr metric is stationary,the Lagrangian is invariant under time translations.The corresponding conserved quantity is the energy E,E[Φ]= ∞r1dr 1−1d(cosϑ) 2π0dϕ∂ΦtΦt−L= (r2+a2)2sin2ϑ−a2Our analysis is based on a few properties of the angular operator Aω,which we now state.For realω,the angular operator Aωclearly is formally selfadjoint on L2(S2). However,this is not sufficient for our purpose,because we need to consider the case that ωis complex.In this case,Aωis a non-selfadjoint operator.Nevertheless,we have the following spectral decomposition,which is proved in[10].Lemma2.1(angular spectral decomposition)For any given c>0,we define the open set U⊂C by the condition|Imω|<cAfter separation (1.6),thereducedwave equation takes the form−∂∂r −1∂cos ϑsin 2ϑ∂sin 2ϑ(aωsin 2ϑ+k )2 Φ=0.(2.10)Under the separation,the above expressions for the charge and energy densities becomeQ =|Φ|2(r 2+a 2)2r 2+a 2 −a 2sin 2ϑ Re ω+k ∆ |ω|2−a 2k 2a 2sin 4ϑ+∆|∂r Φ|2+sin 2ϑ|∂cos ϑΦ|2.(2.12)It is a subtle point to find a scalar product <.,.>which is well-suited to the analysis of the wave equation.It is desirable to choose the scalar product such that the Hamiltonian H is Hermitian (i.e.formally selfadjoint)with respect to it.Since H is the infinitesimal generator of time translations,H is Hermitian w.r.to <.,.>if and only if the inner product <Ψ,Ψ>is time independent for all solutions Ψ=(Φ,i∂t Φ)of the wave equation.This can for example be achieved by imposing that <Ψ,Ψ>should be equal to the energy E corresponding to Ψ.This leads us to introduce a scalar product by polarizing the formula for the energy,(2.4,2.5).We thus obtain the so-called energy scalar product<Ψ,Ψ′>= ∞r 1dr1−1d (cos ϑ) (r 2+a 2)2∂t Φ∂t Φ′+∆∂cos ϑΦ∂cos ϑΦ′+ 1∆∆−a 2sin 2ϑ ω)ΦΦω,λ+2ak r 2+a 2ΦΦω,λ.(2.14)In the special case Ψ=Ψ′,this reduces to<Ψω,λ,Ψω,λ>=2ω ∞r 1dr1−1d (cosϑ)|Φω,λ|2× (r 2+a 2)2∆−1 .(2.15)By construction,the Hamiltonian is Hermitian with respect to the energy scalar product.However,the energy scalar product is in general not positive definite.This is obviousin (2.13)because the factor (sin −2ϑ−a 2/∆)is negative inside the ergosphere.Likewise,the integrand in (2.15)can be negative because the factor ak in the second term in the brackets can have any sign.Apart from the energy,also the charge Q gives rise to a conserved scalar product.It is a natural idea to try to obtain a positive scalar product by taking a suitable linear combination of these two scalar products.Unfortunately,comparing (2.12)and (2.11)one sees that it is impossible to form a non-trivial linear combination of Q and E which is manifestly positive everywhere.One might argue that a suitable linear combination might nevertheless be positive because the positive term ∆|∂r Φ|2+sin 2ϑ|∂cos ϑΦ|2might compensate the negative terms.However,comparing (2.15)with (2.11),one sees that there is a simple relation between the energy scalar product and the charge,<Ψω,λ,Ψω,λ>=2ωQ [Ψω,λ],making it again impossible to form a linear combination such that the integrand of the corresponding scalar product is everywhere positive.Stephen Anco showed that it is indeed impossible to introduce a conserved density for the wave equation which gives rise to a positive definite scalar product [1].We conclude that if we want to consider H as a selfadjoint operator,the underlying scalar product will necessarily be indefinite.But we can clearly consider H as a non-selfadjoint operator on a Hilbert space,and this point of view will indeed be useful for the estimates of Section 4.Our method for constructing a positive scalar product is to simply replace the negative term −a 2/∆in (2.13)by a positive term.More precisely,we introduce the scalar product (.,.)by(Ψ,Ψ′)= ∞r 1dr1−1d (cos ϑ) (r 2+a 2)2∂t Φ∂t Φ′+∆∂cos ϑΦ∂cos ϑΦ′+1∂ϕΦ∂ϕΦ′+(r 2+a 2)2ΦΦ′ .(2.16)We denote the corresponding Hilbert space by H and the norm by . .This norm dom-inates the energy scalar product in the sense that there is a constant c 1>0depending only on the geometry such that the “Schwarz-type”inequality|<Ψ,Ψ′>|≤c 1 Ψ Ψ′ (2.17)holds for all Ψ,Ψ′∈H .We finally bring the Hamiltonian and the above inner products into a more convenient form.First,we introduce the Regge-Wheeler variable u bydu∆,∂∆∂r 2+a 2(2.19)β=−2ak r 2+a 2 (2.20)δ=1r 2+a 2(2.21)as well as the operatorA=1∂u(r2+a2)∂r2+a2∆S2−a2k2 c≤ρbelow).An important special case of a Krein space is when K is positive except on a finite-dimensional subspace,i.e.κ:=dim K−<∞.(3.3) In this case the Krein space is called a Pontrjagin space of indexκ.Classical results ofPontrjagin(see[3,Thms7.2and7.3,p.200]and[16,p.11-12])yield that any selfadjointoperator A on a Pontrjagin space is definitizable,and that it has aκ-dimensional negative subspace which is A-invariant.We now explain how the abstract theory applies to the wave equation in the Kerr ge-ometry.In order to have a spectral theorem,the Hamiltonian must be definitizable.There is no reason why H should be definitizable on the whole space(r1,∞)×S2,and this leads us to consider the wave equation in“finite volume”[r L,r R]×S2with Dirichlet boundary conditions.Thus settingΨ=(Φ,iΦt)and regarding the two components(Ψ1,Ψ2)ofΨasindependent functions,we consider the vector space P rL,r R =(H1,2⊕L2)([r L,r R]×S2)with Dirichlet boundary conditionsΨ1(r L)=0=Ψ1(r R).(3.4) Our definition of H1,2([r L,r R]×S2)coincides with that of the space W1,2((r L,r R)×S2)in[11,Section7.5].Note that we only impose boundary conditions on thefirst componentΨ1ofΨ,which lies in H1,2.According to the trace theorem[7,Part II, Section5.5,Theorem1],the boundary values of a function in H1,2([r L,r R]×S2)are in L2(S2),and therefore we can impose Dirichlet boundary conditions.We endow this vector space with the inner product associated to the energy;i.e.in analogy to(2.13),<Ψ,Ψ′>= r R r L dr 1−1d(cosϑ) (r2+a2)2Ψ2Ψ′2+∆∂cosϑΨ1∂cosϑΨ′1+ 1∆∂rΦ∂rΦ′+sin2ϑsin2ϑ−a2ΦΦ′ .(3.6)Transforming to the variable u,(2.18),and using the representation(2.23),one sees that on the subspace C2([u L,u R]×S2)the inner product(3.6)can be written as<Φ,Φ′>=(Φ,AΦ′)L2([uL,u R]×S2,dµ)(3.7)with A according to (2.22).Here weset u L =u (r L ),u R =u (r R ),and dµis the mea-sure (2.26).A is a Schr¨o dinger operator with smooth potential on a compact domain.Standard elliptic results [20,Proposition 2.1and the remark before Proposition 2.7]yield that H is essentially selfadjoint in the Hilbert space H =L 2([u L ,u R ]×S 2,dµ).It has a purely discrete spectrum which is bounded from below and has no limit points.The corresponding eigenspaces are finite-dimensional,and the eigenfunctions are smooth.Let us analyze the kernel of A .Separating and using that the Laplacian on S 2has eigenvalues −l (l +1),l ∈N 0,A has a non-trivial kernel if and only if for some l ∈N 0,the solution of the ODE−∂∂u +∆r 2+a 2φ(u )=0(3.8)with boundary conditions φ(u R )=0and φ′(u R )=1vanishes at u =u L .Since this φhas at most a countable number of zeros on (−∞,u R ](note that φ(u )=0implies φ′(u )=0because otherwise φwould be trivial),φvanishes at u L only if u L ∈E l with E l countable.We conclude that there is a countable set E =∪l E l such that the kernel of A is trivial unless u L ∈E .Assume that u L /∈E .Then A has no kernel,and so we can decompose H into the positive and negative spectral subspaces,H =H +⊕H −.Clearly,H −is finite-dimensional.Since its vectors are smooth functions,we can consider H −as a subspace of P r L ,r R ,and according to (3.7)it is a negative subspace.Its orthogonal complement in P r L ,r R is contained in H +and is therefore positive.We conclude that P r L ,r R is positive excepton a finite-dimensional subspace.It remains to show that the topology induced by <.,.>is equivalent to the H 1,2-topology.Since on finite-dimensional spaces all norms are equivalent,it suffices to consider for any λ0>0the spectral subspace for λ≥λ0,denoted by H λ0.We choose λ0such that1−λ0≤V 0:=min [r L ,r R ]−a 2k 22 Ψ,A Ψ L 2(dµ)+λ02c Ψ 2H 1,2+V 0−12Ψ 2L 2(dµ)≥1We always choose r L and r R such that P r L ,r R is a Pontrjagin space and that our initialdata is supported in [r L ,r R ]×S 2.We now consider the Hamiltonian (1.13)on the Pontrjagin space P r L ,r R with domainC ∞([r L ,r R ]×S 2)2⊂P r L ,r R .For clarity,we shall often denote this operator by H r L ,r R .Lemma 3.2H r L ,r R has a selfadjoint extension in P r L ,r R .Proof.On the domain of H,the scalar product can be written in analogy to(2.23)as<Ψ,Ψ′>=(Ψ,SΨ′)L2([uL,u R]×S2,dµ),where the operator S acts on the two components ofΨas the matrixS= A001 ,(3.9)where A is again given by(2.22)and dµis the measure(2.26).As shown in Lemma3.1, S has a selfadjoint extension and is invertible.We introduce on C∞0((u L,u R)×S2)2theoperator B by B=|S|−12.The fact that H is symmetric in P rL,r R implies that Bis symmetric in L2([u L,u R]×S2,dµ).A short calculation shows thatB2= |A||A|−12A|A|+β2 .Treating the terms involvingβas a relatively compact perturbation,we readilyfind that B2is selfadjoint on L2([u L,u R]×S2,dµ)with domain D(B2)=D(A)⊕D(A).Con-sequently,the spectral calculus gives us a selfadjoint extension of B with domain D(B)= D(A12).We extend H to the domain D(H):=|S|−12Ψ,B|S|12Φ,|S|12Ψ,S|S|−12Ψ,B˜Ψ)L2(dµ)=(S|S|−12Ψlies in the domain of B and that B|S|12Φ.This implies thatΨ∈D(H)and that HΨ=Φ.with deg p0≤κminimal.Furthermore,we let p be the real polynomial of degree≤2κdefined by p=p0p0(H rL,r R )x,L−>=<x,p0(H rL,r R)L−>=0,(3.10)so thatim p(H rL,r R)⊂im4Resolvent EstimatesIn this section we consider the Hamiltonian H as a non-selfadjoint operator on the Hilbert space H with the scalar product(.,.)according to(2.16).We work either in infinite volume with domain of definition D(H)=C∞0((r1,∞)×S2)2or in thefinite box r∈[r L,r R] with domain of definition given by the functions in C∞((r L,r R)×S2)2which satisfy the boundary conditions(3.4).Some estimates will hold in the same way infinite and infinite volume.Whenever this is not the case,we distinguish betweenfinite and infinite volumewith the subscripts rL,r R and∞,respectively.We always consider afixed k-mode.The next lemma shows that the operator H−ωis invertible if either|Imω|is large or|Imω|=0and|Reω|is large.The second case is more subtle,and we prove it using a spectral decomposition of the elliptic operator A which generates the energy scalar product.This lemma will be very useful in Section7,because it will make it possible to move the contour integrals so close to the real axis that the angular estimates of Lemma2.1 apply.By a slight abuse of notation we use the same notation for H and its closed extension.Lemma4.1There are constants c,K>0such that for allΨ∈D(H)andω∈C,(H−ω)Ψ ≥11+|Reω| Ψ .Proof.For every unit vectorΨ∈D(H),(H−ω)Ψ ≥|(Ψ,(H−ω)Ψ)|≥|Im(Ψ,(H−ω)Ψ)|≥|Imω|−1space L 2(dµ):=L 2(R ×S 2,dµ),with dµaccording to(2.26).Clearly,A is bounded from below,A ≥−c ,and thus σ(A )⊂[−c,∞).For given Λ≫1we let P 0and P Λbe the spectral projectors corresponding to the sets [−c,Λ2)and [Λ2,∞),respectively.We decompose a vector Ψ∈H in the form Ψ=Ψ0+ΨΛwith Ψ0= P 000P 0 Ψ,ΨΛ= P Λ00P ΛΨ.This decomposition is orthogonal w.r.to the energy scalar product,<ΨΛ,Ψ0>= ΨΛ, A 001Ψ0 L 2(dµ)=0.However,our decomposition is not orthogonal w.r.to the scalar product (.,.),because (ΨΛ,Ψ0)= ΨΛ, A +δ001 Ψ0 L 2(dµ)= ΨΛ, δ000Ψ0 L 2(dµ).But at least we obtain the following inequality,|(ΨΛ,Ψ0)|≤c Ψ0 Ψ1Λ L 2(dµ),(4.2)where Ψ1Λdenotes the first component of ΨΛ.Using that Ψ1Λ 2L 2(dµ)=<Ψ1Λ,A −1Ψ1Λ>≤1ΛΨ0 ΨΛ .Choosing Λsufficiently large,we obtain Ψ 2= ΨΛ 2+2Re (ΨΛ,Ψ0)+ Ψ0 2≤4( ΨΛ + Ψ0 )2and thusΨ ≤2( ΨΛ + Ψ0 ).(4.3)Furthermore,we can arrange by choosing Λsufficiently large that <ΨΛ,ΨΛ>= ΨΛ, A 001ΨΛ L 2(dµ)≥12 ΨΛ 2.Next we estimate the inner products <ΨΛ,H Ψ0>,(Ψ0,H ΨΛ)and (Ψ0,H Ψ0).The calculations <ΨΛ,H Ψ0>= ΨΛ, 0A A β Ψ0 L 2(dµ)= ΨΛ, 000β Ψ0 L 2(dµ)(Ψ0,H ΨΛ)= Ψ0, 0A +δA β ΨΛ L 2(dµ)= Ψ0, 0δ0βΨΛ L 2(dµ)|(Ψ0,H Ψ0)|= Ψ0, 0A +δA βΨ0 L 2(dµ) ≤c Ψ0 L 2(dµ)+2 A Ψ10 L 2(dµ) Ψ20 L 2(dµ) A Ψ10 2L 2(dµ)= Ψ0, A 2000 Ψ0 L 2(dµ)≤Λ2 Ψ0, A 001 Ψ0 L 2(dµ)=Λ2 Ψ0 2give us the bounds|<ΨΛ,HΨ0>|≤c ΨΛ Ψ0|(Ψ0,HΨΛ)|≤c Ψ0 ΨΛ|(Ψ0,HΨ0)|≤(c+2Λ) Ψ0 2.Using the above inequalities,we can estimate the inner product<ΨΛ,(H−ω)Ψ>by |<ΨΛ,(H−ω)Ψ>|≥|<ΨΛ,(H−ω)ΨΛ>|−|<ΨΛ,(H−ω)Ψ0>|≥|Imω|c ΨΛ − Ψ0 .(4.4) Next we estimate the inner product(Ψ0,(H−ω)Ψ),|(Ψ0,(H−ω)Ψ)|≥|(Ψ0,(H−ω)Ψ0)|−|(Ψ0,(H−ω)ΨΛ)|≥(|ω|−c−2Λ) Ψ0 2−c 1+|ω|Λ ΨΛ .(4.5) ChoosingΛ=(|ω|−c)/4and increasing c,the inequalities(4.4)and(4.5)give for suffi-ciently large|ω|the bounds(H−ω)Ψ ≥|Imω|2 Ψ0 −c ΨΛ .Multiplying the second inequality by4/|ω|and adding thefirst inequality,we conclude that2 (H−ω)Ψ ≥ |Imω||ω| ΨΛ + Ψ0 .The result now follows from(4.3).1+|Reω| (4.6) with K as in Lemma4.1.Corollary4.2Ifω∈Ω,the operator H−ωis invertible.The corresponding resolventS(ω):=(H−ω)−1satisfies the boundS(ω) ≤cUsing(4.6)in(4.7),we immediately get the boundS(ω) ≤c(1+|Reω|).(4.9) Since S(ω)is a bounded operator,its domain of definition can clearly be chosen to be the whole Hilbert space.We shall assume until the end of this section thatω∈Ω.The next lemma gives detailed estimates for the difference of the resolvents S rL,r R andS∞infinite and infinite volume,respectively.By Qλ(ω)we denote a given projector onto an invariant subspace of the angular operator Aωcorresponding to the spectral parameter λof dimension at most N(see Lemma2.1for details).Lemma4.3For everyΨ∈C∞0((r L,r R)×S2)2and every p∈N,there is a constant C=C(Ψ,p)(independent ofω)such that|<Ψ,[S r L,r R(ω)−S∞(ω)]Ψ>|≤C|Imω|.(4.10) Furthermore,for everyΨ∈C∞0((r L,r R)×S2)2and every p∈N and q≥N,there is a constant C=C(Ψ,p,q)(independent ofωandλ)such that|<Ψ,Qλ[S r L,r R(ω)−S∞(ω)]Ψ>|≤C|Imω| Qλ .(4.11) Proof.By definition of the resolvent,(H−ω)S(ω)Ψ=Ψ.This relation holds both in finite and in infinite volume,and thus((H−ω)[S rL,r R(ω)−S∞(ω)]Ψ)(r,ϑ)=0if r L≤r≤r R.Iterating this identity and using the fact that H and S commute,we see that on[r L,r R]×S2,ωp+1[S rL,r R (ω)−S∞(ω)]Ψ=[S rL,r R(ω)−S∞(ω)]H p+1Ψ.(4.12)Combining this identity with the Schwarz-type inequality (2.17),we obtain|<Ψ,[S r L ,r R (ω)−S ∞(ω)]Ψ>|≤c 1 S r L ,r R (ω)−S ∞(ω) Ψ 2|ωp +1||<Ψ,[S r L ,r R (ω)−S ∞(ω)]Ψ>|≤ <Ψ,[S r L ,r R (ω)−S ∞(ω)]H p +1Ψ> ≤c 1 S r L ,r R (ω)−S ∞(ω) Ψ H p +1Ψ .Since Ψis smooth and has compact support,H p +1Ψalso has these properties.Theestimate (4.9)gives (4.10).In order to prove (4.11),we first combine (4.12)with (2.17)to obtain(1+|ω|p +1)|<Ψ,Q λ(S r L ,r R −S ∞)Ψ>|≤c 1 Q λ S r L ,r R −S ∞ Ψ Ψ + H p +1Ψ .(4.13)Since q is at least as large as the dimension of the invariant subspace corresponding to λ,(A ω−λ)q Q λ=0.Therefore,for every Ψ′∈C ∞0((r L ,r R)×S 2)2,0=<Ψ,(A ω−λ)q Q λΨ′>=<(A ∗ω−λ)q and using (2.17),we obtain|λ|q <Ψ,Q λΨ′> ≤q l =1c l |λ|q −l (A ∗ω)l Ψ χ[r L ,r R ]Q λΨ′ with combinatorial factors c l (here χ[r L ,r R ]is the operator of multiplication by the char-acteristic function).Since the angular operator A ∗ωis according to (1.9)a polynomial inωof degree two,the function (A ∗ω)l Ψis also polynomial in ω,i.e.(A ∗ω)l Ψ=2l p =0ωp Ψp ,where the functions Ψp are composed of Ψand its angular derivatives,as well as the coefficient functions of A ∗ω.This gives the estimate(A ∗ω)l Ψ ≤2l p =0|ω|p Ψp ≤c (1+|ω|2l )with a constant c which depends only on Ψand l .We thus obtain|λ|q<Ψ,Q λΨ′> ≤q l =1c l (Ψ)|λ|q −l (1+|ω|2l ) χ[r L ,r R ]Q λΨ′ .Young’s inequality allows us to compensate the lower powers of λ,|λ|q <Ψ,Q λΨ′> ≤c (q,Ψ)(1+|ω|2q ) χ[r L ,r R ]Q λΨ′ .We now choose Ψ′equal to the left side of (4.12)with p =0and p =r and take the sum of the resulting inequalities.Applying again the Schwarz inequality,we obtain|λ|q (1+|ω|r )|<Ψ,Q λ(S r L ,r R −S ∞)Ψ>|≤c (1+|ω|2q ) Q λ S r L ,r R −S ∞ ( Ψ + H r Ψ ).By choosing r sufficiently large,we can compensate the factor (1+|ω|2q )on the right.More precisely,|λ|q (1+|ω|p +1)|<Ψ,Q λ(S r L ,r R −S ∞)Ψ>|≤c ′ Q λ S r L ,r R −S ∞ Ψ + H p +2q +1Ψ .Adding this inequality to (4.13)and substituting the estimate (4.9)gives (4.11).5Separation of the ResolventIn this section wefixω∈σ(H),so that the resolvent S=(H−ω)−1exists.As in the previous section,we assume that Qλis a given projector onto afinite-dimensional invariant subspace of the angular operator Aωcorresponding to the spectral parameterλ.Our goal is to represent the operator product QλS in terms of the solutions of the radial ODE.According to(1.10)and(1.8),the radial ODE is−∂∂r−(r2+a2)2r2+a2 2+λ R(r)=0,(5.1) whereλis the separation constant.We can assume that k≥0because otherwise we reverse the sign ofω.We again work in the“tortoise variable”u,(2.18),and setφ(r)=r2+a2∂∂u+ ω+ak(r2+a2)2 φr2+a2=0.(5.3)Using that(r2+a2)∂2=−12∂∂u(r2+a2)1∂u2+V(u) φ(u)=0(5.4)with the potentialV(u)=− ω+ak(r2+a2)2+1r2+a2∂2uLemma 5.1The functions (u,u ′):=1∂u 2+V (u )s (u,u ′)=δ(u −u ′).Proof.By definition of the distributional derivative,∞−∞η(u )(−∂2u +V )s (u,u ′)du =∞−∞(−∂2u +V )η(u ) s (u,u ′)du for every test function η∈C ∞0(R ).It is obvious from its definition that the function s (.,u ′)is smooth except at the point u =u ′,where its first derivative has a discontinuity.Thus after splitting up the integral,we can integrate by parts twice to obtain∞−∞(−∂2u +V )η(u ) s (u,u ′)du=u ′−∞η(u )(−∂2u +V )s (u,u ′)du +lim u րu′η(u )∂u s (u,u ′)+∞u ′η(u )(−∂2u+V )s (u,u ′)du −lim u ցu′η(u )∂u s (u,u ′).Since for u =u ′,s is a solution of (5.4),the obtained integrals puting thelimits with (5.7),we get∞−∞(−∂2u +V )η(u ) s (u,u ′)du = lim u րu′−lim u ցu′ η(u )∂u s (u,u ′)=1In what follows we also regard s (u,u ′)as the integral kernel of a corresponding operators ,i.e.(sφ)(u ):=du ′s (u,u ′)φ(u ′)du ′.If Q λprojects onto an eigenspace of A ω,we see from (1.10),(1.7),and (5.2)that(r 2+a 2)−12Q λ(ϑ,ϑ′)δ(u −u ′).(5.8)Loosely speaking,this relation means that the operator product Q λs is an angular modeof the Green’s function of the wave equation.Unfortunately,Q λmight project onto an invariant subspace of A ωwhich is not an eigenspace.In this case,the angular operator has on the invariant subspace the “Jordan decomposition”A ωQ λ=(λ+N )Q λ(5.9)with N =N (ω,λ)a nilpotent operator.Lemma 5.3extends (5.8)to this more general case.In preparation,we need to consider powers of the operator s .Lemma5.2For every l∈N0,the operator s l is well-defined.Its kernel(s l)(u,u′)has regularity C2l−2.Proof.Writing out the operator products with the integral kernel,one sees that the operator s l is obtained from s by iterated convolutions,s p+1(u,u′)= s(u,u′′)s p(u′′,u′)du′′.(5.10)In thefinite box,these convolution integrals are allfinite because s(u,u′)is continuous and the integration range is compact.In infinite volume,the function s(u,u′)decays exponentially as u,u′→±∞(see Corollary6.4),and so the integrals in(5.10)are again finite.Hence s l is well-defined.Let us analyze the regularity of the integral kernel of s l.By definition,s(u,u′)is continuous,and(5.10)immediately shows that the same is true for s p(u,u′).Differen-tiating through(5.10)and applying Lemma5.1,one sees that s p satisfies for p>1the distributional equation−∂2Lemma5.3For givenλ∈σ(Aω)we let g be the operatorg=∞l=0(−N)l s l+1,(5.11)where N is the nilpotent matrix in the Jordan decomposition(5.9).Then(r2+a2)−12Qλ(ϑ,ϑ′)δ(u−u′).(5.12) Note that if Qλprojects onto an eigenspace,N vanishes and thus g=s.Furthermore, since N is nilpotent,the series in(5.11)is actually afinite sum.Thus in view of Lemma5.2, (5.11)is indeed well-defined.Proof of Lemma5.3.Denoting the radial operator with integral kernelδ(u−u′)by11u, we can write the result of Lemma5.1in the compact form(−∂2u+V)s=11u.Hence on the invariant subspace,we can do a Neumann series calculation,(−∂2u+V)g=∞l=0(−N)l(−∂2u+V)s l+1=∞ k=0(−N)l s l=11u−N g,to obtain that(−∂2u+V+N)g=11u.According to(1.10),(1.7),and(5.2),this is equiv-alent to(5.12).。

Top 10 Greatest Physicists of All Time (Based on Foreign English Resources)1. Albert EinsteinWithout a doubt, Albert Einstein tops the list of the greatest physicists of all time. His theory of relativity revolutionized our understanding of space, time, and gravity. Einstein's famous equation, E=mc², demonstrated the relationship between energy and mass, opening up new possibilities in the field of physics.2. Isaac NewtonSir Isaac Newton is another physicist who has left an indelible mark on the scientific world. His laws of motion and universal gravitation laid the foundation for classical mechanics. Newton's work in optics and the development of the reflecting telescope also contributed significantly to the advancement of physics.3. Niels BohrNiels Bohr, a Danish physicist, played a crucial role in the development of quantum mechanics. His model of the atom, which incorporated the concept of quantized energy levels, helped to explain the behavior of electrons and the stability of atoms.4. James Clerk MaxwellScottish physicist James Clerk Maxwell is renowned forhis formulation of the classical theory of electromagnetic radiation. His set of equations, known as Maxwell's equations, unified the understanding of electricity, magnetism, andlight.5. Richard FeynmanRichard Feynman was an American theoretical physicist who made significant contributions to quantum mechanics andparticle physics. His development of the path integral formulation of quantum mechanics and his work on the theoryof quantum electrodynamics earned him a Nobel Prize in Physics.6. Max PlanckGerman physicist Max Planck is considered the father of quantum theory. His discovery of Planck's constant and his proposal that energy is radiated in discrete packets, or quanta, marked the beginning of quantum physics.7. Werner HeisenbergWerner Heisenberg, another prominent figure in quantum mechanics, formulated the uncertainty principle, which states that it is impossible to simultaneously know the exactposition and momentum of a particle.8. Galileo GalileiGalileo Galilei, often referred to as the "Father of Modern Science," made significant contributions to physics, astronomy, and the scientific method. His work on inertia, falling objects, and the laws of motion laid the groundwork for Newton's theories.9. Stephen Hawking10. Marie CurieMarie Curie, a Polishborn physicist and chemist, was the first woman to win a Nobel Prize and remains the only person to win Nobel Prizes in two different sciences (Physics and Chemistry). Her work on radioactivity opened up new avenues in medical research and laid the foundation for the field of atomic physics.Continuing the exploration of the most influential physicists in history, let's delve into the lives and achievements of these extraordinary individuals who have shaped our understanding of the universe.11. Paul DiracPaul Dirac, an English theoretical physicist, is often celebrated for his prediction of the existence of antimatter, a discovery that was later confirmed the experimental work of Carl Anderson. Dirac's formulation of quantum mechanics, particularly the Dirac equation, was pivotal in the development of quantum field theory.12. Michael FaradayMichael Faraday was a British scientist who contributed immensely to the study of electromagnetism. His experiments led to the discovery of electromagnetic induction, the laws of electrolysis, and the invention of the Faraday cage. Faraday's work laid the groundwork for the future development of electric motors and generators.13. Ludwig BoltzmannAustrian physicist Ludwig Boltzmann was a key figure in the development of statistical mechanics and thermodynamics. His insight into the behavior of molecules and his formulation of the Boltzmann equation helped to explain the concepts of entropy and the statistical nature of physical laws.14. Ernest RutherfordErnest Rutherford, a New Zealandborn British physicist, is known as the father of nuclear physics. His groundbreaking gold foil experiment led to the discovery of the atomic nucleus and the understanding that most of an atom's mass is concentrated in a tiny, central nucleus.15. Murray GellMann16. J.J. ThomsonSir Joseph John Thomson, an English physicist, discovered the electron in 1897, demonstrating that atoms are notindivisible and consist of smaller particles. His work on cathode rays and the discovery of the masstocharge ratio of electrons was fundamental in the development of atomic physics.17. Enrico FermiEnrico Fermi, an Italian physicist, was pivotal in the development of nuclear technology. His work on nuclear reactions led to the construction of the first nuclear reactor and the first controlled nuclear chain reaction. Fermi's contributions to quantum theory and particle physics are also noteworthy.18. Lise MeitnerLise Meitner, an AustrianSwedish physicist, was involved in the discovery of nuclear fission. Despite facing discrimination as a woman in science, Meitner's work was crucial in understanding the process which heavy nuclei can split into lighter nuclei, releasing a significant amount of energy.19. Roger PenroseBritish mathematician and physicist Roger Penrose has made significant contributions to the understanding of general relativity and cosmology. His work on black holes, particularly the Penrose process and the Penrose singularitytheorem, has deepened our knowledge of the most extreme phenomena in the universe.20. Kip ThorneKip Thorne, an American theoretical physicist, has been at the forefront of gravitational physics and astrophysics. His work on the detection of gravitational waves, as part of the LIGO collaboration, confirmed a key prediction of Einstein's theory of general relativity and opened a new window into the cosmos.Each of these physicists has left an indelible mark on the field, pushing the boundaries of human knowledge and challenging our understanding of the natural world. Their legacies continue to inspire and guide the next generation of scientists in their quest to uncover the secrets of the universe.As we further unravel the rich tapestry of scientific achievement, let's continue to honor the contributions of more extraordinary physicists whose insights have transformed our understanding of the cosmos and the fundamental forces that govern it.21. Andrei SakharovAndrei Sakharov, a Soviet nuclear physicist, played a crucial role in the development of the Soviet hydrogen bomb. However, he is perhaps best known for his advocacy of civilliberties and human rights in the Soviet Union. His work on the concept of "metric elasticity" also contributed to the field of general relativity.22. ChenNing YangChenNing Yang, a ChineseAmerican physicist, made significant contributions to theoretical physics,particularly in the area of parity nonconservation in weak interactions. His work with TsungDao Lee led to a Nobel Prize in Physics, demonstrating that certain processes are not mirrorsymmetric.23. TsungDao LeeTsungDao Lee, a ChineseAmerican physicist, collaborated with ChenNing Yang to challenge the symmetry principles in physics. Their proposal that weak interactions do not conserve parity was a groundbreaking discovery that revolutionized particle physics.24. Sheldon GlashowSheldon Glashow, an American theoretical physicist, is known for his work on the electroweak theory, which unified the weak nuclear force and electromagnetism. Hiscontributions to particle physics, including the prediction of the W and Z bosons, were recognized with a Nobel Prize in Physics.25. Abdus SalamAbdus Salam, a Pakistani theoretical physicist, sharedthe Nobel Prize in Physics with Sheldon Glashow and Steven Weinberg for their work on the electroweak unification. Salam was instrumental in developing the mathematical frameworkthat described the weak and electromagnetic forces asdifferent aspects of the same force.26. Edward WittenEdward Witten, an American theoretical physicist and mathematician, is often regarded as one of the leadingfigures in string theory. His work has deepened our understanding of the mathematical underpinnings of theuniverse and has earned him numerous accolades, including the Fields Medal.27. Julian Schwinger28. SinItiro Tomonaga29. George GamowGeorge Gamow, a RussianAmerican physicist and cosmologist, made significant contributions to the fields of nuclearphysics and cosmology. He was one of the first to propose the Big Bang theory as the origin of the universe and also made important contributions to the understanding of stellar nucleosynthesis.30. Freeman DysonFreeman Dyson, a BritishAmerican theoretical physicist and mathematician, has made numerous contributions to quantum electrodynamics and solidstate physics. His work on the unification of the electromagnetic and gravitational forces, as well as his speculations on Dyson spheres, have been influential in theoretical physics.These physicists, through their relentless pursuit of knowledge, have not only advanced the frontiers of science but have also inspired a sense of wonder and curiosity in generations of scholars and laypeople alike. Their work continues to be a testament to the power of human intellect and the boundless potential of scientific inquiry.。