Spectral Theory of the Thermal Hamiltonian: 1D Case
SSPECTRAL THEORY OF THE THERMAL HAMILTONIAN:1D CASE
GIUSEPPE DE NITTIS AND VICENTE LENZAbstract. In 1964 J. M. Luttinger introduced a model for the quantumthermal transport. In this paper we study the spectral theory of the Hamil-tonian operator associated to the Luttinger’s model, with a special focusat the one-dimensional case. It is shown that the (so called) thermalHamiltonian has a one-parameter family of self-adjoint extensions and thespectrum, the time-propagator group and the Green function are explicitlycomputed. Moreover, the scattering by convolution-type potentials is ana-lyzed. Finally, also the associated classical problem is completely solved,thus providing a comparison between classical and quantum behavior. Thisarticle aims to be a first contribution in the construction of a completetheory for the thermal Hamiltonian.
Contents1. Introduction 21.1. Physical motivations 21.2. Position of the spectral problem 41.3. Overview on the one-dimensional case 72. The spectral theory of the operator ˝ MSC2010
Primary: 81Q10; Secondary: 81Q05, 81Q15, 33C10.
Keywords.
Thermal Hamiltonian, self-adjoint extensions, spectral theory, scatteringtheory. a r X i v : . [ m a t h - ph ] F e b G. DE NITTIS AND V. LENZ
A.2. Spectral measure 34A.3. Density of states 35Appendix B. Technical tools 36B.1. Some principal value integrals 36B.2. Irregular Kelvin functions 40B.3. Bessel equation and Hankel transform 41References 421. IntroductionThe aim of this introductory section is twice: First of all, we will providethe physical background that motivates the study of the Thermal Hamil-tonian; Secondly, we will present the mathematical problems and the mainresults achieved in this work.1.1.
Physical motivations.
The motion of an electron inside the matter,and subjected to a static magnetic field B , is described by the (one-particle)Hamiltonian H ( A; V ) := K ( A ) + V (1.1)where K ( A ) := 12 m p ` ec A ! : (1.2)The parameters m and e describe the mass and the charge of the electron,respectively. The constant c is the (in vacum) speed of the light . Thestatic (effective) potential V takes care of the interaction of the electronwith the atomic structure of the matter and causes only elastic scattering .The magnetic field enters in the kinetic term K ( A ) through its vectorpotential A according to the equation B = r ˆ A . In Quantum Mechanicsthe Hamiltonian H ( A; V ) is interpreted as a differential operator actingon the Hilbert space L ( R d ) , where the differential part is provided bythe momentum operator p := ` (cid:125) i r , (cid:125) being the Planck constant . Thepotentials V = V ( x ) and A = A ( x ) are functions of the position operator x = ( x ; : : : ; x d ) , and act as multiplication operators.The transport phenomena in the matter are analyzed by studing theresponse of the system to an external perturbation F = F ( x ) [Lut1, Lut2].In the stationary regime, that is when all the transient effects due tothe switching-on of the perturbation are suppressed, the system reacts bygenerating a (stationary) drift current . The latter can be computed (atleast in the linear response regime, see e. g. [DL]) starting from the fulldynamics generated by the perturbed Hamiltonian H ( A; F; V ) := H ( A; F ) +
V : (1.3)In (1.3) the “free” Hamiltonian H ( A; F ) := K ( A ) + F (1.4)describes the motion of an electron that moves in the empty space underthe influence of the (external) fields generated by A and F . The potential V in (1.3) describes the interaction with the matter which generates elasticscattering of the particle. Once the “free” dynamics generated by H ( A; F ) PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 3 is known, one can study the influence of the matter by means of the scattering theory [RS3, Yaf, Kat] for the pair of operators H ( A; F ) and H ( A; F; V ) .The best studied case concerns the response of the system to the pertur-bation induced by a uniform electric field E = ( E ; : : : ; E d ) . In this casethe perturbation is described by the electrostatic potential F E ( x ) := ` eE ´ x = ` e ( E x + : : : + E d x d ) and the associate perturbed Hamiltonian takes the form H ( A; F E ; V ) := H Stark ( A ) + V (1.5)where the “free” part is given by H Stark ( A ) := K ( A ) ` eE ´ x (1.6)according to (1.4). The operator H Stark ( A ) is known as (magnetic) StarkHamiltonian . The non-magnetic case H Stark ( A = 0) = p m ` eE ´ x has beenextensively studied since the dawn of the Quantum Mechanics. Among thevast literature, we will refer to [AH] for a concise and rigorous presentationof the spectral theory of H Stark (0) and the related scattering theory whenthe background potential V is taken in consideration. The spectral theoryof H Stark ( A ) in presence of a uniform magnetic field is discussed in [DP,ADF], among others.In order to study the thermal transport in the matter, Luttinger proposeda model which allows a “mechanical” derivation of the thermal coefficients[Lut2]. Such a model has been then applied and generalized successfullyby other authors like in [SS, VMT]. The essential point of the Luttinger’smodel is that the effect of the thermal gradient in the matter is replacedby a “fictitious” gravitational field, which can be easily described by aperturbation of the Hamiltonian in the spirit of (1.3) and (1.4). Moreprecisely, one assumes that the particle is subject to a force which hasthe direction of the thermal gradient r (where is the distribution oftemperature) and which is proportional to the local content of energydivided by c (in view of the mass-energy equivalence). The latter is givenby the Hamiltonian (1.1) itself. Such a thermal-gravitational field is givenby the potential F T : = 12 " r ´ x ! H ( A; V ) + H ( A; V ) r ´ x ! = r ´ n H ( A; V ) ; x o (1.7)where the anti-commutator f ; g between H ( A; V ) and x is needed to make F T formally self-adjoint ( i. e. symmetric). The total perturbed Hamilton-ian H ( A; F T ; V ) , computed according to (1.3), can be written as H ( A; F T ; V ) = H T ( A ) + W ( V ) (1.8) G. DE NITTIS AND V. LENZ where the “free” part, called (magnetic) thermal Hamiltonian , is given by H T ( A ) := K ( A ) + r ´ n K ( A ) ; x o (1.9)and the effective gravitational-matter potential reads W ( V ) := r ´ x ! V : (1.10)The thermal Hamiltonian H T ( A ) is the analog of the Stark Hamiltonianwhen the system is perturbed by the gravitational-thermal field instead ofthe electric field. For this reason, it seems natural to look for the extensionof the results valid for the Stark Hamiltonian ( e. g. [AH, DP, ADF]) tothe case of the thermal Hamiltonian. This consists of two consecutiveproblems: (i) the analysis of the spectral theory of the “free” operator H T ( A ) ; (ii) the study of the scattering theory for the pair H T ( A ) and H ( A; F T ; V ) . Both othese problems seem not to have been studied yetin the literature, at least to the best of our knowledge. For this reasonwe devote this work at the analysis of the questions (i) and (ii) above, inthe one-dimensional case. The multi-dimensional case will be treated ina future work.1.2. Position of the spectral problem.
In order to formulate the prob-lems sketched above in a rigorous mathematical setting we will make somesimplifications. The most relevant concerns the absence of the magneticfield. From here on, unless otherwise indicated, we will fix A = 0 . Itis worth mentioning that this is not a major restriction as long as one isinterested only the one-dimensional regime. Indeed, in one spatial dimen-sion the magnetic field is a pure gauge and can be removed with a unitarytransformation .As usual in mathematics, we will normalize all the physical units: m = (cid:125) = c = e = 1 . Moreover, we will denote with – := jr j > the strengthof the thermal gradient and with ‚ := – ` r S d ` its direction. Withthese simplifications the thermal Hamiltonian reads H T ” H T ( –; ‚ ) := p + – n p ; ‚ ´ x o : (1.11)The expression (1.11) is formal without the specification of the domain ofdefinition of H T . However, H T is evidently well defined on the space of thecompactly supported smooth function C c ( R d ) or on the Schwartz space S ( R d ) . On these dense domains the operator (1.11) acts as “ H T ” ( x ) := ` (1 + – ‚ ´ x ) (´ )( x ) ` – ( ‚ ´ r )( x ) (1.12)where ´ := P dj =1 @ x j denotes the Laplacian and ‚ ´ r := P dj =1 ‚ j @ x j . Wecan simplify the last expression with the help of two unitary transformations An equivalently appropriate name for H T ( A ) could be (magnetic) LuttingerHamiltonian . This fact can be interpreted as a consequence of the
Stone-von Neumann theorem (see e. g. [Ros]). Indeed, in one spatial dimension the pair x , ı f := p + f ( x ) necessarily meetsthe canonical commutation rule and so it is unitarily equivalent to the canonical pair x; p . PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 5 of the Hilbert space L ( R d ) . The first one is the rotation ( R ‚ )( x ) := “ O ` ‚ x ” ; L ( R d ) (1.13)where the orthogonal matrix O ‚ meets the condition O ‚ ‚ = (1 ; ; : : : ; .A short computation shows that “ R ‚ H T R ˜ ‚ ” ( x ) = ` (1 + – x ) (´ )( x ) ` – @ @x ( x ) ; where x denotes the first component of the position vector x = ( x ; x ? ) R d and x ? := ( x ; : : : ; x d ) R d ` is its orthogonal complement. Evi-dently, the rotation O ‚ has the role of aligning the thermal-gravitationalfield along the x -axis . The second transformation is the translation ( S – )( x ; x ? ) := x ` – ; x ? ! ; L ( R d ) (1.14)and a direct calculation provides “ S – R ‚ H T R ˜ ‚ S ˜ – ” ( x ) = – ` x (´ )( x ) ` @ @x ( x ) ! : The operator on the brackets ( T )( x ) := ` x (´ )( x ) ` @ @x ( x ) (1.15)agrees with the formal anti-commutator T ” n p ; x o := 12 “ p x + x p ” (1.16)when evaluated on sufficiently regular functions like ( R d ) . With aslight abuse of notation, we will often use the representation (1.16) forthe operator T , instead of the more precise definition (1.15).The unitary equivalence between H T and T implies that the spectraltheory of the thermal Hamiltonian H T can be completely recovered fromthe spectral theory of the operator T . For this reason, one is led to theproblem of determining if the operator T , initially defined by (1.15) on thedense domain S ( R d ) , admits self-adjoint extensions and, in that case, tocompute the related spectra.For technical reasons, it results easier to face the equivalent problemsin the Fourier space. Let F : L ( R d ) ! L ( R d ) be the Fourier transformdefined (just to fix the convention) by ( F )( k ) := 1(2 ı ) d Z R d d x e ` i k ´ x ( x ) on the dense subspace L ( R d ) \ L ( R d ) . Let ˝ := F T F ˜ be theFourier transformed version of the operator (1.15). A direct computationshows that for ( R d ) “ ˝ ” ( x ) := i " x ( x ) + x @ @x ( x ) ; (1.17) Clearly, in dimension d = 1 the thermal-gravitational field is trivially aligned with theonly spatiual axis and therefore R ‚ reduces to the identity. G. DE NITTIS AND V. LENZ where x := P dj =1 x j . The operator defined by (1.17) agrees with theformal expression ˝ ” ` n x ; p o := ` “ x p + p x ” (1.18)on sufficiently regular functions .The representation (1.18) is quite intriguing if one compares the operator ˝ with the typical generator of C -groups associated to C -flows [ABG,Chapter 4]. At first glance, it would seem that the general theory of C -groups applies to ˝ . However, a closer inspection to the R -flow associatedto ˝ shows that this is not the case in general (see Section 2.4 for moredetails). Therefore, the question of the self-adjointness of ˝ needs to beinvestigated with other tools.The first fundamental question is whether the operator ˝ , initially de-fined by (1.17) on S ( R d ) , admits self-adjoint extensions or not. This isfortunately true and easily demonstrable. Indeed, it is straightforward tocheck that ˝ , as defined by (1.17), is symmetric (hence closable ) on S ( R d ) , i. e. h ˝ ; ’ i = h ; ˝ ’ i ; ; ’ ( R d ) : This observation allows us to identify ˝ with its closure (still denoted withthe same symbol) defined on the domain D := S ( R d ) jj jj ˝ (1.19)obtained by the closure of S ( R d ) with respect to the graph-norm jj jj := jj jj + jj ˝ jj : The existence of self-adjoint extensions of ˝ is justified by the von Neu-mann’s criterion [RS2, Theorem X.3]. Let ˇ be the anti-unitary operatoron L ( R d ) defined by (ˇ )( x ) = ( ` x ) . The domains C c ( R d ) or S ( R d ) are left unchanged by ˇ and a direct check shows that ˇ˝ = ˝ˇ on thesedomains. This is sufficient to claim that: Proposition 1.1.
The closed symmetric operator ˝ with domain D admitsself-adjoint extensions. Proposition 1.1 allows a precise definition of the family of thermal Hamil-tonians.
Definition 1.1 (Thermal Hamiltonian) . Let ˝ „ be a given self-adjoint ex-tension of the operator ˝ with domain D (˝ „ ) ff D . Let F ( –; ‚ ) := F S – R ‚ be the unitary operator given by the product of the Fourier trans-form F , the translation S – defined by (1.14) and the rotation R ‚ definedby (1.13). Then, the associated thermal Hamiltonian is the self-adjointoperator H T;„ ( –; ‚ ) := – F ( –; ‚ ) ˜ ˝ „ F ( –; ‚ ) ; – > ; ‚ S d ` defined on the domain D ( H T;„ ) := F ( –; ‚ ) ˜ [ D (˝ „ )] . Formula (1.18) can be formally derived from (1.16) by using the well known transfor-mations of the canonical operators F p j F ˜ = x j and F x j F ˜ = ` p j for all j = 1 ; : : : ; d . PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 7
Definition (1.1) reduces the question of the spectral theory of the thermalHamiltonian to the analysis of the self-adjoint realizations of the operator ˝ . This is usually done by studying the deficiency subspaces K ˚ := Ker( i ˇ ˝ ˜ ) : The existence of the conjugation ˇ for ˝ implies the equality of the defi-ciency indices n ˚ := dim( K ˚ ) [RS2, Theorem X.3] which in turn ensuresthe existence of self-adjoint extensions. In order to build the spaces K ˚ and to compute n ˚ , one needs to solve the equations ˝ ˜ = ˚ i which,in view of (1.17), is equivalent of finding the weak solutions [RS1, SectionV.4] to the differential equations ( x + x ? ) @ @x ( x ; x ? ) + ( x ˇ ( x ; x ? ) = 0 L ( R d ) \ S ( R d ) where S ( R d ) is the space of tempered distributions . This problem willbe solved for the one-dimensional case in Section 2.2.1.3. Overview on the one-dimensional case.
In Section 2.2, it is shownthat the differential operator (1.17), in one spatial dimension ( d = 1) ,admits a family of self-adjoint realizations parametrized by the angle „ S (see Theorem 2.1). As a consequence, the domains C c ( R ) or S ( R ) can not be cores for ˝ (in contrast to [ABG, Proposition 4.2.3]). However, it turnsout that all these self-adjoint realizations ˝ „ are equivalent in the sensethat there are unitary operators L „ such that ˝ „ = L „ ˝ L ˜ „ . This factimmediately implies the independence of the spectrum by the particularself-adjoint realization. In particular, it results that the spectrum of everyextension ˝ „ is purely absolutely continuous and coincides with the realaxis, i. e. ff “ ˝ „ ” = ff a : c : “ ˝ „ ” = R ; „ S : (1.20)We are now in position to state our first main result. Let us just recallthat in dimension d = 1 the only relevant parameter in the definition ofthe thermal Hamiltonian is – > since no rotation R ‚ is required ( cf . Note3). Then, according to Definition 1.1, we can define the family of one-dimensional thermal Hamiltonians as H T;„ ( – ) := – ( F S – ) ˜ ˝ „ ( F S – ) ; – > ; „ S : In view of the unitary equivalence of the various realizations ˝ „ it followsthat all the one-dimensional thermal Hamiltonians with a given couplingconstant – > are unitarily equivalent. For this reason we can focus onthe special realization with „ = 0 . Theorem 1.1 (Spectral theory in d = 1 ) . For every – > , and up to aunitary equivalence, there is a unique one-dimensional thermal Hamiltonianon L ( R ) defined by H T ” H T ( – ) := – ( F S – ) ˜ ˝ ( F S – ) : (1.21) Similarly, one can consider weak solutions in L ( R d ) \ D ( R d ) where D ( R d ) ff S ( R d ) is the space of distributions. G. DE NITTIS AND V. LENZ
The operator H T is self-adjoint on its domain D ( H T ) := ( F S – ) ˜ [ D (˝ )] and has purely absolutely continuous spectrum given by ff “ H T ” = ff a : c : “ H T ” = R independently of – > . The proof of Theorem (1.1) is a corollary Theorem 2.1 and of Definition1.1. For the determionation of the spectrum one uses the invariance of thespectrum under unitary equivalences and the spectral mapping theorem .The operator H T , defined by (1.21), will be called the standard realiza-tion of the one-dimensional thermal Hamiltonian (with coupling constant – > ). Theorem 1.1 expresses the fact that in dimension d = 1 thereis a “unique” thermal Hamiltonian, at least in the sense that all relevant physical quantities , which by definition must be invariant under unitaryequivalences, can be calculated from H T .Theorem 1.1 can be complemented with some more precise information.First of all, it is possible to have a precise description of the domain D ( H T ) ( cf . Section 3.1). Let Q ( R ) := ( L ( R ) ˛˛˛˛ Z R d x x j ( x ) j < + ) (1.22)be the natural domain of the position operator. Let ( B – )( x ) := Z R d y B x + 1 – ; y ! ( y ) (1.23)be the unitary operator with integral kernel B ( x; y ) := i sgn ( x ) ` sgn( y )2 J „ q j xy j « (1.24)where sgn( x ) := x j x j if x = 00 if x = 0 is the sign function and J is the 0-th Bessel function of the first kind [GR]. Then, it holds true that D ( H T ) = B – [ Q ( R )] : Moreover D ( H T ) contains a dense core for H T given by D ( H T ) : = S ( R ) + C [ » ]= ’ L ( R ) ˛˛˛˛ ’ = + c» ; ( R ) ; c C ff and on this core H T acts according to ( cf . Proposition 3.1) “ H T ( + c» ) ” ( x ) := ` (1 + –x ) ( x ) ` – ( x ) + c» ( x ) (1.25) The kernel (1.24) is reminiscent of the Hankel transform of order -th . This aspect isbriefly discussed in Section B.3. PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 9 where the (normalized) functions » and » are explicitly given by » ( x ) : = ` vuut ı sgn x + 1 – ! kei vuut˛˛˛˛˛ x + 1 – ˛˛˛˛˛1CA » ( x ) : = vuut ı ker vuut˛˛˛˛˛ x + 1 – ˛˛˛˛˛1CA (1.26)and kei( x ) and ker( x ) are the irregular Kelvin functions of -th order (seeSection B.2 and references therein). It is worth noting that the function » introduces a jump discontinuity around the critical point x c = ` – ` . TheHamiltonian H T , acting on » , produces the wavefunction » which showsa logarithmic divergence around x c . A similar singular behavior around thecritical point x c is detectable also in the classical dynamics ( cf . Section4).The unitary propagator U T ( t ) := e ` i tH T acts as an integral operator ( U T ( t ) )( x ) := Z R d y U –t x + 1 – ; y + 1 – ! ( y ) ; t R n f g (1.27)with kernel given by ( cf . Proposition 3.2) U fi ( x; y ) := sgn( x ) + sgn( y )i 2 fi e i ( x + y ) fi J fi q j xy j ! (1.28)for all fi R nf g . Finally, the knowledge of the unitary propagator allowsto compute the resolvent R “ ( H T ) := ( H T ` “ ) ` ; “ C n R by means of the Laplace transformation (see Section 3.3). It turns outthat also R “ ( H T ) is an integral operator ( R “ ( H T ) )( x ) := 1 – Z R d y Z “– x + 1 – ; y + 1 – ! ( y ) ; (1.29)with kernel Z ¸ ( x; y ) given explicitly by the (long) formulas (3.7) and (3.8).Theorem (1.1) provides also the first step for the one-dimensional scat-tering theory of the thermal Hamiltonian. Indeed, one infers from Theorem(1.1) that H T does not admit bounded states and so generate a “free-like”dynamics. In this work only the scattering theory for a special type of convolution perturbations is discussed. The scattering theory for (physi-cal) perturbations given by gravitational-matter potentials of type (1.10)presents several technical complications and will be treated in a separatedwork. By convolution perturbation we mean an integral operator W g actingon L ( R ) as ( W g )( x ) := Z R d y g ( x ` y ) ( y ) (1.30)where the kernel is chosen as g L ( R ) . Let us denote by H T;g := H T + W g (1.31)the perturbed operator. As usual the wave operators for the pair ( H T ; H T;g ) are defined by ˙ ˚ g := s ` lim t !˚ e i tH T;g e ` i tH T (1.32) where the limit is meant in the strong sense. In Section 3.4 the followingresult will be proven. Theorem 1.2 (Scattering theory for convolution perturbations in d = 1 ) . Let g L ( R ) and W g the associated convolution perturbation defined by (1.30) . Then: (i) The perturbed operator H T;g defined by (1.31) is self-adjoint withdomain D ( H T ) and ff “ H T;g ” = ff a : c : “ H T;g ” = R : Let ^ g be the Fourier transform of g and assume that there are constants " > and C > such that j ^ g ( x ) j (cid:54) Cx for all j x j < " . Then: (ii) The wave operators ˙ ˚ g defined by (1.32) exist and are complete; (iii) The S-matrix S g := (˙ + g ) ˜ ˙ ` g is a constant phase given by S g = e ` i p ı– R R d s ^ g ( s ) s : Structure of the paper.
Section 2 is devoted to the study of the spectraltheory of the auxiliary operator ˝ in the one-dimensional case. The spec-tral theory of the one-dimensional thermal Hamiltonian H T is discussed inSection 3 along with a subsection on the scattering theory by a convolution-type potentials. The classical dynamics of the thermal Hamiltonian (inany dimension) is studied in Section 4. Finally Appendix A and AppendixB contain some review material and some technical computations neededto make the present work self contained. Acknowledgements.
GD’s research is supported by the grant
FondecytRegular - 1190204. GD and VL are indebted to Claudio Fernández, MariusMăntoiu and Serge Richard for many stimulating discussions.2. The spectral theory of the operator ˝ We already know from the general discussion in Section 1.2 that theoperator ˝ defined by (1.17) (or formally by (1.18)) is symmetric andin turn closable. Moreover, Proposition 1.1 ensures that ˝ admits self-adjoint extensions. While, on the one hand, these results are valid in everydimension, in this section we will classify all the self-adjoint extensions of ˝ in dimension d = 1 and we will describe the the spectral theory for thisfamily of operators.2.1. Equivalence with the momentum operator.
In dimensional d = 1 the operator ˝ is initially defined by “ ˝ ” ( x ) = i " x ( x ) + x d d x ( x ) = i x dd x [ x ( x )] ( R ) : (2.1)The last equality allows us to identify ˝ ” ` xpx on sufficiently regular functions. PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 11
The operator (2.1) is symmetric, hence closable, and its closure (stilldenoted with ˝ ) has domain D given by (1.19). In order to give a moreprecise characterization of D we will benefit from the transformation ( I )( x ) := 1 x x ! ; L ( R ) : Lemma 2.1. I is a unitary involution.Proof. A direct computation shows that jj I jj = Z R d xx ˛˛˛˛˛ x !˛˛˛˛˛ = ` Z + d x ! ˛˛˛˛˛ x !˛˛˛˛˛ = ` Z `1 + d s j ( s ) j = jj jj : Then I , initially defined on every “good enough” dense domain, extendsto an isometry on the whole L ( R ) . From its very definition, it followsthat I = . This shows that I is an involution, and in particular it isinvertible. As a consequence I is also unitary. (cid:3) Instead of ˝ let us consider the transformed operator } := I ˝ I (2.2)defined on the domain D ( } ) := I [ D ] . We use the standard notation H k (˙) := W k; (˙) L (˙) for the k -th Sobolev space with respect tothe open set ˙ „ R . Let H ( R ) := n ffi H ( R ) ˛˛˛ ffi (0) = 0 o be the space of the Sobolev functions on R vanishing in x = 0 . Let uspoint out that the latter requirement makes sense since Sobolev functionson R are uniquely identifiable with continuous functions [Bre, Theorem8.2]. In view of this remark we will tacitly identify Sobolev functions withtheir continuous representative so that the following inclusions H ( R ) H ( R ) C ( R ) hold. Proposition 2.1.
The closed symmetric operator } defined by (2.2) coin-cides with the momentum operator on H ( R ) , namely ( }ffi )( x ) = ` i ffi ( x ) ; ffi ( } ) = H ( R ) where ffi is the weak derivative of ffi .Proof. The unitarity of I implies that the graph norms of } and ˝ arerelated by jj ffi jj } = jj Iffi jj ˝ for all ffi ( } ) . This gives D ( } ) = I " S ( R ) jj jj ˝ = I [ S ( R )] jj jj } : Let ffi I [ S ( R )] . Since Iffi ( R ) , one infers from (2.1) that (˝ Iffi )( x ) = i x dd x [ x ( Iffi )( x )] = i x dd x " ffi x ! = ` i x d ffi d x x ! : Therefore ( }ffi )( x ) = ( I (˝ Iffi ))( x ) = ` i d ffi d x ( x ) For the theory of Sobolev spaces we refer the reader to [Bre, Chapter 8 & Chapter 9]. acts as the momentum operator on I [ S ( R )] . This implies that the domainof the closed operator } is given by the closure of I [ S ( R )] with respectthe Sobolev norm jj ffi jj H := jj ffi jj + jj ffi jj . Let C c ( R n f g ) be the setof smooth functions having compact support separated from the origin. Itholds true that C c ( R n f g ) I [ S ( R )] H ( R ) : (2.3)Indeed, let C c ( R n f g ) supported in [ ` b; ` a ] [ [ a; b ] and ffi := I . A direct inspection shows that ffi is a smooth function supported in [ ` a ` ; ` b ` ] [ [ b ` ; a ` ] . This allows to conclude that I [ C c ( R n f g )] „C c ( R n f g ) . By exploiting the involutive character of I one gets I [ C c ( R nf g )] = C c ( R n f g ) S ( R ) and in turn C c ( R n f g ) I [ S ( R )] . Forthe second inclusion let us take ffi I ( S ( R )) so that ffi ( x ) = x ` ( x ` ) for some ( R ) . Clearly, ffi is smooth in R n f g and extends to asmooth function on R such that ffi ( n ) (0) = 0 for all n N . In particular ffi H ( R ) , implying the second inclusion I [ S ( R )] H ( R ) . To concludethe proof it is enough to show that the closure of the space C c ( R n f g ) with respect to the Sboolev norm jj jj H is (identifiable with) H ( R ) . Let R + := (0 ; + ) and R ` := ( `1 ; and observe that C c ( R n f g ) jj jj H = C c ( R ` ) jj jj H ˘ C c ( R + ) jj jj H = W ; ( R ` ) ˘ W ; ( R + ) = H ( R ) (2.4)where the notation for W ; (˙) was borrowed from [Bre, Section 8.3].The last equality in (2.4) is a consequence of the fact that every elementof W ; ( R ˚ ) can be uniquely identified with a continuous function thatvanishes on the boundary x = 0 [Bre, Theorem 8.12]. The identification(2.4), along with the double inclusion (2.3), implies the desired result D ( } ) = I [ S ( R )] jj jj H = H ( R ) . (cid:3) The first consequence of Proposition 2.1 is a precise description of thedomain of the closed operator ˝ , i. e. D = I [ D ( } )] = ( L ( R ) ˛˛˛˛ ( x ) = 1 x ffi x ! ; ffi H ( R ) ) : (2.5)Unlike the functions in H ( R ) , the elements of the domain D are generallynot continuous and can show singularities in x = 0 . An example is thefunction ffi ( x ) := (1 + x ) ` e ` x which is evidently an element of H ( R ) .Its image ( x ) := ( Iffi )( x ) = ( x + x ) ` e ` x is divergent in x = 0 . Onthe other hand elements of D have a decay at infinity which is at least oforder 1. Proposition 2.2.
Let . Then it holds true that lim j x j!1 ( x ( x )) = 0 : Proof.
The claim follows from the characterization (2.5) which provides lim x !˚1 ( x ( x )) = lim t ! ˚ ffi ( t ) = ffi (0) = 0 : In the last equality, the continuity of ffi H ( R ) is used. (cid:3) PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 13
Classification of self-adjoint extensions.
We are now in positionto study the self-adjoint realizations of ˝ . In view of the unitary transform I this is the same of studing the self-adjoint realization of the singular momentum operator } . The latter is a classical problem strongly relatedwith the study of singular delta interactions for one-dimensional Diracoperators [GS, BD, CMP] (see also [AGHG, Appendix J]). Proposition 2.3.
The closed symmetric operator } has deficiency indicesequal to 1. Therefore, the self-adjoint extensions of } are in one-to-one correspondence with the angles „ S ’ [0 ; ı ) . The self-adjointextension } „ has domain D ( } „ ) := ’ L ( R ) ˛˛˛˛ ’ = ffi + c” „ ; ffi H ( R ) ; c C ff where ” „ ( x ) := e `j x j e i sgn( x ) „ and acts has } „ ( ffi + c” „ ) := ` i ffi + c” „ + ı : (2.6) Finally, } agrees with the standard momentum operator p with domain H ( R ) .Proof. Since C c ( R n f g ) = C c ( R ` ) ˘ C c ( R + ) is dense (with respect tothe graph norm) in the domain of } , a standard argument shows that theadjoint operator } ˜ acts as the weak derivative on its domain D ( } ˜ ) := H ( R ` ) ˘ H ( R + ) (see e. g. [RS1, Section VII.2]). The eigenvalue equations } ˜ ffi ˚ = ˚ i ffi ˚ for the deficiency subspaces correspond to the differentialequations ffi = ˇ ffi ˚ which admit in D ( } ˜ ) the unique (normalized) weaksolutions ffi + ( x ) := p ` x if x > if x < ; ffi ` ( x ) := if x > p + x if x < : According to the von Neumann’s theory for self-adjoint extensions ( cf . [RS2,Section X.1]) one has that the self-adjoint extensions of } are parametrizedby the unitary maps from K + = C [ ffi + ] ’ C to K ` = C [ ffi ` ] ’ C . Thelater are identified by the angle „ S ’ [0 ; ı ) according to U „ ffi + :=e ` i „ ` . From the general theory [RS2, Theorem X.2] one has that thedomain of the self-adjoint extension } „ is made by functions of the type ffi + c ( ffi + + e ` i „ ffi ` ) = ffi + c” „ with ffi H ( R ) and c; c C suitablecomplex coefficients. The action of } „ on the elements of its domain isgiven by } „ “ ffi + c ( ffi + + e ` i „ ffi ` ) ” = ` i ffi + i c ( ffi + ` e ` i „ ffi ` ) which translates into equation (2.6) in terms of the function ” „ . Evidently,the standard momentum operator p is a self-adjoint extension of } since H ( R ) H ( R ) . This extension corresponds to } in view of the fact that ” H ( R ) . (cid:3) Although the symmetric operator } admits several self-adjoint realiza-tions, all these realizations are in a sense equivalent. To express this factin a precise way we need to introduce the family of unitary operators L „ defined by ( L „ )( x ) := e i sgn( x ) „ ( x ) ; L ( R ) : Proposition 2.4.
The unitary operators L „ intertwine all the self-adjointrealizations of the operator } . More precisely one has that } „ = L „ pL ˜ „ ; „ S where p = } is the standard momentum operator. As a consequence onehas that ff ( } „ ) = ff a : c : ( } „ ) = R ; „ S : Proof.
Let us shows that D ( } „ ) = L „ [ H ( R )] . Every H ( R ) can bedecomposed as ( ` (0) ” ) + (0) ” . Evidently ffi := ` (0) ” H ( R ) , L „ ffi H ( R ) and L „ ” = ” „ . Therefore, L „ ( } „ ) whichimplies L „ [ H ( R )] „ D ( } „ ) . On the other hand every ’ ( } „ ) canbe decomposed as ’ = L „ ( L ` „ ffi + c” ) whit L ` „ ffi H ( R ) , and in turn ( L ` „ ffi + c” ) H ( R ) proving the inverse inclusion D ( } „ ) „ L „ [ H ( R )] .Now, let ’ ( } „ ) . By exploiting the decomposition used above one has ( L „ pL ˜ „ ) ’ = L „ ( pL ` „ ffi + cp” ) = ` i ffi + c” „ + ı where we used ( p” )( x ) = ” ( x ) e i sgn( x ) ı and pL ` „ ffi = L ` „ pffi in view of ffi H ( R ) . Hence, a comparison with (2.6) shows that L „ pL ˜ „ = } „ onthe domain D ( } „ ) . (cid:3) Remark 2.1.
The unitary equivalence of the different realizations } „ canbe understood in terms of the celebrated Stone-von Neumann theorem (see e. g. [Ros]). Indeed, a direct computation shows that ( x } „ ` } „ x ) ’ = i ’ ; ’ c ( R n f g ) and C c ( R n f g ) = C c ( R ` ) ˘ C c ( R + ) is dense in L ( R ` ) ˘ L ( R + ) = L ( R ) . Therefore, by continuous extension, one can unambiguously definethe commutation relation [ x; } „ ] = i which means that the pair ( x; } „ ) satisfies the canonical commutation relation . As a result, the Stone-vonNeumann theorem ensures that } „ is unitarily equivalent to the standardmomentum p . (cid:74) Proposition 2.3 provides the key result for the complete description ofthe self-adjoint extensions of ˝ . Theorem 2.1 (Self-adjoint extensions: one-dimensional case) . The self-adjoint extensions of the closed symmetric operator ˝ initially defined by (2.1) are in one-to-one correspondence with the angles „ S . The self-adjoint extension ˝ „ has domain D (˝ „ ) : = ’ L ( R ) ˛˛˛˛ ’ = + c“ „ ; ; c C ff where “ „ ( x ) := 1 x e ` j x j e i sgn( x ) „ and acts has ˝ „ ( + c“ „ ) := ˝ + c“ „ + ı : All the self-adjoint realizations are unitarily equivalent, i. e. ˝ „ = L „ ˝ L ˜ „ for all „ S . Finally one has that ff (˝ „ ) = ff a : c : (˝ „ ) = R ; „ S : PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 15
Proof.
This is a direct consequence of the unitary equivalence establishedin Proposition 2.1 which allows to define the self-adjoint realizations of ˝ by ˝ „ := I} „ I . Therefore, the statement is nothing more than a rephrasingof Proposition 2.3 and Proposition 2.4. The formula ˝ „ = L „ ˝ L ˜ „ isjustified by the commutation relation L „ I = IL „ . (cid:3) In view of the unitary equivalence among all the self-adjoint realizationsof ˝ we can focus the attention only in a “preferred” realization. Definition 2.1 (Standard realization) . We will call ˝ = ˝ „ =0 the stan-dard self-adjoint realization of the operator initially defined by (2.1).2.3. Boundary triplets.
The problem of the determination of the self-adjoint realizations of } or ˝ can be investigated also with the theory ofthe boundary triplets [Sch, Chapter 14]. Let us start with the operator } and its adjoint } ˜ . According to [Sch, Definition 14.2], a boundary tripletfor } ˜ is a triplet ( H ; ` ; ` ) made by an Hilbert space H and linear maps ` ; ` from D ( } ˜ ) to H that satisfy the abstract Green’s identity h } ˜ ’; i ` h ’; } ˜ i = h ` ’; ` i H ` h ` ’; ` i H ; ’; ( } ˜ ) and the mapping D ( } ˜ ) ’ (` ’; ` ’ ) H ˆ H is surjective. Sincethe operator } ˜ acts as the weak derivative on its domain D ( } ˜ ) := H ( R ` ) ˘ H ( R + ) ; an integration by parts provides h } ˜ ’; i ` h ’; } ˜ i = i „ ’ (0 ` ) (0 ` ) ` ’ (0 + ) (0 + ) « where ’ (0 ˚ ) := lim x ! ˚ ’ ( x ) and similarly for (0 ˚ ) . A comparison withthe abstract Green’s identity shows that the triplet ( H ; ` ; ` ) can be fixedin the following way: H := C ; ` ’ := ’ (0 + ) ` ’ (0 ` )i p ; ` ’ := ’ (0 + ) + ’ (0 ` ) p : The surjectivity condition is obviously satisfied. Observe that
Ker(` ) \ Ker(` ) = H ( R ) = D ( } ) . The self-adjoint extensions of } are in one-to-one correspondence with the self-adjoint operators on H = C [Sch,Theorem 14.10]. More precisely, the self-adjoint extensions of } can beparametrized by a real number ‚ R [ f1g which defines a restriction } ‚ := } ˜ j D ‚ where the domain D ‚ D ( } ˜ ) is defined by D ‚ : = ’ ( } ˜ ) ˛˛˛˛ ‚ ` ’ = ` ’ ff = ( ’ ( } ˜ ) ˛˛˛˛ e ` i arctan “ ‚ ” ’ (0 + ) = e i arctan “ ‚ ” ’ (0 ` ) ) : (2.7)A comparison with Proposition 2.3 shows that the self-adjoint extensions } „ and } ‚ are related by the equation „ ( ‚ ) = arctan „ ‚ « . In particular,the standard momentum is identified by ‚ = which corresponds to „ = 0 . The definition (2.7) provides the description of the domain of } „ in therms of boundary conditions . The same can be done for the the self-adjoint extensions ˝ „ with the help of the unitary operator I . A directcomputation shows that D (˝ „ ) : = ’ I [ D ( } ˜ )] ˛˛˛˛ e ` i „ ( x’ )(+ ) = e + i „ ( x’ )( `1 ) ff where ( x’ )( ˚1 ) := lim x !˚1 x’ ( x ) .2.4. Unitary propagator.
Let V „ ( t ) := e ` i t ˝ „ ; t R (2.8)be the unitary propagator defined by the self-adjoint operator ˝ „ on L ( R ) .The description of V „ ( t ) is provided in the following theorem. Theorem 2.2.
Let V „ ( t ) be the unitary group defined by (2.8) . It holdstrue that “ V „ ( t ) ” ( t ) = e i „ “ ` sgn(1 ` tx ) ” sgn( x ) ` tx x ` tx ! ; L ( R ) : Proof.
We can use the unitary equivalence ˝ „ = IL „ pL ˜ „ I proved in Section2.2. This implies that V „ ( t ) = IL „ e ` i tp L ˜ „ I along with the well-knownfact ( e ` i tp )( x ) = ( x ` t ) . The proof of the claim follows by a directcomputation. (cid:3) For each t R let us consider the map f t : R [ f1g ! R [ f1g definedby f t ( x ) := x ` tx if x R n f t ` g1 if x = t ` ` t ` if x = : (2.9)with the convention that ˚ ` ” 1 . The family of these maps defines an R -flow in the sense that the following relations hold: f = Id f t ‹ f t = f t + t f ` t = f ` t t; t ; t R : (2.10)The flow f t allows to rewrite the action of V „ ( t ) in the form “ V „ ( t ) ” ( t ) = e i2 “ ` sgn(1 ` tx ) ”“ sgn( x ) „ + ı ” q ( @ x f t )( x ) ( f t ( x )) : (2.11)When „ = ı the exponential prefactor is 1 and equation (2.11) agrees withthe definition of the C -group associated to the flow f t as defined in [ABG,Section 4.2]. It is interesting to notice that the flow f t is not of class C and the generator of the flow F ( x ) := d f t d t ˛˛˛˛˛ t =0 ( x ) = x has an unbounded first derivative. Therefore the flow f t doesn’t meetthe conditions of [ABG, Lemma 4.2.2 & Proposition 4.2.3]. The latterfact explains why [ABG, Proposition 4.2.3] doesn’t apply to the operator ˝ ” ` ( pF ( x ) + F ( x ) p ) which indeed is not essentially self-adjoint on C c ( R ) . PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 17
Resolvent and Green function.
The resolvent of the of the operator ˝ „ can be derived from the resolvent of the standard momentum operator p by exploiting the various unitary equivalences described in Section 2.2.For every “ C n R the resolvent of ˝ „ at “ is defined as R “ (˝ „ ) := (˝ „ ` “ ) ` = L „ I ( p ` “ ) ` IL ˜ „ : (2.12)The next results shows that R “ (˝ „ ) is an integral operator. Proposition 2.5.
Let “ := › ˚ i ‹ C n R with ‹ > . The resolvent R “ (˝ „ ) acts as “ R “ (˝ „ ) ” ( x ) = Z R d y R „“ ( x; y ) ( y ) ; L ( R ) with kernel given by R „› ˚ i ‹ ( x; y ) := e i “ sgn( x ) ` sgn( y ) ” „ ˇ i xy ˆ ˚ x ` y !! e i › “ x ` y ” e ` ‹ ˛˛˛ x ` y ˛˛˛ where ˆ is the Heaviside function . Proof.
The integral kernel R “ of the resolvent of ˝ can be obtained fromthe Green’s function G “ of the standard momentum operator (see AppendixA.1). A direct computation provides “ R “ (˝ ) ” ( x ) = “ I ( p ` “ ) ` I ” ( x ) = 1 x Z R d y G “ x ; y ! y y ! : The explicit expression of G “ given in (A.1) and a change of variable in theintegral provide R “ ( x; y ) := 1 xy G “ x ; y ! : Since L „ is a multiplication operator, the relation between the kernels for „ = 0 and „ = 0 is simply given by R „“ ( x; y ) := e i “ sgn( x ) ` sgn( y ) ” „ R “ ( x; y ) = e i “ sgn( x ) ` sgn( y ) ” „ xy G “ x ; y ! : This concludes the proof. (cid:3)
It is worth noting that that along the diagonal one has R „“ ( x; x ) = sgn(Im( “ )) i2 x for all “ C n R and „ S . The
Heaviside function is defined by ˆ( x ) := if x > if x = 00 if x < : Spectral measure and density of states.
Let — „ be the spec-tral measure of the operator ˝ „ associated with the normalized state L ( R ) . We know from Theorem 2.1 that ˝ „ as a purely absolutelycontinuous spectrum which coincides with R . This implies that the spectralmeasure — „ is purely absolutely continuous. More precisely one has that — „ ( d › ) := f „ ( › ) d › with f „ L ( R ) a non-negative function. The next result provides adescription of f „ . Proposition 2.6.
Let — „ be the spectral measure of the operator ˝ „ as-sociated with the (normalized) state L ( R ) . Then — „ is absolutelycontinuous with respect to the Lebesgue measure d › in R and — „ ( d › ) := j b ffi „ ( › ) j d › (2.13) where b ffi „ := F ( ffi „ ) is the Fourier transform of the function ffi „ ( x ) := ( L ˜ „ I )( x ) = e ` i sgn( x ) „ x x ! : Proof.
From the unitary equivalence ˝ „ = IL „ pL ˜ „ I one gets F „ ( “ ) := h ; (˝ „ ` “ ) ` i = h ; IL „ ( p ` “ ) ` L ˜ „ I i = F pffi „ ( “ ) : Following the arguments in Appendix A.2 on gets lim ‹ ! + ı Im “ F „ ( › + i ‹ ) ” = f pffi „ ( › ) = j b ffi „ ( › ) j where the last equality is justified by (A.3). This concludes the proof. (cid:3) In order to define the integrated density of states (IDOS) of ˝ „ let usstart by introducing the spectral projections P „› of ˝ „ defined by P „› := ffl [0 ;› ] (˝ „ ) if › > ffl [ ›; (˝ „ ) if › < : Let ( Q ˜ )( x ) = ffl ˜ ( x ) ( x ) be the projection which restricts the functions L ( R ) on the interval ˜ = [ a; b ] . Let us introduce the function N „ ˜ : R ! R defined by N „ ˜ ( › ) := sgn( › ) j ˜ j Tr “ P „› Q ˜ ” : (2.14)Definition (2.14) is well posed in view of the following result: Lemma 2.2.
Let ˜ := [ a; b ] with ab > . The operator P „› Q ˜ is traceclass and N „ ˜ ( › ) = 1 ab › ı independently of „ .Proof. By combining the spectral theorem with the unitary equivalencebetween ˝ „ and p one gets that P „› = L „ IP › IL ˜ „ where P › := ffl [0 ;› ] ( p ) if › > ffl [ ›; ( p ) if › < : (2.15) PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 19
This means that P „› Q ˜ = L „ ( IP › IQ ˜ ) L ˜ „ = L „ I ( P › IQ ˜ I ) IL „ . Thus, toprove that P „› Q ˜ is trace-class it is sufficient to prove that P › ( IQ ˜ I ) istrace-class. Let b > a > or a < b < . A direct computation shows that ( IQ [ a;b ] I )( x ) = ffl [ a;b ] x ! ( x ) = ffl [ b ` ;a ` ] ( x ) ( x ) ; L ( R ) ; namely IQ ˜ I = Q ~˜ with ~˜ := [ b ` ; a ` ] . This implies that P › ( IQ ˜ I ) = P › Q ~˜ is trace-class in view of [RS3, Theorem XI 20]. Moreover, one hasthat N „ ˜ ( › ) = j ~˜ jj ˜ j sgn( › ) j ~˜ j Tr “ P „› Q ˜ ” = a ` ` b ` b ` a N p ~˜ ( › ) where N p ~˜ ( › ) is the local density of states for the operator p in the region ~˜ . The proof follows by using Lemma A.2. (cid:3) The quantity N „ ˜ ( › ) measures the volumetric density of states up tothe energy › localized in the region ˜ . States with negative energy arecounted as “negative” states. Lemma 2.2 shows that this number is nothomogeneous in space. One can ask how this number changes for fixedvolume in function of the spatial localization. Let ‘ > and set ˜ x;‘ :=[ x; x + ‘ ] when x > or ˜ x;‘ := [ x ` ‘; x ] when x < . Then N „ ˜ x;‘ ( › ) := 1 x + j x j ‘ › ı : Since the density decreases as x ` in function of the spatial localizationand as ‘ ` in function of the volume one immediately concludes that themajority of states are concentrated around x = 0 with a divergent density.Ultimately, the spatial inhomogeneity of N „ ˜ is a consequence of thefact that ˝ „ breaks the invariance under spatial translations. To define adensity of states on the thermodynamic limit a precise prescription on howto carry out the spatial average is necessary. Let us define the principalvalue integral density of states (pv-IDOS) as pv ` N „ ( › ) := lim L !1 L L `
1) sgn( › ) Tr “ P › Q L ” where Q L := Q [ ` L;L ] ` Q [ ` L ` ;L ` ] . From Lemma (2.2) one immediatelygets that pv ` N „ ( › ) := › ı :
3. The spectral theory of the thermal HamiltonianThe thermal Hamiltonian H T is defined by equation (1.21) as the conju-gation of ˝ through the unitary F S – . For this reason the spectral theoryof H T (summarized by Theorem 1.1) is equivalent to the spectral theoryof ˝ studied in Section 2. The next section is mainly devoted to thetranslation of the results obtained for ˝ to H T by exploiting the preciseform of the unitary F S – . Description of the domain.
By construction the domain of H T isgiven by D ( H T ) := ( F S – ) ˜ [ D (˝ )] = ( S ˜ – B F ˜ )[ H ( R )] with B := F ˜ I F . The last equality is justified by D (˝ ) = I [ D ( } )] and D ( } ) = H ( R ) . It is known that the Fourier transform of H ( R ) isthe domain of the position operator [RS2, Chapter IX] defined by (1.22).Therefore, the domain H T is made by functions in Q ( R ) transformed bythe operator S ˜ – B . The operators B and B – := S ˜ – B have a description interms of integral kernels. Lemma 3.1.
On the dense domain L ( R ) \ L ( R ) the operator B = F ˜ I F acts as an integral operator with kernel given by (1.24) . As a consequence B – := S ˜ – B acts according to (1.23) .Proof. Let us start with the computation of the kernel of B acting on L ( R ) \ L ( R ) . Then F ( ) L ( R ) \C ( R ) , namely F ( ) is a square-integrable continuous function that vanishes at infinity. For every n N ,let ffl I n be the characteristic function of the interval I n := [ ` n; ` n ` ] [ [ n ` ; n ] Since F ( ) ` F ( ) ffl I n = F ( ) ffl I cn , where I cn is the complementof I n , one can prove that F ( ) ffl I n ! F ( ) in the L -topology when n ! + . Thus, the unitarity of the Fourier transform implies that n ! inthe L -topology where n := F ˜ ( F ( ) ffl I n ) = ˜ F ˜ ( ffl I n ) and ˜ denotesthe convolution. Since B is a unitary operator one gets B n ! B withrespect to the L -topology. An explicit computation provides ( B n )( x ) = ( F ˜ I F n )( x )= ( F ˜ I F ( ˜ F ˜ ffl I n )) ( x )= ( F ˜ I ( F ) ffl I n )( x )= 1 p ı Z R d u e i ux u ( F ) u ! ffl I n u ! = 12 ı Z I n d u e i ux u Z R d y e ` i yu ( y ) ! where in the last two equalities we used the fact that I ( F ) ffl I n and are L -functions (this justifies the use of the integral representation of F and F ˜ ) and the equality ffl I n ( u ` ) = ffl I n ( u ) . Since the function g x ( y; u ) := u e i xu e ` i yu ( y ) is absolutely integrable in R ˆ I n one can invoke theFubini-Tonelli theorem to change the order of integration. This provides ( B n )( x ) = 12 ı Z R d y ( y ) I n d u e i xu e ` i yu u : (3.1)Corollary B.1 says that lim n !1 Z I n d u e i xu e ` i yu u = 2 ı B ( x; y ) : And ˛˛˛˛˛˛Z I n d u e i xu e ` i yu u ˛˛˛˛˛˛ (cid:54) ı PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 21 for all n > n . In view of the bound above one can use the Lebesgue’sdominated convergence theorem in (3.1) providing the formula lim n ! + ( B n )( x ) = Z + d y B ( x; y ) ( y ) : (3.2)Equation (3.2) says that B n converges pointwise to the integral in theright-hand side. Since B n converges to B in the L -topology it fol-lows there exists a subsequence B n k which converges pointwise (almosteverywhere) to B [Bre, Theorem 4.9 (a)]. Then the unicity of the limitassures that B coincides with the right-hand side of (3.2). The last partof the proof follows from the explicit computation ( B – )( x ) = ( B ) x + 1 – ! = Z R d y B x + 1 – ; y ! ( y ) which provides equation (1.23). (cid:3) Remark 3.1.
Lemma 3.1 states that B – can be expressed as an integraloperator only on the dense domain L ( R ) \ L ( R ) . For function in L ( R ) n L ( R ) in principle, we do not have the right to write B – using the integral kernel. However, in the following, we will tacitly usethe following convention ( B – )( x ) ” lim R !1 Z + R ` R d y B x + 1 – ; y ! ( y ) ; if L ( R ) n L ( R ) : This identification must be understood as follows: (i) The product R := ffl [ ` R; + R ] is in L ( R ) \ L ( R ) and so B – R can be computed (pointwise)through the integral formula; (ii) R ! , and in turn B – R ! B – , inthe L -topology; (iii) Then, the identification above makes sense almosteverywhere on subsequences [Bre, Theorem 4.9 (a)]. (cid:74) Lemma 3.1 allows to describe the domain of H T as follows: D ( H T ) = ( L ( R ) ˛˛˛ ( x ) = Z R d y B – ( x; y ) ffi ( y ) ; ffi ( R ) ) : An explicit computation (made of several changes of integration variable)shows that the generic element in D ( H T ) has the form ( x ) = 1 x + – Z + d s J ( p s ) ffi sx + – ; ffi ( R ) : From (1.19) and Theorem 2.1 one infers that S ( R ) D D (˝ ) and S ( R ) + C [ “ ] is a core for ˝ . Since ( F S – ) ˜ [ S ( R )] = S ( R ) in view ofthe invariance of the Schwartz space under the Fourier transform and thetranslations, it follows that D ( H T ) := S ( R ) + C [ » ] is a core for H T , with » := ( B – F ˜ ) ” (the function ” is described inProposition 2.3). Moreover, the unitary transform B – F ˜ and Proposition2.3 also justify (1.25) with » := ( B – F ˜ ) ” ı . Proposition 3.1.
The functions » and » are given by the formulas (1.26) . Proof.
Let ” ( x ) = e `j x j and ” ı ( x ) = i sgn( x ) e `j x j . The inverse Fouriertransforms of these functions are given by ( F ˜ ” )( x ) = vuut ı
11 + x ; ( F ˜ ” ı )( x ) = ` vuut ı x x : Since F ˜ ” L ( R ) \ L ( R ) , the transformed function B F ˜ ” can becomputed via the integral kernel of B . Then Lemma B.2 provides ( B F ˜ ” )( x ) = ` i vuut ı sgn( x ) kei „ q j x j « : Since F ˜ ” L ( R ) n L ( R ) , the transformed function B F ˜ ” as to becomputed according to the prescription of Remark 3.1. In this case onehas ( B F ˜ ” ı )( x ) = ` vuut ı lim R ! + Z + R ` R d y B ( x; y ) y y : However, as shown in the proof of Lemma B.2, the integrant is absolutelyintegrable for every values of x . This allows to forget the limit and onegets ( B F ˜ ” ı )( x ) = i vuut ı ker „ q j x j « : Finally a translation by S ˜ – and a multiplication by ` i provide the formulas(1.26). (cid:3) Remark 3.2 (Other self-joined extensions) . As for the operator ˝ discussedin Section 2, also the thermal Hamiltonian H T admits a family of unitarilyequivalent self-adjoint extension parametrized by „ S , and defined by H T;„ := – ( F S – ) ˜ ˝ „ ( F S – ) : Since ˝ „ = L „ ˝ L ˜ „ one obtains that H T;„ is related to the standard thermalHamiltonian H T by the unitary equivalence H T;„ := N „ H T N ˜ „ where N „ := ( F S – ) ˜ L „ ( F S – ) . An explicit computation provides that N „ := cos „ ! ` sin „ ! H where H denotes the Hilbert transform defined by ( H )( x ) := 1 ı Z R d y ( y ) x ` y over sufficiently regular functions , and with the integral taken as aCauchy principal value. (cid:74) Unitary propagator.
Let U T ( t ) := e ` i tH T the unitary propagator associated with the self-adjoint operator H T . Us-ing the various unitary equivalences that connect H T with the momentumoperator p one has that U T ( t ) = B – “ F ˜ e ` i –tp F ” B ˜ – = S ˜ – ( B e i –tx B ) S – PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 23 where in the last equality we used F ˜ p F = ` x . With the help of Lemma3.1 we can compute the integral kernel of U T ( t ) . Proposition 3.2.
On the dense domain L ( R ) \ L ( R ) the unitary propa-gator U T ( t ) with ( t = 0 ) acts as an integral operator with kernel given by (1.27) and (1.28) .Proof. Let us start by computing the kernel of A fi := B e i fi x B , with fi R n f g , on L ( R ) \ L ( R ) . ( A fi )( x ) := lim R ! + Z + R ` R d y e i fi y B ( x; y ) Z R d s B ( y; s ) ( s ) ! The integral in the variable y is meant in the sense of a principal valuein view of Remark 3.1. For every x; fi R the function g ( x;fi ) ( s; y ) :=e i fi y B ( x; y ) B ( y; s ) ( s ) is absolutely integrable in R ˆ [ ` R; + R ] since j g ( x;fi ) j (cid:54) j B ( x; y ) jj ( s ) j . Then, we can invoke the Fubini-Tonelli theoremto change the order of integration ( A fi )( x ) = lim R ! + Z R d s A Rfi ( x; s ) ( s ) (3.3)where A Rfi ( x; s ) := Z + R ` R d y e i fi y B ( x; y ) B ( y; s ) : For xs = 0 the change of variables u := ` xy provides A Rfi ( x; s ) := sgn( s ) + sgn( x )2 x Z + R j x j d u e ` i fix u J (2 p u ) J vuut j s jj x j p u : By using formula [GR, eq. 6.615] one gets lim R ! + A Rfi ( x; s ) = ` i sgn( s ) + sgn( x )2 fi e i x + sfi I ` i 2 fi q j xs j ! : Finally, the well known relations I ( ˚ i x ) = J ( ˇ x ) = J ( x ) valid for x (cid:62) provide lim R ! + A Rfi ( x; s ) = U fi ( x; s ) (3.4)where the kernel U fi is defined by (1.28). Equation (3.4) is valid also inthe singular cases xs = 0 . For instance, for x = 0 on gets after the usualchange of coordinates lim R ! + A Rfi (0 ; s ) = 12 s Z + d u e ` i fis u J “ p u ” = U fi (0 ; s ) where the last equality is justified by [GR, eq. 6.614 (1)]. The case s = 0 is similar. In view of (3.4) we have the pointwise convergence lim R ! + A Rfi ( x; s ) ( s ) = U fi ( x; s ) ( s ) : and since j U fi ( x; s ) j (cid:54) j fi j ` for all ( x; s ) R one has that the function s A Rfi ( x; s ) ( s ) is definitively dominated by the integrable function s
7! j fi j ` ( s ) (provided fi = 0 ). This fact allows to use the Lebesgue’sdominated convergence theorem in (3.3), providing in this way ( A fi )( x ) = Z R d s U fi ( x; s ) ( s ) : (3.5)Formula (1.27) is obtained by observing that U T ( t ) = S ˜ – A –t S – . (cid:3) Resolvent and Green function.
The resolvent of H T can be com-puted as the Laplace transform of the unitary propagator U T ( t ) accordingto the well known formula [Kat, eq. (1.28), p. 484]. For every “ C n R let R “ ( H T ) := ( H T ` “ ) ` be the resolvent of H T . Then, it holds true that R “ ( H T ) = i Z + d t e i “t U T ( t ) ; Im( “ ) > (3.6)where the integral is interpreted as a strong Riemann integral lim ff ! + R ff .The resolvent for Im( “ ) < can be obtained from the relation R “ ( H T ) = R “ ( H T ) ˜ . The formula (3.6) is helpful to compute the integral kernel of R “ ( H T ) .One can take advantage of the unitary equivalence U T ( t ) = S ˜ – A –t S – used in Proposition 3.2 to obtain R “ ( H T ) = – ` S ˜ – Z “– S – with Z ¸ := i Z + d fi e i ¸fi A fi ; Im( ¸ ) > : Let L ( R ) \ L ( R ) . With the integral kernel of A fi provided in (3.5)one can write ( Z ¸ )( x ) := i lim ff ! + Z ff d fi e i ¸fi Z R d s U fi ( x; s ) ( s ) ! : Since the J ( fi ` ) ‰ p fi if fi ! one can check that the function h x ( fi; s ) := e i ¸fi U fi ( x; s ) ( s ) meets the conditions of the Fubini-Tonellitheorem for the change of the order of integration. Moreover, one cantake care of the limit in ff with the help of the Lebesgue’s dominatedconvergence theorem. At the end of these manipulations one gets ( Z ¸ )( x ) := Z R d s Z ¸ ( x; s ) ( s ) with kernel Z ¸ ( x; s ) : = i Z + d fi e i ¸fi U fi ( x; s )= “ sgn( x ) + sgn( y ) ” F ¸ ( x; y ) (3.7)where F ¸ ( x; y ) := 12 Z + d fi e i “ ¸fi + ( x + s ) fi ” fi J fi q j xs j ! : Setting ¸ := j ¸ j e i ffi , < ffi < ı , the last integral can be integrated caseby case using the Macdonald’s and Nicholson’s formulas [Erd1, Section7.7.6] or [MO, Section III, p. 98]. A different way of calculating the kernel(3.7) is sketched at the end of Appendix B.3. In both cases, after sometedious calculations, one gets F ¸ ( x; y ) := I q j ¸ j min fj x j ; j y jg e i » ffi ` ı “ sgn( x )+1 ”– 1CA ˆ K q j ¸ j min fj x j ; j y jg e i » ffi ` ı “ sgn( x )+1 ”– 1CA : (3.8) PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 25
It is also possible to check directly that the kernel Z ¸ ( x; s ) inverts in adistributional sense the operator T ` ¸ .3.4. Scattering by a convolution potential.
Let g L ( R ) and considerthe associated convolution potential W g defined by (1.30). Since W g is abounded operator of norm k W g k = k g k the perturbed operator H T;g := H T + W g is well defined as a self-adjoint operator on the domain D ( H T ) as a consequence of the Kato-Rellich theorem [RS2, Theorem X.12]. Theit makes sense to consider the scattering theory of the pair ( H T ; H T;g ) .In view of the unitary equivalence p = – I F S – H T S ˜ – F ˜ I between themomentum operator and H T we can equivalently study the scattering the-ory of the pair ( p; p g ) . where p g := p + M g is the perturbation of themomentum given by the potential M g := 1 – I F S – W g S ˜ – F ˜ I :
Lemma 3.2.
The potential M g is the multiplication operator defined by ( M g )( x ) := p ı– ^ g x ! ( x ) ; ( R ) : where ^ g denotes the Fourier transform of g .Proof. By construction the convolution is invariant under translations.This means that S – W g S ˜ – = W g . Moreover the Fourier transform of aconvolution gives a multiplication operator ( F W g F ˜ )( x ) := p ı ^ g ( x ) ( x ) ; ( R ) where ^ g denotes the Fourier transform of g . The proof is completed byobserving recalling the definition of the involution I . (cid:3) We are now in position to provide the proof of Theorem 1.2.
Proof (of Theorem 1.2).
Since g L ( R ) then ^ g C ( R ) (continuousfunctions vanishing at infinity) in view of the Riemann-Lebesgue Lemma[RS2, Theorem IX.7]. This implies that the function x ^ g “ x ” belongsto C ( R ) \ L ( R ) . As a result the multiplicative potential M g is boundedwith norm k M g k = p ı– k ^ g k and the conditions of [Kat, Example 3.1, p.530] are satisfied. Then, one obtain that p and p g are unitarily equivalent.This also implies the unitary equivalence of H T and H T;g , and in turn item(i) of claim. In [Kat, Example 3.1, p. 530] it is also proven the existenceand the completeness for the waves operators associated to the pair ( p; p g ) under the assumption that of the existence of the the improper integrals lim x ! + Z + x d s ^ g s ! = lim x ! + Z x d s ^ g ( s ) s lim x ! ` Z x `1 d s ^ g s ! = lim x !`1 Z x d s ^ g ( s ) s : This requires that ^ g ! fast enough when s ! ˚ . This is guaranteedby the (not optimal) conditions required in the theorem statement. Invok-ing once again the unitary equivalence between p and H T one obtain theexistence and the completeness for the waves operators associated to thepair ( H T ; H T;g ) , proving in this way item (ii). Also for item (iii), in [Kat, Example 3.1, p. 530] is proven that the S -matrix for the pair ( p; p g ) is acomplex number given by S g := e ` i p ı– R R d x ^ g ( x ) := e ` i p ı– R R d s ^ g ( s ) s : Since a complex number is unchanged by unitary equivalences it followsthat S g is also the S -matrix for the pair ( H T ; H T;g ) . (cid:3)
4. The classical dynamicsIn this last section we will study the classical dynamics induced by a ther-mal gradient. The classic analogue of the Luttinger’s model is providedby the the Hamiltonian function H T ( x; p ) := (1 + – ‚ ´ x ) p m = K ( p ) + – T ‚ ( x; p ) ; (4.1)with parameters – > and ‚ S d ` . The Hamiltonian H T can be seen asthe sum of the Hamiltonian of a free d -dimensional particle of mass mK ( p ) := p m = 12 m N X j =1 p j coupled through the coupling constant – > with the thermal potential T ‚ ( x; p ) := ( ‚ ´ x ) K ( p ) = p m d X j =1 ‚ j x j along the direction ‚ S d ` . The coupling constant has the dimension ofthe inverse of a distance, namely – = ‘ ` with ‘ > the typical length of the thermal field. Therefore, the limit – ! describes the situationin which the typical length of the field is much larger than the typicallength of the system ( e. g. the size of the particle). The potential T ‚ is anexample of what is known as a generalized potential , namely a potentialwhich depends not only on the position but also on the the velocity.4.1. Hamiltonian Formalism and Newton equation.
The Hamilton equa-tions associated to (4.1) read _ x = + r p H T = (1 + – ‚ ´ x ) m p _ p = `r x H T = ` – p m ‚ : (4.2)The first equation can be inverted out of the critical plane ¨ c := n x R d j ‚ ´ x + ‘ = 0 o (4.3)and provides p ( x; _ x ) = m (1 + – ‚ ´ x ) _ x (4.4)One can restore the usual relation p = m T _ x between momentum andvelocity by introducing the position-dependent mass (PDM) m T ( x ) := m (1 + – ‚ ´ x ) : PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 27
It is interesting to notice that the Hamiltonian (4.1) can be rewritten as H T ( x; p ) = p m T ( x ) ; (4.5)namely as the Hamiltonian of a free particle with a PDM. The secondequation of (4.2) can be rewritten as _ p = ` – r x T ‚ : (4.6)A straightforward computation allows to derive the Newton’s laws from(4.2): m x = – ( ‚ ´ _ x ) p ` – (1 + – ‚ ´ x ) p m ‚ : After introducing (4.4) in the las expression one obtains the Newton’sequation m x = – F T ( x; _ x ) where the thermal force (which has the dimensions of a force times adistance) is given by F T ( x; _ x ) = m T ( x ) ( ‚ ´ _ x ) _ x ` _ x ‚ : (4.7)A way of interpreting this Newton’s Equation is to say that the motionof the PDM-particle is influenced by the effect of its own internally self-produced force field generated by the spatial dependence of the mass.The relation between the force F T and the potential T ‚ can be deducedby observing that ` – r x T ‚ = ` – m T ( x ) m _ x ‚ (4.8)in view of the (4.6), (4.2) and (4.4), respectively. After some manipulationand the use of equation (4.4) one gets F T ( x; p ) = `r x T ‚ ( x; p ) + R T ( x; p ) (4.9)which shows that the thermal force is not simply given by `r x T ‚ , as forordinary conservative forces, but it includes an extra reacting term R T ( x; p ) := dd t ( ‚ ´ x ) p = m dd t ( r p T ‚ ( x; p )) (4.10)which is generally not aligned with the direction ‚ of the field.4.2. Qualitative analysis.
Let us start with the analysis of the qualita-tive behavior of the solution of the Hamiltonian system (4.1). To simplifythe study let us fix convenient notations. The unit vector ‚ can be com-pleted to an orthonormal basis by adding other d ` orthonormal vectors e ; : : : ; e d ` . This allows to fix the generalized coordinates x := ‚ ´ x , x j := e j ´ x , and the generalized momenta p := ‚ ´ p , p j := e j ´ p with j = 1 ; : : : ; d ` . In this coordinates the Hamiltonian (4.1) reads H T ( x ; p ; : : : ; p d ) = (1 + –x ) p m (4.11) and the Hamilton equations (4.2) become _ x j = (1 + – x ) p j m _ p j = ` ‹ ;j – p m j = 0 ; : : : ; d ` : (4.12)The integration of the equations for the “orthogonal” components of themomentum immediately leads to p j ( t ) = } j = const : ; j = 1 ; : : : ; d ` : This can be seen as a consequence of the Noether’s theorem applied tothe invariance under translations of the Hamiltonian H T along all thedirections orthogonal to ‚ . Let us introduce the constant of motion } ? := d ` X j =1 } j which quantifies the momentum in the orthogonal plane to the directionof the thermal field. The square of the momentum at any time takes theform p ( t ) = p ( t ) + } ? : (4.13)The value of the parameter } ? strongly determines the behavior of thesolutions of the system (4.12). To see this, one can observe that theHamiltonian H T is time-independent and therefore the Noether’s theoremprovides a further constant of motion, i. e. the (total) energy E := (1 + – % ) } + } ? m which is completely specified by the initial conditions % := x ( t = 0) ; } := p ( t = 0) : The constraint H T ( x ( t ) ; p ( t )) = E ; t R (4.14)can be used to obtain the equation x ( t ) = 1 – mE p ( t ) ` = } + } ? p ( t ) + } ? – + % ! ` – ; (4.15)which provides the time evolution of x once it is known the form of p ( t ) and the initial conditions % and } ; } ; : : : ; } N ` . In addition to this, theconstraint (4.14) also provides useful information for a qualitative studyof the trajectory x ( t ) of the particle. A comparison between (4.11) and(4.14) shows that the sign of E only depends on the quantity –% .More precisely, one has that ˚ E (cid:62) , ˚ % (cid:62) ˇ ‘ : Thus, the critical plane ¨ c R N separates the space into two regionslabelled by the sign of the energy E . The full trajectory x ( t ) of theparticle is fully contained in only one of these two half-spaces accordingto the initial position % along the direction ‚ at the initial time t = 0 . PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 29
Moreover, the trajectory can touch the critical plane only at the cost of adivergence in the value of the total momentum, p ! 1 .The existence of this critical impenetrable plane can be justified on thebasis of the Newton’s law m x j = – F T;j where the force (4.7) is given forcomponents by F T;j = E ` (1 + –x ) } ? m if j = 0(1 + –x ) p } j m if j = 1 ; : : : ; d ` : (4.16)In the derivation of (4.16) from (4.7) we made use of (4.5) along with m T _ x = p and the conservation laws (4.13) and (4.14). The component F T; is proportional to E very close to the critical plane ( –x ‰ ) andforce the particle to stay inside the half-space where the particle was at theinitial time. When } ? = 0 the component F T; changes sign sufficiently farfrom the critical plane and begins to attract the particle towards ¨ c . Thissuggests that the motion of the particle must be bounded in the direction ‚ provided that the momentum has a non-vanishing component orthogonalto ‚ at the initial time. The components F T; ; : : : ; F T;d ` are due to thereaction term R T (4.10). The conservation of the energy implies that j p j / j –x j ` for x ! ` ‘ . Therefore the orthogonal components of F T vanish when the particle approaches the critical plane.4.3. Exceptional solutions.
The Hamilton equations (4.12) (or equiv-alently (4.2)) admit the exceptional family of solutions p ( t ) = 0 and x ( t ) = % for all t R parametrized by all the possible initial positions % R d n ¨ c not belong to the critical plane. In this case the particle isat every moment at rest in a configuration of total zero energy E = 0 .This is not surprising even though the particle is immersed in the thermalfield. In fact the force F T produced by the field vanishes when p = 0 . Ifat the initial time one has } j = 0 for all j = 0 ; : : : ; d ` and % = ` ‘ ,then p = 0 for all t R (as a consequence of energy conservation) andtherefore the particle is not subject to any force. This allows the particleto stay in equilibrium forever at the position % .Another family of exceptional solutions is again described by x ( t ) = % for all t R with the initial positions % ¨ c . Also in this case the particleremains at rest in a configuration of total zero energy E = 0 . However,since the particle lies in the critical plane the total momentum is notforced to be zero. While the component of the momentum orthogonal to ‚ is constant and quantified by } ? the component p ( t ) evolves in timeaccording to the Hamilton equation (4.12) (with solutions (4.24) if } ? = 0 or (4.17) when } ? = 0 ).4.4. The general solution.
Let us derive the general solution of theHamiltonian system (4.12) under the generic assumption } ? = 0 . In thiscase the differential equation for p reads _ p = ` – p + } ? m and is solved by p ( t ) = } ? tan ffi ` – } ? m t ! (4.17)where ffi := arctan „ } } ? « is determined by the initial conditions. Equation(4.17) shows that p ( t ) diverges periodically at the critical times t ( n )c := t c + nT , n Z , where t c := (2 ffi ` ı ) ‘m} ? ; T := 2 ı ‘m} ? and ‘ = – ` .From (4.17) and (4.13) one immediately gets p ( t ) = } ? cos “ ffi ` – } ? m t ” and after some manipulations, equation (4.15) provides x ( t ) = % + A – cos ffi ` – } ? m t ! ` cos( ffi ) (4.18)where we the amplitude A – is given by A – := ‘ mE } ? = ‘ + % cos( ffi ) : Equation (4.18) shows that the motion along the direction ‚ is bounded andmore precisely is confined between the critical plane ¨ c which is reachedperiodically at the critical times t ( n )c and the extremal plane ¨ e := x R d j ‚ ´ x = % + } } ? ! ( ‘ + % ) (4.19)which is reached periodically at the extremal times t ( n )e := t e + nT where t e := 2 ffi ‘m} ? .By inserting the solution (4.18) in the differential equations for the othercomponents of the position one gets _ x j ( t ) = – } j m A – cos ffi ` – } ? m t ! ; j = 1 ; : : : ; d ` : For each j , the corresponding differential equation is integrated by x j ( t ) = % j + – } j m A – t ` A – } j } ? " sin ffi ` – } ? m t ! ` sin(2 ffi ) : (4.20)Evidently the motion in the directions e j is unbounded when } j = 0 dueto the linear term in t which describes a uniform motion with constantvelocity v j;– := –A – } j m .Let us introduce the unit vector (cid:23) := } ` ? P d ` j =1 } j e j . By construction (cid:23) is orthogonal to ‚ and } := } ‚ + } ? (cid:23) describes the initial momentumof the particle at t = 0 . From (4.18) and (4.20) one gets that x ( t ) = % + A – “ f ( t ) ‚ + f ? ( t ) (cid:23) ” (4.21) PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 31 with % := %‚ + P d ` j =1 j e j the initial position and f ( t ) := cos ffi ` – } ? m t ! ` cos( ffi ) f ? ( t ) := – } ? m t ` " sin ffi ` – } ? m t ! ` sin(2 ffi ) : Equation (4.21) shows that the motion of the particle is essentially two-dimensional. In fact the orbit x ( t ) lies entirely in the affine plane spannedby — and (cid:23) and passing through the initial position . Remark 4.1 (2D-case) . In view of (4.21) the general motion of a particlein the thermal field is a two-dimensional motion provided that the initialmomentum is not aligned with the direction of the field. Therefore, onecan always identify the direction ‚ of the field and the direction (cid:23) of theorthogonal component of the initial momentum with the x -axis and the y -axis of R , respectively. This allows us to use the “cozy” notation x ( t ) and y ( t ) for the two projections of the trajectory along the direction ‚ y (cid:23) ,respectively. Let } = ( } x ; } y ) be the components of the initial momentumprojected along the two coordinate direction ‚ and (cid:23) . Let us considerhere the special situation in which the total momentum is completelyorthogonal to ‚ . This means that } = } x = 0 and } ? = j } y j = j } j .This also implies that ffi = arctan(0) = 0 and A – = ‘ + % x with % x = % is the x -component of the initial position % = ( % x ; % y ) . In this case theequations of motion for the position simplify to x ( t ) = % x + ( ‘ + % x ) cos – j } j m t ! ` ;y ( t ) = % y + ( ‘ + % x ) " – j } j m t + 12 sin – j } j m t ! : (4.22)The time evolution of the momentum is described by the equations p x ( t ) = `j } j tan „ – j } j m t « and p y ( t ) = } y . (cid:74) The one-dimensional case.
As discussed at the end of Section 4.4(see Remark 4.1) the general motion of a particle in the thermal fieldis two-dimensional whenever } ? = 0 . Therefore the condition } ? = 0 , } = 0 corresponds to considering the one-dimensional case. In fact, underthese conditions, one immediately gets from (4.12) that p j ( t ) = } j = 0 for all j = 1 ; : : : ; d ` . This in turn implies _ x j = 0 for j = 1 ; : : : ; d ` and so x j ( t ) := % j = const : ; j = 1 ; : : : ; d ` : This means that the only possible motion could take place exclusively inthe direction ‚ , namely it is one-dimensional.Without loss of generality let us assume that % = : : : = % d ` = 0 which means that x j ( t ) = 0 = p j ( t ) for all j = 1 ; : : : ; d ` . Given that,the only interesting degrees of freedom are x and p and we can simplifythe notation identifying x with x and p with p . With this notation the (non-trivial) one-dimensional system of Hamilton equations reads _ x = (1 + – x ) pm _ p = ` – p m : (4.23)The equation for the momentum immediately integrated by p ( t ) = ‘ } } m t + ‘ (4.24)with } = p (0) the initial momentum. Notice that the value of the mo-mentum diverges at the critical time t c := ` ‘ m} .The time evolution of the position can be derived directly from equation(4.15) which, after some algebraic manipulation, provides x ( t ) = ( ‘ + % ) ‘ } m t + ‘ ! ` ‘ (4.25)with % = x (0) the initial position. The long time behavior of the trajectoryis determined by the sign of the coefficient of t in (4.25), namely by thesign of ‘ + % . It follows that lim j t j!1 x ( t ) = ˚1 if ˚ % > ˇ ‘ : The turning time in which the velocity changes sign is determined by _ x ( t ) =0 and a simple computation shows that this time coincides with the criticaltime t c . Moreover, one has that x ( t c ) = ` ‘ independently of the initialvalue % = ` ‘ . In conclusion the critical plane ¨ c separates the spaceinto two regions and the trajectory x ( t ) is fully contained in only one ofthese two half-spaces according to the initial position % . Moreover, thetrajectory can touch the critical plane only once at the critical time t c .These results are in accordance with the qualitative analysis of Section4.2.4.6. The Lagrangian Formalism.
By using the Legendre transformation L T ( x; _ x ) = _ x ´ p ` H T ( x; p ) one can compute the Lagrangian of thesystem: L T ( x; _ x ) := 12 m (1 + – ‚ ´ x ) _ x = m T ( x ) _ x : (4.26)Expressions of the type (4.26) are well studied in the literature under thename of quasi-free PDM Lagrangian (see [MM, BDGP, Mu] and referencestherein). The canonical momentum p ( x; _ x ) := r _ x L T ( x; _ x ) = m T ( x ) _ x is exactly that given by equation (4.4). To compute the Euler-Lagrangeequations of motion we need also r x L T ( x; _ x ) = r x m T ( x ) _ x ` – m T ( x ) m _ x ‚ : A comparison with (4.8) shows that r x L T = _ p = `r x H T PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 33 and this assures that the Euler-Lagrange equation dd t ( r _ x L T ) = r x L T is equivalent to the Hamilton system (4.2). An explicit computation pro-vides dd t ( r _ x L T ) = m T ( x ) x + dd t ( m T ( x )) _ x = m T ( x ) x ` – m T ( x ) m ( ‚ ´ _ x ) _ x and putting all the pieces together one gets m T ( x ) x = – m T ( x ) m ( ‚ ´ _ x ) _ x ` – m T ( x ) m _ x ‚ (4.27)which is equivalent to the Newton’s equation m x = –F T with the force(4.7).In the one-dimensional it is useful to use the change of Lagrangian co-ordinates ( x; _ x ) ( q; _ q ) implemented by x ( q ) := e –q ` – ; _ x ( q; _ q ) := – e –q _ q : The inverse is given by q ( x ) := 1 » log x + 1 – ! and shows that the change of coordinates between x and q is one-to-oneonly when x (cid:62) ` ‘ . However, as seen in Section 4.2, this is exactly therange of values of interest for the problem. With this change of coordinatesthe Lagrangian becomes L T ( q; _ q ) := m – e –q _ q : (4.28)and the associated Euler-Lagrange equation reads q := ` – _ q : This equation immediately provides the time-behavior of the generalizedvelocity _ q ( t ) := _ q _ q – t and a further integration gives q ( t ) := q + 2 – log q – t ! where q ; _ q are the initial conditions. By coming back to the originalvariable one can recover the expression (4.25) for x ( t ) .Appendix A. Spectral theory of the momentum operatorLet p = ` i dd x be the momentum operator with domain H ( R ) L ( R ) .and purely absolutely continuous spectrum ff ( p ) = ff a : c : ( p ) = R . A.1.
Green’s function.
With the help of the Fourier transform F onegets [RS2, IX.29] “ ( p ` “ ) ` ” ( x ) = Z R d y G “ ( x; y ) ( y ) where the Green’s function of p is given by G “ ( x; y ) := 1 p ı F ˜ " k ` “ ( x ` y ) : A straightforward computation involving contours integrals in the complexplane provides G › ˚ i ‹ ( x; y ) = ˚ i ˆ “ ˚ ( x ` y ) ” e i › ( x ` y ) e ` ‹ j x ` y j (A.1)with › R and ‹ > .A.2. Spectral measure.
Let — A be the spectral measure of the self-adjoint operator A associated with the state L ( R ) . The function F A : C n R ! C defined by the scalar product F A ( “ ) := h ; ( A ` “ ) ` i = Z R d — A ( › ) 1 › ` “ is called the Borel-Stieltjes transformation of the finite Borel measures — A . Since Im “ F A ( “ ) ” = Im( “ ) Z R d — A ( › ) 1 j › ` “ j it follows that F A : C + ! C + is is a holomorphic map from the upper halfplane C + into itself. Such functions are called Herglotz or Nevanlinnafunctions (see [DK, Section 1.4] or [AW, Appendix]). A classical resultby de la Vallée-Poussin assures that the limit F A ( › ) := lim ‹ ! + F A ( › +i ‹ ) exists and is finite for Lebesgue-almost every › R . Moreover, theabsolutely continuous part of the spectral measure — A can be recoveredfrom the imaginary part of F A ( ) according to the classical formula [DK,Theorem 1.4.16.] — A j a : c : ( d › ) = f A ( › ) d › with f A ( › ) := lim ‹ ! + ı Im “ F A ( › + i ‹ ) ” : (A.2)In the case A = p is the standard momentum operator one knows thatthe spectral measure is purely absolutely continuous, i. e. — p = — p j a : c : . Bythe help fo the Fourier transform F one obtains that F p ( › + i ‹ ) := h ; ( p ` ( › + i ‹ ) ) ` i = Z R d k j b ( k ) j ( k ` › ) ` i ‹ where b := F ( ) is the Fourier transform of . The application of theformula (A.2) provides f p ( › ) = lim ‹ ! + Z R d k ı ‹ ( k ` › ) + ‹ j b ( k ) j = j b ( › ) j PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 35 where in the last equality one used that ı ‹x + ‹ converges in the distribu-tional sense to ‹ ( x ) when ‹ ! + . In this way one recovers the well-knownresult — p ( d › ) = j b ( › ) j d › : (A.3)A.3. Density of states.
For › R let P › , be the spectral projection of p associated with the energy › according to (2.15). Let ( Q L )( x ) = ffl [ ` L;L ] ( x ) ( x ) be the projection which restricts the functions L ( R ) on the set [ ` L; L ] . The quantity N pL ( › ) := sgn( › )2 L Tr ( P › Q L ) is well defined since P › Q L is trace-class in view of [RS3, Theorem XI 20]. Lemma A.1.
For every › R and L > it hols true that N pL ( › ) = › ı : Proof.
By introducing the local Fourier basis supported in [ ` L; L ] Ln ( x ) := ffl [ ` L;L ] ( x ) p L e i ı nL x ; n Z one obtains that N pL ( › ) = sgn( › )2 L X n Z h Ln ; P › Ln i = 12 L X n Z Z › — p Ln ( d › )= 12 L X n Z Z › d › j b Ln ( › ) j = Z › d › g L ( › ) (A.4)where g L ( › ) : = 12 L X n Z j b Ln ( › ) j = 12 ı X n Z sin( ›L ` ın ) ›L ` ın ! = 12 ı sin( ›L ) ı ! X n Z ›Lı ` n ! ` : (A.5)Observe that the exchange between the sum and the integral in the lastequality of (A.4) is justified by the monotone convergence theorem andthe computation (A.5). The formula P n Z ( a ` n ) ` = ( ı sin( a ) ) [] provides g L ( › ) = ı independently of L . (cid:3) The integrated density of states (IDOS) N p : R ! R is defined by thelimit N p ( › ) := lim L ! + N pL ( › ) : From Lemma A.1 one gets that N p ( › ) = › ı = Z › d › g ( › ) where the last equality emphasizes the fact that N p can be obtained byintegrating the constant density of states (DOS) g ( › ) := ı .The definition of the IDOS can be generalized allowing sequences ofincreasing sets less symmetric than [ ` L; L ] . This essentially boils downon the invariance of p under translations. Lemma A.2.
For every › R and every interval ˜ := [ a; b ] R of finitevolume j ˜ j = b ` a it hols true that N p ˜ ( › ) := sgn( › ) j ˜ j Tr ( P › Q ˜ ) = › ı where Q ˜ is the projection on ˜ .Proof. Set L := b ` a and d := ` a + b . Let U d be the unitary operatordefined by ( U d )( x ) := ( x ` d ) . A simple calculation provides U d Q ˜ U ˜ d = Q [ ` L;L ] ” Q L . From the invariance of the trace under unitary equivalencesand the fact that P › and U d commute one gets N p ´ ( › ) = sgn( › ) j ˜ j Tr “ U d P › Q ˜ U ˜ d ” = sgn( › )2 L Tr ( P › Q L ) = N pL ( › ) : The claim follows from Lemma A.1.
Remark A.1 (The DOS of the Laplacian) . The IDOS of the momentum p and of the Laplacian p are easily related by observing that ffl [0 ;› ] ( x ) = ffl [ `p ›; p › ] ( x ) . From this relation one deduces N p ( › ) = N p “ p › ” ` N p “ ` p › ” = 2 N p “ p › ” = p ›ı ; › (cid:62) : The last equality allows to recover the well-known formula for the DOSof the Laplacian which is given by g (2) ( › ) := ı p › . (cid:74) Appendix B. Technical toolsB.1.
Some principal value integrals.
The central argument of this ap-pendix is the study of the following principal value integral P Z R d u f ( u ) := lim R ! + r ! + Z I R;r d u f ( u ) where I R;r := [ ` R; ` r ] [ [+ R; + r ] for all R > r > . Lemma B.1.
Let G ˚ s ( u ) := e i s ( u ˚ u ) u ; s R : Then the principal value of G ˚ s is given by P Z R d u G ˚ s ( u ) = i (1 ˚ ı sgn( s ) J (2 j s j ) (B.1) where J is the 0-th Bessel function of the first kind.Proof. For the trivial case s = 0 one has that G ˚ ( u ) = u ` and Z I R;r d uu = 0 ; R > r > since the function u ` is odd and the integration domain I R;r is symmet-ric with respect the origin. It follows that the principal value of G ˚ isidentically zero according to (B.1). For s = 0 we one has the symmetry G ˚`j s j ( u ) = ` G ˚j s j ( ` u ) PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 37 which provides P Z R d u G ˚`j s j ( u ) = P Z R d( ` u ) G ˚j s j ( ` u ) = `P Z R d u G ˚j s j ( u ) : (B.2)The relation (B.2) guarantees that we can focus only on the case s > .In this case the computation of the principal value of G ˚ s requires theCauchy’s residue theorem. The function G ˚ s has a holomorphic extensionto every bounded open subset of C n f g and has a singularity in . Let usstart by computing the residue of G ˚ s . From the formula of the generatingfunction for Bessel functions [GR, eq. 8.511 (1)] one obtains the Laurentserie G ` s ( u ) = X n Z J n ( i 2 s ) u n ` = X n Z i n I n (2 s ) u n ` where the J n are the Bessel function of the first kind and the I n ( z ) :=( ` i ) n J n ( i z ) are the modified Bessel functions of the first kind. TheLaurent serie for G + s can be derived from the relation G + s ( u ) = i G `` i s ( i u ) and provides G + s ( u ) = X n Z i n J n (2 s ) u n ` : By definition, the residue of G ˚ s is the coefficient of its Laurent series for n = ` . This provides Res u =0 ( G ` s ) = I (2 s ) ; Res u =0 ( G + s ) = J (2 s ) : From the Cauchy’s residue theorem one gets i 2 ı Res u =0 ( G ˚ s ) = I ` R;r d z G ˚ s ( z ) = I R;r + Z C + R + Z C ` r d z G ˚ s ( z ) where ` R;r is a positively (counterclockwise) oriented simple closed curvecomposed by the union of the domain I R;r on the real line, the semicircle C ` r := f r e i „ j „ [ ` ı; g in the lower half-plane and the semicircle C + R := f R e i „ j „ [0 ; ı ] g in the upper half-plane. An explicit computa-tion provides Z C + R d z G ˚ s ( z ) = i Z + ı d „ e i s ( R ˚ R ` ) cos „ e ` s ( R ˇ R ` ) sin „ : and consequently one has the following estimate ˛˛˛˛˛˛Z C + R d z G ˚ s ( z ) ˛˛˛˛˛˛ (cid:54) Z + ı d „ e ` s ( R ˇ R ` ) sin „ : Since e ` s ( R ˇ R ` ) sin „ ! when R ! + for all „ (0 ; ı ) , it followsfrom the Lebesgue’s dominated convergence theorem that lim R ! + Z C + R d z G ˚ s ( z ) = 0 : (B.3)A similar computation for the integral along C ` r provides Z C ` r d z G ˚ s ( z ) = i Z ` ı d „ e i s ( r ˚ r ` ) cos „ e ` s ( r ˇ r ` ) sin „ : After the change of coordinate „
7! ` „ one gets ˛˛˛˛˛˛Z C ` r d z G ˚ s ( z ) ˛˛˛˛˛˛ (cid:54) Z + ı d „ e ˇ s ( r ` ˚ r ) sin „ : The latest inequality along with the Lebesgue’s dominated convergencetheorem provides lim r ! + Z C ` r d z G + s ( z ) = 0 (B.4)but we didn’t get a similar result for G ` s ( z ) . Putting together (B.3), (B.4)and the formula of the residue theorem one gets P Z R d u G + s ( u ) = i 2 ıJ (2 s ) ; s > (B.5)Finally,from both estimates and the residue the following uniform boundfor r < < R is obtained ˛˛˛˛˛˛Z I R;r d z G + s ( z ) ˛˛˛˛˛˛ (cid:54) ı For the case s < the relation (B.2) immediately provides P Z R d u G + s ( u ) = ` i 2 ıJ (2 j s j ) ; s < : (B.6)Equations (B.5) and (B.6) together, provide the proof of the formula (B.1)for G + s (which automatically includes also the case s = 0 discussed at thebeginning). The case of G ` s can be managed by the following applicationof the Cauchy’s residue theorem I ˚ R;r d z G ` s ( z ) = I R;r + Z C + R ` Z C + r d z G ` s ( z ) where ˚ R;r is a positively (counterclockwise) oriented simple closed curvecomposed by the union of the domain I R;r on the real line the semicircles C + R := f R e i „ j „ [0 ; ı ] g and C + r := f r e i „ j „ [0 ; ı ] g both in theupper half-plane. The zero on the right-hand side is justified by the factthat ˚ R;r does not enclose the singularity of G ` s ( z ) and the negative signon the last integral is due to the fact that the semicircle C + r ha the oppositeorientation with respect to C + R . Equation (B.3) takes care of the integralover C + R . The integral over C + r can be controlled by observing that ˛˛˛˛˛˛Z C + r d z G ˚ s ( z ) ˛˛˛˛˛˛ (cid:54) Z ı d „ e ` s ( r + r ` ) sin „ : and, in turn lim r ! + Z C + r d z G ` s ( z ) = 0 ; s > (B.7)as a consequence of the Lebesgue’s dominated convergence theorem. Puttingtogether (B.3), (B.7) in the Cauchy’s residue formula one gets P Z R d u G ` s ( u ) = 0 ; s > : (B.8) PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 39
Similarly to case (a), an analogous bound can also be obtained. Combiningboth results we have ˛˛˛˛˛˛Z I R;r d z G ˚ s ( z ) ˛˛˛˛˛˛ (cid:54) ı: (B.9)These same results also hold also for s < in view of the relation B.2. (cid:3) Corollary B.1.
The formula P Z R d u e i xu e ` i yu u = i 2 ı sgn( x ) ` sgn( y )2 ! J „ q j xy j « holds true for all ( x; y ) R . Moreover the following uniform bound ˛˛˛˛˛˛Z I R;r d u e i xu e ` i yu u ˛˛˛˛˛˛ (cid:54) ı (B.10) Is valid x; y R .Proof. Let us start by considering the singular situations xy = 0 . The case x = 0 = y corresponds to P Z R d uu = 0 as proved at the beginning of Lemma B.1. The case y = 0 is proportionalto the (well known) Fourier transform of the function u ` and provides P Z R d u e i xu u = `p ı F u ! = i ı sgn( x ) : The case x = 0 can be treated with the change of variables u
7! ` v ` which provides P Z R d u e ` i yu u = `P Z R d v e i yv v = ` i ı sgn( y ) : The non singular situation xy = 0 can be separated in two different cases:(a) xy > , and (b) xy < . Case (a).
Let a := p xy . Then, after the change of variables v := a j y j u ,one has Z I R;r d u e i xu e ` i yu u = Z I R ;r d v e i x j y j a v e ` i sgn( y ) av v = Z I R ;r d v G ` s ( v ) where R := a j y j ` R , r := a j y j ` r and s = a sgn( y ) . Then, Lemma B.1provides P Z R d u e i xu e ` i yu u = P Z R d v G ` s ( v ) = 0 : Case (b).
Let b := q j xy j . Then, after the change of variables v := b j y j u ,one has Z I R;r d u e i xu e ` i yu u = Z I R ;r d v e i x j y j b v e ` i sgn( y ) bv v = Z I R ;r d v G + s ( v ) where R := b j y j ` R , r := b j y j ` r and s = ` b sgn( y ) . Again Lemma B.1provides P Z R d u e i xu e ` i yu u = P Z R d v G + s ( v ) = ` i 2 ı sgn( y ) J (2 q j xy j ) : The observation that ` y ) = sgn( x ) ` sgn( y ) when xy < completesthis case. The uniform bound (B.10) is deduced directly from (B.9) andthe particular case x = y = 0 (cid:3) B.2.
Irregular Kelvin functions.
A reference for the (irregular) Kelvinfunctions is [OMS, Chapter 55]. Here we are interested only on the irreg-ular functions of -th order ker( x ) := ker (cid:23) =0 ( x ) ; kei( x ) := kei (cid:23) =0 ( x ) : We are interested in the behavior of these functions on the half line R + :=[0 ; + ) . Both ker( x ) and kei( x ) have an exponential decay of the type ‰ r ı x e ` x p when x ! + . The function kei( x ) is regular in the originwhere it takes the value kei(0) = ` ı . The function ker( x ) diverges at theorigin as ‰ ` log( x ) . In particular one has that both the Kelvin functionsare in L ( R + ) . The importance of the Kelvin functions for the present workis related to the next result. Lemma B.2.
Let B ( x; y ) the kernel (1.24) . Then, the following formulashold true: Z R d y B ( x; y )1 + y = ` i 2 sgn( x ) kei „ q j x j «Z R d y B ( x; y ) y y = ` i 2 ker „ q j x j « Proof.
After the change of variable s := xy one gets I ( x ) := Z R d y B ( x; y )1 + y = i x Z `1 d s J „ q j s j « x + s : A second change of variable s := ` t provides I ( x ) = i 2 x Z + d t t J (2 t ) x + t = i 2 sgn( x ) Z + d t q j x j t q j x j J q j x j t p j x j ! t p j x j ! + 1= ` i 2 sgn( x ) kei „ q j x j « where the last equality is justified by [OMS, eq. 55:3:6].The second formula can be proved with similar changes of variable and PECTRAL THEORY OF THE THERMAL HAMILTONIAN: 1D CASE 41 one gets I ( x ) : = Z R d y B ( x; y ) y y = i Z `1 d s J „ q j s j « sx + s = ` i 2 Z + d t t J (2 t ) x + t = ` i 2 Z + d t q j x j t q j x j J q j x j t p j x j ! t p j x j ! + 1= ` i 2 ker „ q j x j « where the last equality comes from [OMS, eq. 55:3:5]. (cid:3) B.3.
Bessel equation and Hankel transform.
According to (1.15), theeigenvalue equation associated to the one-dimensional version of the op-erator T is x d d x ( x ) + d d x ( x ) = ` k ( x ) ; k R : (B.11)The change of coordinates x ( u; k ) := u k produces d ffi d u ( u ) + 1 u d ffi d u ( u ) + ffi ( u ) = 0 (B.12)where ffi ( u ) := ( x ( u; k )) . The (B.12) are the Bessel’s equations of order -th and the solutions are the function J ( u ) and Y ( u ) in the standardcase and K ( u ) and I ( u ) in the modified case. The only solution whichhas no singularity is the J . With this information, a physical (a.k.a. nonsingular) solution of (B.11) in the case k > is k> ( x ) := ffl [0 ; + ) ( x ) J „ q j kx j « ; while in the opposite case k < is k< ( x ) := ffl ( `1 ; ( x ) J „ q j kx j « ; where ffl I is the characteristic function of the interval I . In the case k = 0 the general solution of (B.11) is c log( j x j ) + c , then the physical solutioncan be chosen as the constant solution k =0 ( x ) := 1 : These solutions are not in L ( R ) but they meet the (generalized) normal-ization condition Z R d x k ( x ) k ( x ) = ‹ ( k ` k ) in view of [GR, 6.512 (8)]. Let f L ( R ) and define the generalizedeigenfunction expansion f ( x ) : = Z R d k k ( x ) f ( k )= ffl ( `1 ; ( x ) ( H ` f )( x ) + ffl [0 ; + ) ( x ) ( H + f )( x ) where ( H ˚ f )( x ) := Z + d k J „ q j kx j « f ( ˚ k ) are (a variant of) the Hankel transform of f [Erd3, p. 3]. For f L ( R ) it is possible to prove that f L ( R ) . In this way the Hankel transformcan be used to generalize the Fourier theory for the operator (1.16).As a final remark, it is worth observing that the kernel (3.7) of theresolvent ( T ` ¸ ) ` can be obtained by the expansion on the basis k according to Z ¸ ( x; y ) = Z R d k k ( x ) k ( y ) k ` ¸ ; ¸ C n R : The last expression can be integrated by means of the formulas [GR, eq.6.541 (1)]. References [ABG] Amrein, W. O.; Boutet de Monvel, A.; Georgescu, V.: C -Groups, CommutatorMethods and Spectral Theory of N-Body Hamiltonians . Birkhäuser, Basel,1996[ADF] Assel, R.; Dimassi, M.; Fernandez, C.: Some Remarks on the Spectrum of theMagnetic Stark Hamiltonians . Mathematics and Statistics , 101-104 (2014)[AGHG] Albeverio, S.; Gesztesy, F.; Høegh-Krohn, R.; Gesztesy, H.: Solvable modelsin quantum mechanics . AMS, Providence, 1988[AH] Avron, J. E.; Herbst, I. W.:
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E-mail address : [email protected] (V. Lenz) Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile,Santiago, Chile E-mail address ::