Spectral Theory of the Thermal Hamiltonian: 1D Case

In 1964 J. M. Luttinger introduced a model for the quantum thermal transport. In this paper we study the spectral theory of the Hamiltonian operator associated to the Luttinger's model, with a special focus at the one-dimensional case. It is shown that the (so called) thermal Hamiltonian has a one-parameter family of self-adjoint extensions and the spectrum, the time-propagator group and the Green function are explicitly computed. Moreover, the scattering by convolution-type potentials is analyzed. Finally, also the associated classical problem is completely solved, thus providing a comparison between classical and quantum behavior. This article aims to be a first contribution in the construction of a complete theory for the thermal Hamiltonian.

The aim of this introductory section is twice: First of all, we will provide the physical background that motivates the study of the Thermal Hamiltonian; Secondly, we will present the mathematical problems and the main results 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 ( (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. The static (effective) potential V takes care of the interaction of the electron with the atomic structure of the matter and causes only elastic scattering.
The magnetic field enters in the kinetic term K(A) through its vector potential A according to the equation B = rˆA. In Quantum Mechanics the Hamiltonian H(A; V ) is interpreted as a differential operator acting on the Hilbert space L 2 (R d ), where the differential part is provided by the momentum operator p :=` i r, being the Planck constant. The potentials V = V (x) and A = A(x) are functions of the position operator x = (x 1 ; : : : ; x d ), and act as multiplication operators.
The transport phenomena in the matter are analyzed by studing the response of the system to an external perturbation F = F (x) [Lut1,Lut2]. In the stationary regime, that is when all the transient effects due to the switching-on of the perturbation are suppressed, the system reacts by generating a (stationary) drift current. The latter can be computed (at least in the linear response regime, see e. g. [DL]) starting from the full dynamics generated by the perturbed Hamiltonian describes the motion of an electron that moves in the empty space under the influence of the (external) fields generated by A and F . The potential V in (1.3) describes the interaction with the matter which generates elastic scattering of the particle. Once the "free" dynamics generated by H 0 (A; F ) 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 0 (A; F ) and H(A; F; V ).
The best studied case concerns the response of the system to the perturbation induced by a uniform electric field E = (E 1 ; : : : ; E d ). In this case the perturbation is described by the electrostatic potential 2m`e E´x has been extensively studied since the dawn of the Quantum Mechanics. Among the vast literature, we will refer to [AH] for a concise and rigorous presentation of the spectral theory of H Stark (0) and the related scattering theory when the background potential V is taken in consideration. The spectral theory of 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 proposed a model which allows a "mechanical" derivation of the thermal coefficients [Lut2]. Such a model has been then applied and generalized successfully by other authors like in [SS, VMT]. The essential point of the Luttinger's model is that the effect of the thermal gradient in the matter is replaced by a "fictitious" gravitational field, which can be easily described by a perturbation of the Hamiltonian in the spirit of (1.3) and (1.4). More precisely, one assumes that the particle is subject to a force which has the direction of the thermal gradient r# (where # is the distribution of temperature) and which is proportional to the local content of energy divided by c 2 (in view of the mass-energy equivalence). The latter is given by the Hamiltonian (1.1) itself. Such a thermal-gravitational field is given by the potential (1.10) The thermal Hamiltonian H T (A) is the analog of the Stark Hamiltonian when the system is perturbed by the gravitational-thermal field instead of the electric field. For this reason, it seems natural to look for the extension of the results valid for the Stark Hamiltonian (e. g. [AH, DP, ADF]) to the case of the thermal Hamiltonian. This consists of two consecutive problems: (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 yet in the literature, at least to the best of our knowledge. For this reason we devote this work at the analysis of the questions (i) and (ii) above, in the one-dimensional case. The multi-dimensional case will be treated in a future work.
1.2. Position of the spectral problem. In order to formulate the problems sketched above in a rigorous mathematical setting we will make some simplifications. The most relevant concerns the absence of the magnetic field. From here on, unless otherwise indicated, we will fix A = 0. It is worth mentioning that this is not a major restriction as long as one is interested only the one-dimensional regime. Indeed, in one spatial dimension the magnetic field is a pure gauge and can be removed with a unitary transformation 2 .
As usual in mathematics, we will normalize all the physical units: 2m = = c = e = 1. Moreover, we will denote with -:= jr#j > 0 the strength of the thermal gradient and with ‚ := -`1r# 2 S d`1 its direction. With these simplifications the thermal Hamiltonian reads (1.11) The expression (1.11) is formal without the specification of the domain of definition of H T . However, H T is evidently well defined on the space of the compactly supported smooth function C 1 c (R d ) or on the Schwartz space S(R d ). On these dense domains the operator (1.11) acts as (1.12) where´:= P d j=1 @ 2 xj denotes the Laplacian and ‚´r := P d j=1 ‚ j @ xj . We can simplify the last expression with the help of two unitary transformations 1 An equivalently appropriate name for H T (A) could be (magnetic) Luttinger

Hamiltonian.
2 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 meets the canonical commutation rule and so it is unitarily equivalent to the canonical pair x; p.
of the Hilbert space L 2 (R d ). The first one is the rotation where the orthogonal matrix O ‚ meets the condition O ‚ ‚ = (1; 0; : : : ; 0). A short computation shows that where x 1 denotes the first component of the position vector x = (x 1 ; x ? ) 2 R d and x ? := (x 2 ; : : : ; x d ) 2 R d`1 is its orthogonal complement. Evidently, the rotation O ‚ has the role of aligning the thermal-gravitational field along the x 1 -axis 3 . The second transformation is the translation and a direct calculation provides The operator on the brackets agrees with the formal anti-commutator when evaluated on sufficiently regular functions like 2 S(R d ). With a slight abuse of notation, we will often use the representation (1.16) for the operator T , instead of the more precise definition (1.15).
The unitary equivalence between H T and T implies that the spectral theory of the thermal Hamiltonian H T can be completely recovered from the spectral theory of the operator T . For this reason, one is led to the problem of determining if the operator T , initially defined by (1.15) on the dense domain S(R d ), admits self-adjoint extensions and, in that case, to compute the related spectra.
For technical reasons, it results easier to face the equivalent problems in the Fourier space. Let F : L 2 (R d ) ! L 2 (R d ) be the Fourier transform defined (just to fix the convention) by on the dense subspace 2 L 1 (R d ) \ L 2 (R d ). Let˝:= F T F˜be the Fourier transformed version of the operator (1.15). A direct computation shows that for 2 S(R d ) (1.17) 3 Clearly, in dimension d = 1 the thermal-gravitational field is trivially aligned with the only spatiual axis and therefore R ‚ reduces to the identity. x 2 p 1 + p 1 x 2 " (1.18) on sufficiently regular functions 4 .
The representation (1.18) is quite intriguing if one compares the operator with the typical generator of C 0 -groups associated to C 1 -flows [ABG,Chapter 4]. At first glance, it would seem that the general theory of C 0groups applies to˝. However, a closer inspection to the R-flow associated to˝shows that this is not the case in general (see Section 2.4 for more details). Therefore, the question of the self-adjointness of˝needs to be investigated with other tools.
The first fundamental question is whether the operator˝, initially defined by (1.17) on S(R d ), admits self-adjoint extensions or not. This is fortunately true and easily demonstrable. Indeed, it is straightforward to check that˝, as defined by (1.17) This observation allows us to identify˝with its closure (still denoted with the same symbol) defined on the domain obtained by the closure of S(R d ) with respect to the graph-norm jj jj 2 := jj jj 2 + jj˝ jj 2 : The existence of self-adjoint extensions of˝is justified by the von Neumann's criterion [RS2,Theorem X.3]. Letˇbe the anti-unitary operator on L 2 (R d ) defined by (ˇ )(x) = (`x). The domains C 1 c (R d ) or S(R d ) are left unchanged byˇand a direct check shows thatˇ˝=˝ˇon these domains. This is sufficient to claim that: Proposition 1.1. The closed symmetric operator˝with domain D 0 admits self-adjoint extensions.
Proposition 1.1 allows a precise definition of the family of thermal Hamiltonians.
Definition 1.1 (Thermal Hamiltonian). Let˝" be a given self-adjoint extension of the operator˝with domain D(˝") ff D 0 . Let F (-; ‚) := F S -R ‚ be the unitary operator given by the product of the Fourier transform F , the translation Sdefined by (1.14) and the rotation R ‚ defined by (1.13). Then, the associated thermal Hamiltonian is the self-adjoint operator 4 Formula (1.18) can be formally derived from (1.16) by using the well known transformations of the canonical operators F p j F˜= x j and F x j F˜=`p j for all j = 1; : : : ; d.
Definition (1.1) reduces the question of the spectral theory of the thermal Hamiltonian to the analysis of the self-adjoint realizations of the operator . This is usually done by studying the deficiency subspaces The existence of the conjugationˇfor˝implies the equality of the deficiency indices n˚:= dim(K˚) [RS2,Theorem X.3] which in turn ensures the existence of self-adjoint extensions. In order to build the spaces Kå nd to compute n˚, one needs to solve the equations˝˜ =˚i which, in view of (1.17), is equivalent of finding the weak solutions [RS1,Section V.4] to the differential equations where S 0 (R d ) is the space of tempered distributions 5 . This problem will be solved for the one-dimensional case in Section 2.2.
1.3. Overview on the one-dimensional case. In Section 2.2, it is shown that the differential operator (1.17), in one spatial dimension (d = 1), admits a family of self-adjoint realizations parametrized by the angle " 2 S 1 (see Theorem 2.1). As a consequence, the domains C 1 c (R) or S(R) can not be cores for˝(in contrast to [ABG,Proposition 4.2.3]). However, it turns out that all these self-adjoint realizations˝" are equivalent in the sense that there are unitary operators L " such that˝" = L "˝0 L" . This fact immediately implies the independence of the spectrum by the particular self-adjoint realization. In particular, it results that the spectrum of every extension˝" is purely absolutely continuous and coincides with the real axis, i. e. ff "˝"" = ff a:c: We are now in position to state our first main result. Let us just recall that in dimension d = 1 the only relevant parameter in the definition of the thermal Hamiltonian is -> 0 since no rotation R ‚ is required (cf. Note 3). Then, according to Definition 1.1, we can define the family of onedimensional thermal Hamiltonians as In view of the unitary equivalence of the various realizations˝" it follows that all the one-dimensional thermal Hamiltonians with a given coupling constant -> 0 are unitarily equivalent. For this reason we can focus on the special realization with " = 0.
is the space of distributions.

The operator H T is self-adjoint on its domain D(H T ) := (F S -)˜[D(˝0)]
and has purely absolutely continuous spectrum given by = ff a:c: The proof of Theorem (1.1) is a corollary Theorem 2.1 and of Definition 1.1. For the determionation of the spectrum one uses the invariance of the spectrum under unitary equivalences and the spectral mapping theorem.
The operator H T , defined by (1.21), will be called the standard realization of the one-dimensional thermal Hamiltonian (with coupling constant -> 0). Theorem 1.1 expresses the fact that in dimension d = 1 there is a "unique" thermal Hamiltonian, at least in the sense that all relevant physical quantities, which by definition must be invariant under unitary equivalences, 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 be the natural domain of the position operator. Let x is the sign function and J 0 is the 0-th Bessel function of the first kind 6 [GR]. Then, it holds true that Moreover D(H T ) contains a dense core for H T given by where the (normalized) functions » 0 and » 1 are explicitly given by and kei(x) and ker(x) are the irregular Kelvin functions of 0-th order (see Section B.2 and references therein). It is worth noting that the function » 0 introduces a jump discontinuity around the critical point x c =`-`1. The Hamiltonian H T , acting on » 0 , produces the wavefunction » 1 which shows a logarithmic divergence around x c . A similar singular behavior around the critical point x c is detectable also in the classical dynamics (cf. Section 4).
The unitary propagator U T (t) := e`i tHT acts as an integral operator with kernel given by (cf. Proposition 3.2) for all fi 2 Rnf0g. Finally, the knowledge of the unitary propagator allows to compute the resolvent R " (H T ) := (H T`" 1)`1 ; " 2 C n R by means of the Laplace transformation (see Section 3.3). It turns out that also R " (H T ) is an integral operator (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 scattering 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 (physical) perturbations given by gravitational-matter potentials of type (1.10) presents several technical complications and will be treated in a separated work. By convolution perturbation we mean an integral operator W g acting on 2 L 2 (R) as where the kernel is chosen as g 2 L 1 (R where the limit is meant in the strong sense. In Section 3.4 the following result will be proven. Theorem 1.2 (Scattering theory for convolution perturbations in d = 1). Let g 2 L 1 (R) and W g the associated convolution perturbation defined by (1.30). Then: Letĝ be the Fourier transform of g and assume that there are constants " > 0 and C > 0 such that jĝ(x)j Cx 3 2 for all jxj < ". 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 Structure of the paper. Section 2 is devoted to the study of the spectral theory of the auxiliary operator˝in the one-dimensional case. The spectral theory of the one-dimensional thermal Hamiltonian H T is discussed in Section 3 along with a subsection on the scattering theory by a convolutiontype potentials. The classical dynamics of the thermal Hamiltonian (in any dimension) is studied in Section 4. Finally Appendix A and Appendix B contain some review material and some technical computations needed to make the present work self contained.
2. The spectral theory of the operatorW e already know from the general discussion in Section 1.2 that the operator˝defined by (1.17) (or formally by (1.18)) is symmetric and in turn closable. Moreover, Proposition 1.1 ensures that˝admits selfadjoint extensions. While, on the one hand, these results are valid in every dimension, 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 this family of operators.
2.1. Equivalence with the momentum operator. In dimensional d = 1 the operator˝is initially defined by The last equality allows us to identify "`xpx on sufficiently regular functions.
The operator (2.1) is symmetric, hence closable, and its closure (still denoted with˝) has domain D 0 given by (1.19). In order to give a more precise characterization of D 0 we will benefit from the transformation Lemma 2.1. I is a unitary involution.
Proof. A direct computation shows that ds j (s)j 2 = jj jj 2 : Then I, initially defined on every "good enough" dense domain, extends to an isometry on the whole L 2 (R). From its very definition, it follows that I 2 = . This shows that I is an involution, and in particular it is invertible. As a consequence I is also unitary.
Instead of˝let us consider the transformed operator be the space of the Sobolev functions on R vanishing in x = 0. Let us point out that the latter requirement makes sense since Sobolev functions on R are uniquely identifiable with continuous functions [Bre,Theorem 8.2]. In view of this remark we will tacitly identify Sobolev functions with their continuous representative so that the following inclusions H 1 0 (R)  H 1 (R)  C(R) hold.
Proposition 2.1. The closed symmetric operator } defined by (2.2) coincides with the momentum operator on H 1 0 (R), namely Proof. The unitarity of I implies that the graph norms of } and˝are related by jjffijj } = jjIffijj˝for all ffi 2 D(}). This gives acts as the momentum operator on I[S(R)]. This implies that the domain of the closed operator } is given by the closure of I[S(R)] with respect the Sobolev norm jjffijj 2 H 1 := jjffijj 2 + jjffi 0 jj 2 . Let C 1 c (R n f0g) be the set of smooth functions having compact support separated from the origin. It holds true that For the second inclusion let us take ffi 2 I(S(R)) so that ffi(x) = x`1 (x`1) for some 2 S(R). Clearly, ffi is smooth in R n f0g and extends to a smooth function on R such that ffi (n) (0) = 0 for all n 2 N. In particular To conclude the proof it is enough to show that the closure of the space C 1 c (R n f0g) with respect to the Sboolev norm jj jj H 1 is (identifiable with) H 1 0 (R). Let R + := (0; +1) and R`:= (`1; 0) and observe that where the notation for W 1;2 0 (˙) was borrowed from [Bre,Section 8.3]. The last equality in (2.4) is a consequence of the fact that every element of W jj jj H 1 = H 1 0 (R). The first consequence of Proposition 2.1 is a precise description of the domain of the closed operator˝, i. e.
Unlike the functions in H 1 0 (R), the elements of the domain D 0 are generally not continuous and can show singularities in x = 0. An example is the function ffi(x) := (1 + x 2 )`1 3 e`1 x 2 which is evidently an element of H 1 0 (R).
On the other hand elements of D 0 have a decay at infinity which is at least of order 1.
Proof. The claim follows from the characterization (2.5) which provides In the last equality, the continuity of ffi 2 H 1 0 (R) is used.

Classification of self-adjoint extensions.
We are now in position to 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 related with the study of singular delta interactions for one-dimensional Dirac operators [GS, BD, CMP] (see also [AGHG, Appendix J]).
is dense (with respect to the graph norm) in the domain of }, a standard argument shows that the adjoint operator }˜acts as the weak derivative on its domain D(}˜) := H 1 (R`)˘H 1 (R + ) (see e. g. [RS1, Section VII.2]). The eigenvalue equations }˜ffi˚=˚i ffi˚for the deficiency subspaces correspond to the differential equations ffi 0 =ˇffi˚which admit in D(}˜) the unique (normalized) weak solutions 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 parametrized by the unitary maps from The later are identified by the angle " 2 S 1 ' [0; 2ı) according to U " ffi + := e`i " `. From the general theory [RS2,Theorem X.2] one has that the domain of the self-adjoint extension } " is made by functions of the type ffi + c 0 (ffi + + e`i " ffi`) = ffi + c" " with ffi 2 H 1 0 (R) and c; c 0 2 C suitable complex coefficients. The action of } " on the elements of its domain is given by 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 1 0 (R)  H 1 (R). This extension corresponds to } 0 in view of the fact that " 0 2 H 1 (R).
Although the symmetric operator } admits several self-adjoint realizations, all these realizations are in a sense equivalent. To express this fact in a precise way we need to introduce the family of unitary operators L " defined by Proposition 2.4. The unitary operators L " intertwine all the self-adjoint realizations of the operator }. More precisely one has that } " = L " pL" ; " 2 S 1 where p = } 0 is the standard momentum operator. As a consequence one has that ff(} " ) = ff a:c: (} " ) = R ; 8 " 2 S 1 : . By exploiting the decomposition used above one has Hence, a comparison with (2.6) shows that L " pL" = } " on the domain D(} " ).
Remark 2.1. The unitary equivalence of the different realizations } " can be understood in terms of the celebrated Stone-von Neumann theorem (see e. g. [Ros]). Indeed, a direct computation shows that ' 2 C 1 c (R n f0g) and C 1 c (R n f0g) = C 1 c (R`)˘C 1 c (R + ) is dense in L 2 (R`)˘L 2 (R + ) = L 2 (R). Therefore, by continuous extension, one can unambiguously define the commutation relation [x; } " ] = i 1 which means that the pair (x; } " ) satisfies the canonical commutation relation. As a result, the Stone-von Neumann theorem ensures that } " is unitarily equivalent to the standard momentum p.
Proposition 2.3 provides the key result for the complete description of the self-adjoint extensions of˝.
Theorem 2.1 (Self-adjoint extensions: one-dimensional case). The selfadjoint extensions of the closed symmetric operator˝initially defined by (2.1) are in one-to-one correspondence with the angles " 2 S 1 . The selfadjoint extension˝" has domain and acts has˝" ( + c" " ) :=˝ + c" "+ı : All the self-adjoint realizations are unitarily equivalent, i. e.˝" = L "˝0 L" for all " 2 S 1 . Finally one has that ff(˝") = ff a:c: (˝") = R ; 8 " 2 S 1 : Proof. This is a direct consequence of the unitary equivalence established in Proposition 2.1 which allows to define the self-adjoint realizations ofb y˝" := I} " I. Therefore, the statement is nothing more than a rephrasing of Proposition 2.3 and Proposition 2.4. The formula˝" = L "˝0 L" is justified by the commutation relation L " I = IL " .
In view of the unitary equivalence among all the self-adjoint realizations of˝we can focus the attention only in a "preferred" realization.
Definition 2.1 (Standard realization). We will call˝0 =˝" =0 the standard self-adjoint realization of the operator initially defined by (2.1).
2.3. Boundary triplets. The problem of the determination of the selfadjoint realizations of } or˝can be investigated also with the theory of the boundary triplets [Sch,Chapter 14]. Let us start with the operator } and its adjoint }˜. According to [Sch,Definition 14.2], a boundary triplet for }˜is a triplet (H;`0;`1) made by an Hilbert space H and linear maps 0 ;`1 from D(}˜) to H that satisfy the abstract Green's identity and the mapping D(}˜) 3 ' 7 ! (`0';`1') 2 HˆH is surjective. Since the operator }˜acts as the weak derivative on its domain an integration by parts provides where '(0˚) := lim x!0˚' (x) and similarly for (0˚). A comparison with the abstract Green's identity shows that the triplet (H;`0;`1) can be fixed in the following way: H := C; The surjectivity condition is obviously satisfied. Observe that Ker(`0) \ Ker(`1) = H 1 0 (R) = D(}). The self-adjoint extensions of } are in oneto-one correspondence with the self-adjoint operators on H = C [Sch,Theorem 14.10]. More precisely, the self-adjoint extensions of } can be parametrized by a real number ‚ 2 R [ f1g which defines a restriction (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 ‚ = 1 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 direct computation shows that be the unitary propagator defined by the self-adjoint operator˝" on L 2 (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 holds true that Proof. We can use the unitary equivalence˝" = IL " pL" I proved in Section 2.2. This implies that V " (t) = IL " e`i tp L" I along with the well-known fact ( e`i tp )(x) = (x`t). The proof of the claim follows by a direct computation.
For each t 2 R let us consider the map f t : with the convention that˚0`1 " 1. The family of these maps defines an R-flow in the sense that the following relations hold: The flow f t allows to rewrite the action of V " (t) in the form When " = ı the exponential prefactor is 1 and equation (2.11) agrees with the definition of the C 0 -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 1 and the generator of the flow has an unbounded first derivative. Therefore the flow f t doesn't meet the conditions of [ [ABG,Proposition 4.2.3] doesn't apply to the operator "`1 2 (pF (x) + F (x)p) which indeed is not essentially self-adjoint on C 1 c (R).

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 " 2 C n R the resolvent of˝" at " is defined as R " (˝") := (˝"`"1)`1 = L " I (p`"1)`1 IL" : ( 2.12) The next results shows that R " (˝") is an integral operator.
with kernel given by hereˆis the Heaviside function. 8 Proof. The integral kernel R 0 " of the resolvent of˝0 can be obtained from the Green's function G 0 " of the standard momentum operator (see Appendix A.1). A direct computation provides The explicit expression of G 0 " given in (A.1) and a change of variable in the integral provide Since L " is a multiplication operator, the relation between the kernels for " = 0 and " 6 = 0 is simply given by This concludes the proof.
It is worth noting that that along the diagonal one has R " " (x; x) = sgn(Im(")) i 2x 2 for all " 2 C n R and " 2 S 1 . 8 The Heaviside function is defined byˆ(x) := 2.6. Spectral measure and density of states. Let -" be the spectral measure of the operator˝" associated with the normalized state 2 L 2 (R). We know from Theorem 2.1 that˝" as a purely absolutely continuous spectrum which coincides with R. This implies that the spectral measure -" is purely absolutely continuous. More precisely one has that -" ( d›) := f " (›) d› with f " 2 L 1 (R) a non-negative function. The next result provides a description of f " .
Proposition 2.6. Let -" be the spectral measure of the operator˝" associated with the (normalized) state 2 L 2 (R). Then -" is absolutely continuous with respect to the Lebesgue measure d› in R and In order to define the integrated density of states (IDOS) of˝" let us start by introducing the spectral projections P " › of˝" defined by Proof. By combining the spectral theorem with the unitary equivalence between˝" and p one gets that P " › = L " IP › IL" where This means that P " › Q˜= L " (IP › IQ˜)L" = L " I(P › IQ˜I)IL " . Thus, to prove that P " › Q˜is trace-class it is sufficient to prove that P › (IQ˜I) is trace-class. Let b > a > 0 or a < b < 0. A direct computation shows that namely IQ˜I = Q~with~:= [b`1; a`1]. This implies that P › (IQ˜I) = P › Q~is trace-class in view of [RS3,Theorem XI 20]. Moreover, one has that where N p (›) is the local density of states for the operator p in the regioñ . The proof follows by using Lemma A.2.
The quantity N " (›) measures the volumetric density of states up to the energy › localized in the region˜. States with negative energy are counted as "negative" states. Lemma 2.2 shows that this number is not homogeneous in space. One can ask how this number changes for fixed volume in function of the spatial localization. Let ' > 0 and set˜x ;' : : Since the density decreases as x`2 in function of the spatial localization and as '`1 in function of the volume one immediately concludes that the majority of states are concentrated around x = 0 with a divergent density.
Ultimately, the spatial inhomogeneity of N " is a consequence of the fact that˝" breaks the invariance under spatial translations. To define a density of states on the thermodynamic limit a precise prescription on how to carry out the spatial average is necessary. Let us define the principal value integral density of states (pv-IDOS) as  Equation (3.2) says that B n converges pointwise to the integral in the right-hand side. Since B n converges to B in the L 2 -topology it follows there exists a subsequence B nk which converges pointwise (almost everywhere) to B [Bre, Theorem 4.9 (a)]. Then the unicity of the limit assures that B coincides with the right-hand side of (3.2). The last part of the proof follows from the explicit computation which provides equation (1.23).
Remark 3.1. Lemma 3.1 states that Bcan be expressed as an integral operator only on the dense domain 2 L 2 (R) \ L 1 (R). For function in 2 L 2 (R) n L 1 (R) in principle, we do not have the right to write Busing the integral kernel. However, in the following, we will tacitly use the following convention This identification must be understood as follows: (i) The product R := ffl [`R;+R] is in L 2 (R) \ L 1 (R) and so B -R can be computed (pointwise) through the integral formula; (ii) R ! , and in turn B -R ! B -, in the L 2 -topology; (iii) Then, the identification above makes sense almost everywhere on subsequences [Bre,Theorem 4 Proof. Let " 0 (x) = e`j xj and " ı (x) = i sgn(x) e`j xj . The inverse Fourier transforms of these functions are given by Since F˜" 0 2 L 2 (R) \ L 1 (R), the transformed function BF˜" 0 can be computed via the integral kernel of B. Then Lemma B.2 provides Since F˜" 1 2 L 2 (R) n L 1 (R), the transformed function BF˜" 1 as to be computed according to the prescription of Remark 3.1. In this case one has B(x; y) y 1 + y 2 : However, as shown in the proof of Lemma B.2, the integrant is absolutely integrable for every values of x. This allows to forget the limit and one gets Finally a translation by S-and a multiplication by`i provide the formulas (1.26).
Remark 3.2 (Other self-joined extensions). As for the operator˝discussed in Section 2, also the thermal Hamiltonian H T admits a family of unitarily equivalent self-adjoint extension parametrized by " 2 S 1 , and defined by Proposition 3.2. On the dense domain L 2 (R) \ L 1 (R) the unitary propagator U T (t) with (t 6 = 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 2 R n f0g, on 2 L 2 (R) \ L 1 (R). Finally, the well known relations I 0 (˚i x) = J 0 (ˇx) = J 0 (x) valid for where the kernel U fi is defined by (1.28). Equation (3.4) is valid also in the singular cases xs = 0. For instance, for x = 0 on gets after the usual change of coordinates 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 and since jU fi (x; s)j jfi j`1 for all (x; s) 2 R 2 one has that the function s 7 ! A R fi (x; s) (s) is definitively dominated by the integrable function s 7 ! jfi j`1 (s) (provided fi 6 = 0). This fact allows to use the Lebesgue's dominated convergence theorem in (3.3), providing in this way (3.5) Formula (1.27) is obtained by observing that U T (t) = S-A -t S -.

Resolvent and Green function. The resolvent of H T can be computed as the Laplace transform of the unitary propagator U T (t) according
to the well known formula [Kat, eq. (1.28), p. 484]. For every " 2 C n R let R " (H T ) := (H T`" 1)`1 be the resolvent of H T . Then, it holds true that where the integral is interpreted as a strong Riemann integral lim ff!+1 R ff 0 . The resolvent for Im(") < 0 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 ). Setting¸:= j¸j e i ffi , 0 < ffi < ı, the last integral can be integrated case by case using the Macdonald's and Nicholson's formulas [Erd1,Section 7.7.6] 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 some tedious calculations, one gets (3.8) It is also possible to check directly that the kernel Z¸(x; s) inverts in a distributional sense the operator T`¸1.
3.4. Scattering by a convolution potential. Let g 2 L 1 (R) and consider the associated convolution potential W g defined by (1.30). Since W g is a bounded operator of norm kW g k = kgk 1 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]. The it makes sense to consider the scattering theory of the pair (H T ; H T ;g ).
In view of the unitary equivalence p = 1 -IF S -H T S-F˜I between the momentum operator and H T we can equivalently study the scattering theory of the pair (p; p g ). where p g := p + M g is the perturbation of the momentum given by the potential Lemma 3.2. The potential M g is the multiplication operator defined by whereĝ 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 a convolution gives a multiplication operator whereĝ denotes the Fourier transform of g. The proof is completed by observing recalling the definition of the involution I.
We are now in position to provide the proof of Theorem 1.2.
Proof (of Theorem 1.2). Since g 2 L 1 (R) thenĝ 2 C 0 (R) (continuous functions vanishing at infinity) in view of the Riemann-Lebesgue Lemma [RS2,Theorem IX.7]. This implies that the function x 7 !ĝ " 1 x " belongs to C(R) \ L 1 (R). As a result the multiplicative potential M g is bounded with norm kM g k = p 2ı -kĝk 1 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 existence and 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 This requires thatĝ ! 0 fast enough when s ! 0˚. This is guaranteed by the (not optimal) conditions required in the theorem statement. Invoking once again the unitary equivalence between p and H T one obtain the existence and the completeness for the waves operators associated to the pair (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 a complex number given by Since a complex number is unchanged by unitary equivalences it follows that S g is also the S-matrix for the pair (H T ; H T ;g ).

The classical dynamics
In this last section we will study the classical dynamics induced by a thermal gradient. The classic analogue of the Luttinger's model is provided by the the Hamiltonian function ‚ j x j along the direction ‚ 2 S d`1 . The coupling constant has the dimension of the inverse of a distance, namely -= '`1 with ' > 0 the typical length of the thermal field. Therefore, the limit -! 0 describes the situation in which the typical length of the field is much larger than the typical length of the system (e. g. the size of the particle). The potential T ‚ is an example of what is known as a generalized potential, namely a potential which depends not only on the position but also on the the velocity. (4.2) The first equation can be inverted out of the critical planë and provides One can restore the usual relation p = m T _ x between momentum and velocity by introducing the position-dependent mass (PDM) It is interesting to notice that the Hamiltonian (4.1) can be rewritten as namely as the Hamiltonian of a free particle with a PDM. The second equation of (4.2) can be rewritten as A straightforward computation allows to derive the Newton's laws from (4.2): After introducing (4.4) in the las expression one obtains the Newton's equation where the thermal force (which has the dimensions of a force times a distance) is given by A way of interpreting this Newton's Equation is to say that the motion of the PDM-particle is influenced by the effect of its own internally selfproduced force field generated by the spatial dependence of the mass. The relation between the force F T and the potential T ‚ can be deduced by observing that`- in view of the (4.6), (4.2) and (4.4), respectively. After some manipulation and the use of equation (4.4) one gets which shows that the thermal force is not simply given by`r x T ‚ , as for ordinary conservative forces, but it includes an extra reacting term which is generally not aligned with the direction ‚ of the field.
4.2. Qualitative analysis. Let us start with the analysis of the qualitative behavior of the solution of the Hamiltonian system (4.1). To simplify the study let us fix convenient notations. The unit vector ‚ can be completed to an orthonormal basis by adding other d`1 orthonormal vectors e 1 ; : : : ; e d`1 . This allows to fix the generalized coordinates x 0 := ‚´x, x j := e j´x , and the generalized momenta p 0 := ‚´p, p j := e j´p with j = 1; : : : ; d`1. In this coordinates the Hamiltonian (4.1) reads H T (x 0 ; p 1 ; : : : ; p d ) = (1 + -x 0 ) p 2 2m (4.11) and the Hamilton equations (4.2) become j = 0; : : : ; d`1 : (4.12) The integration of the equations for the "orthogonal" components of the momentum immediately leads to p j (t) = } j = const: ; j = 1; : : : ; d`1 : This can be seen as a consequence of the Noether's theorem applied to the invariance under translations of the Hamiltonian H T along all the directions orthogonal to ‚. Let us introduce the constant of motion which quantifies the momentum in the orthogonal plane to the direction of the thermal field. The square of the momentum at any time takes the form p 2 (t) = p 2 0 (t) + } 2 ? : (4.13) The value of the parameter } ? strongly determines the behavior of the solutions of the system (4.12). To see this, one can observe that the Hamiltonian H T is time-independent and therefore the Noether's theorem provides a further constant of motion, i. e. the (total) energy 2m which is completely specified by the initial conditions % 0 := x 0 (t = 0) ; } 0 := p 0 (t = 0) : The constraint H T (x(t); p(t)) = E 0 ; 8 t 2 R (4.14) can be used to obtain the equation which provides the time evolution of x 0 once it is known the form of p 2 0 (t) and the initial conditions % 0 and } 0 ; } 1 ; : : : ; } N`1 . In addition to this, the constraint (4.14) also provides useful information for a qualitative study of the trajectory x(t) of the particle. A comparison between (4.11) and (4.14) shows that the sign of E 0 only depends on the quantity 1 + -% 0 . More precisely, one has that Thus, the critical plane¨c  R N separates the space into two regions labelled by the sign of the energy E 0 . The full trajectory x(t) of the particle is fully contained in only one of these two half-spaces according to the initial position % 0 along the direction ‚ at the initial time t = 0.
Moreover, the trajectory can touch the critical plane only at the cost of a divergence in the value of the total momentum, p 2 ! 1.
The existence of this critical impenetrable plane can be justified on the basis of the Newton's law m x j = -F T ;j where the force (4.7) is given for components by (1 + -x 0 ) p 0 } j m if j = 1; : : : ; d`1 : (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 ;0 is proportional to E 0 very close to the critical plane (1+-x 0 ‰ 0) and force the particle to stay inside the half-space where the particle was at the initial time. When } 2 ? 6 = 0 the component F T ;0 changes sign sufficiently far from the critical plane and begins to attract the particle towards¨c. This suggests that the motion of the particle must be bounded in the direction ‚ provided that the momentum has a non-vanishing component orthogonal to ‚ at the initial time. The components F T ;1 ; : : : ; F T ;d`1 are due to the reaction term R T (4.10). The conservation of the energy implies that jp 0 j / j1 + -x 0 j`1 2 for x 0 !`'. 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 equivalently (4.2)) admit the exceptional family of solutions p(t) = 0 and x(t) = % for all t 2 R parametrized by all the possible initial positions % 2 R d n¨c not belong to the critical plane. In this case the particle is at every moment at rest in a configuration of total zero energy E 0 = 0. This is not surprising even though the particle is immersed in the thermal field. In fact the force F T produced by the field vanishes when p = 0. If at the initial time one has } j = 0 for all j = 0; : : : ; d`1 and % 0 6 =`', then p 2 = 0 for all t 2 R (as a consequence of energy conservation) and therefore the particle is not subject to any force. This allows the particle to stay in equilibrium forever at the position %.
Another family of exceptional solutions is again described by x(t) = % for all t 2 R with the initial positions % 2¨c. Also in this case the particle remains at rest in a configuration of total zero energy E 0 = 0. However, since the particle lies in the critical plane the total momentum is not forced to be zero. While the component of the momentum orthogonal to ‚ is constant and quantified by } ? the component p 0 (t) evolves in time according to the Hamilton equation (4.12) (with solutions (4.24) if } ? = 0 or (4.17) when } ? 6 = 0).

The general solution.
Let us derive the general solution of the Hamiltonian system (4.12) under the generic assumption } ? 6 = 0. In this case the differential equation for p 0 reads and is solved by where ffi := arctan " }0 }?
« is determined by the initial conditions. Equation (4.17) shows that p 0 (t) diverges periodically at the critical times t and ' = -`1.
From (4.17) and (4.13) one immediately gets and after some manipulations, equation (4.15) provides where we the amplitude Ais given by Equation (4.18) shows that the motion along the direction ‚ is bounded and more precisely is confined between the critical plane¨c which is reached periodically at the critical times t which is reached periodically at the extremal times t (n) e := t e + nT where t e := 2ffi 'm }? . By inserting the solution (4.18) in the differential equations for the other components of the position one gets ; j = 1; : : : ; d`1 : For each j, the corresponding differential equation is integrated by (4.20) Evidently the motion in the directions e j is unbounded when } j 6 = 0 due to the linear term in t which describes a uniform motion with constant velocity v j;-:= -A -}j 2m . Let us introduce the unit vector := }`1 ? P d`1 j=1 } j e j . By construction is orthogonal to ‚ and } := } 0 ‚ + } ? describes the initial momentum of the particle at t = 0. From (4.18) and (4.20) one gets that (4.21) with % := %‚ + P d`1 j=1  j e j the initial position and Equation (4.21) shows that the motion of the particle is essentially twodimensional. In fact the orbit x(t) lies entirely in the affine plane spanned by -and and passing through the initial position .
Remark 4.1 (2D-case). In view of (4.21) the general motion of a particle in the thermal field is a two-dimensional motion provided that the initial momentum is not aligned with the direction of the field. Therefore, one can always identify the direction ‚ of the field and the direction of the orthogonal component of the initial momentum with the x-axis and the y-axis of R 2 , 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 , respectively. Let } = (} x ; } y ) be the components of the initial momentum projected along the two coordinate direction ‚ and . Let us consider here the special situation in which the total momentum is completely orthogonal to ‚. This means that } 0 = } x = 0 and } ? = j} y j = j}j. This also implies that ffi = arctan(0) = 0 and A -= ' + % x with % x = % 0 is the x-component of the initial position % = (% x ; % y ). In this case the equations of motion for the position simplify to The time evolution of the momentum is described by the equations p x (t) = j}j tan " -j}j 2m t « and p y (t) = } y .
4.5. 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 field is two-dimensional whenever } ? 6 = 0. Therefore the condition } ? = 0, } 0 6 = 0 corresponds to considering the one-dimensional case. In fact, under these conditions, one immediately gets from (4.12) that p j (t) = } j = 0 for all j = 1; : : : ; d`1. This in turn implies _ x j = 0 for j = 1; : : : ; d`1 and so x j (t) := % j = const: ; j = 1; : : : ; d`1 : This means that the only possible motion could take place exclusively in the direction ‚, namely it is one-dimensional.
Without loss of generality let us assume that % 1 = : : : = % d`1 = 0 which means that x j (t) = 0 = p j (t) for all j = 1; : : : ; d`1. Given that, the only interesting degrees of freedom are x 0 and p 0 and we can simplify the notation identifying x 0 with x and p 0 with p. With this notation the (non-trivial) one-dimensional system of Hamilton equations reads (4.23) The equation for the momentum immediately integrated by (4.24) with } = p(0) the initial momentum. Notice that the value of the momentum diverges at the critical time t c :=`' 2m } . The time evolution of the position can be derived directly from equation (4.15) which, after some algebraic manipulation, provides (4.25) with % = x(0) the initial position. The long time behavior of the trajectory is determined by the sign of the coefficient of t 2 in (4.25), namely by the sign of ' + %. It follows that 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 critical time t c . Moreover, one has that x(t c ) =`' independently of the initial value % 0 6 =`'. In conclusion the critical plane¨c separates the space into two regions and the trajectory x(t) is fully contained in only one of these two half-spaces according to the initial position %. Moreover, the trajectory can touch the critical plane only once at the critical time t c . These results are in accordance with the qualitative analysis of Section 4.2.
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 the system: Expressions of the type (4.26) are well studied in the literature under the name of quasi-free PDM Lagrangian (see [MM, BDGP, Mu] and references therein). The canonical momentum is exactly that given by equation (4.4). To compute the Euler-Lagrange equations of motion we need also A comparison with (4.8) shows that r x L T = _ p =`r x H T and this assures that the Euler-Lagrange equation is equivalent to the Hamilton system (4.2). An explicit computation provides and putting all the pieces together one gets 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 coordinates (x; _ x) 7 ! (q; _ q) implemented by The inverse is given by and shows that the change of coordinates between x and q is one-to-one only when x `'. However, as seen in Section 4.2, this is exactly the range of values of interest for the problem. With this change of coordinates the Lagrangian becomes where q 0 ; _ q 0 are the initial conditions. By coming back to the original variable one can recover the expression (4.25) for x(t).
Appendix A. Spectral theory of the momentum operator Let p =`i d dx be the momentum operator with domain H 1 (R)  L 2 (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 one gets [RS2,IX.29] " (p`"1)`1 where the Green's function of p is given by A straightforward computation involving contours integrals in the complex plane provides with › 2 R and ‹ > 0.
A.2. Spectral measure. Let -A be the spectral measure of the selfadjoint operator A associated with the state 2 L 2 (R). The function F A : C n R ! C defined by the scalar product is called the Borel-Stieltjes transformation of the finite Borel measures -A . Since it follows that F A : C + ! C + is is a holomorphic map from the upper half plane C + into itself. Such functions are called Herglotz or Nevanlinna functions (see [DK,Section 1.4] or [AW, Appendix]). A classical result by de la Vallée-Poussin assures that the limit F A (›) := lim ‹!0 + F A (› + i ‹) exists and is finite for Lebesgue-almost every › 2 R. Moreover, the absolutely continuous part of the spectral measure -A can be recovered from the imaginary part of F A () according to the classical formula [DK,Theorem 1.4.16.] -A j a:c: In the case A = p is the standard momentum operator one knows that the spectral measure is purely absolutely continuous, i. e. -p = -p j a:c: . By the help fo the Fourier transform F one obtains that where in the last equality one used that 1 ı ‹ x 2 +‹ 2 converges in the distributional sense to ‹(x) when ‹ ! 0 + . In this way one recovers the well-known result p ( d›) = j b (›)j 2 d› : A.3. Density of states. For › 2 R let P › , be the spectral projection of p associated with the energy › according to ( is well defined since P › Q L is trace-class in view of [RS3,Theorem XI 20]. Observe that the exchange between the sum and the integral in the last equality of (A.4) is justified by the monotone convergence theorem and the computation (A.5). The formula 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 2 are easily related by observing that ffl [0;›] (x 2 ) = ffl [`p›; p ›] (x). From this relation one deduces The last equality allows to recover the well-known formula for the DOS of the Laplacian which is given by g (2 Then the principal value of Gs is given by where J 0 is the 0-th Bessel function of the first kind. Proof. For the trivial case s = 0 one has that G0 (u) = u`1 and Z IR;r du u = 0 ; 8 R > r > 0 since the function u`1 is odd and the integration domain I R;r is symmetric with respect the origin. It follows that the principal value of G0 is identically zero according to (B.1). For s 6 = 0 we one has the symmetry G˚j sj (u) =`Gj sj (`u) which provides The relation (B.2) guarantees that we can focus only on the case s > 0.
In this case the computation of the principal value of Gs requires the Cauchy's residue theorem. The function Gs has a holomorphic extension to every bounded open subset of C n f0g and has a singularity in 0. Let us start by computing the residue of Gs . From the formula of the generating function for Bessel functions [GR,eq. 8.511 (1)] one obtains the Laurent serie Gs (u) = where`R ;r is a positively (counterclockwise) oriented simple closed curve composed by the union of the domain I R;r on the real line, the semicircle Cr := fr e i " j " 2 [`ı; 0]g in the lower half-plane and the semicircle C + R := fR e i " j " 2 [0; ı]g in the upper half-plane. An explicit computation provides d" e i s(R˚R`1) cos " e`s (RˇR`1) sin " : and consequently one has the following estimatę Since e`s (RˇR`1) sin " ! 0 when R ! +1 for all " 2 (0; ı), it follows from the Lebesgue's dominated convergence theorem that where˚R ;r is a positively (counterclockwise) oriented simple closed curve composed by the union of the domain I R;r on the real line the semicircles C + R := fR e i " j " 2 [0; ı]g and C + r := fr e i " j " 2 [0; ı]g both in the upper half-plane. The zero on the right-hand side is justified by the fact that˚R ;r does not enclose the singularity of Gs (z) and the negative sign on the last integral is due to the fact that the semicircle C We are interested in the behavior of these functions on the half line R + := [0; +1). Both ker(x) and kei(x) have an exponential decay of the type ‰ r ı 2x e`x p 2 when x ! +1. The function kei(x) is regular in the origin where it takes the value kei(0) =`ı 4 . The function ker(x) diverges at the origin as ‰`log(x). In particular one has that both the Kelvin functions are in L 2 (R + ). The importance of the Kelvin functions for the present work is related to the next result. where ffl I is the characteristic function of the interval I. In the case k = 0 the general solution of (B.11) is c 1 log(jxj) + c 2 , then the physical solution can be chosen as the constant solution k=0 (x) := 1 : These solutions are not in L 2 (R) but they meet the (generalized) normalization condition Z R dx k (x) k 0 (x) = ‹(k`k 0 ) in view of [GR,6.512 (8)]. Let f 2 L 1 (R) and define the generalized eigenfunction expansion are (a variant of) the Hankel transform of f [Erd3,p. 3]. For f 2 L 2 (R) it is possible to prove that f 2 L 2 (R). In this way the Hankel transform can 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 the resolvent (T`¸1)`1 can be obtained by the expansion on the basis k according to The last expression can be integrated by means of the formulas [GR, eq. 6.541 (1)].