Perturbation Theory for the Thermal Hamiltonian: 1D Case
aa r X i v : . [ m a t h - ph ] F e b PERTURBATION THEORY FOR THE THERMAL HAMILTONIAN: 1D CASE
GIUSEPPE DE NITTIS AND VICENTE LENZA
BSTRACT . This work continues the study of the thermal Hamiltonian , initially proposed by J.M. Luttinger in 1964 as a model for the conduction of thermal currents in solids. The previouswork [DL] contains a complete study of the “free” model in one spatial dimension along with apreliminary scattering result for convolution-type perturbations. This work complements the resultsobtained in [DL] by providing a detailed analysis of the perturbation theory for the one-dimensionalthermal Hamiltonian. In more detail the following result are established: the regularity and decayproperties for elements in the domain of the unperturbed thermal Hamiltonian; the determinationof a class of self-adjoint and relatively compact perturbations of the thermal Hamiltonian; the proofof the existence and completeness of wave operators for a subclass of such potentials.
MSC 2010 : Primary: 81Q10; Secondary: 81Q05, 81Q15, 33C10.
Keywords : Thermal Hamiltonian, self-adjoint extensions, spectral theory, scattering theory. C ONTENTS
1. Introduction 12. Analysis of the domain 52.1. Basic facts about the unperturbed operator 52.2. Regularity and decay 73. Perturbations by potential 113.1. Self-adjoint perturbations 113.2. Relatively compact perturbations 124. Scattering theory 15References 161. I
NTRODUCTION
In order to study the thermal transport in the matter, J. M. Luttinger proposed in 1964 a modelwhich allows a “mechanical” derivation of the thermal coefficients [Lut]. Such a model has beeneventually studied and generalized successfully in various later works such as [SS, VMT]. Theessential insight of the Luttinger’s model is to describe the effect of the thermal gradient in thematter by a fictitious gravitational field which affects the dynamics of a charged particle movingin a background material.In absence of thermal fields, and ignoring all physical constants, the dynamics of a one-dimensionalquantum particle is described by the Hamiltonian(1.1) h V := p + V where p := − i dd x is the momentum operator and V is the background (or electrostatic) potential which takes care of the interaction of the particle with the atomic structure of the matter. In the Date : February 23, 2021. absence of interaction with matter ( V = 0 ) the dynamics is simply described by h = p . Theeffect of the thermal field is introduced in the model by a thermal potential which is proportionalto the local content of energy. Since the latter is given by the Hamiltonian (1.1) itself, one endswith the following (effective) model(1.2) H T,V := h V + λ { h V , x } known as thermal Hamiltonian or Luttinger’s Hamiltonian . We will refer to [DL, Section 1.1],and references therein, for more details on the physical justification of (1.2). Here, it is worth topoint out that mathematically the thermal potential is introduced by the anti-commutator { , } between h V and the position operator x , and that the parameter λ > describes the strength ofthe thermal field.The Hamiltonian (1.2) can be rearranged in the form(1.3) H T,V := H T + W V where(1.4) H T := h + λ { h , x } is the thermal Hamiltonian in absence of a background potential and(1.5) W V ( x ) := (1 + λx ) V ( x ) , x ∈ R is the resulting potential that combines the effects of the thermal field and the electrostatic in-teraction with the matter. The study of the spectral properties of the Hamiltonian H T has beenthe central argument of [DL]. The main aim of this work is to provide a satisfactory descrip-tion of the spectral theory for the perturbed Hamiltonian H T,V and to derive the scattering theory[RS3, Kat, Yaf] for the pair ( H T , H T,V ) for a sufficiently general class of background potentials V .Before describing the new results, let us recall some essential facts about the “unperturbed”operator H T . On sufficiently regular functions ψ : R → C the operator H T acts as follows(1.6) ( H T ψ )( x ) = − (1 + λx ) ψ ′′ ( x ) − λψ ′ ( x ) where ψ ′ and ψ ′′ are the first and the second derivatives of ψ , respectively. However, when re-stricted to the Schwartz space S ( R ) , H T turns out to be symmetric but not essentially self-adjoint.In fact, the operator initially defined on S ( R ) by (1.6) admits a one-parameter family of self-adjoint extensions. However, it turns out that all these self-adjoint extensions are unitarily equiv-alent and, without loss of generality, one can focus on a specific “canonical” realization [DL,Theorem 1.1]. Such a realization is obtained by considering the dense domain(1.7) D T, := S ( R ) + C [ κ ] and the prescription(1.8) (cid:0) H T ( ψ + cκ ) (cid:1) ( x ) = ( H T ψ )( x ) + cλκ ( x ) , ψ ∈ S ( R ) where the term H T ψ is given by (1.6) and(1.9) κ ( x ) : = − r π sgn (cid:18) x + 1 λ (cid:19) kei s(cid:12)(cid:12)(cid:12)(cid:12) x + 1 λ (cid:12)(cid:12)(cid:12)(cid:12)! κ ( x ) : = r π ker s(cid:12)(cid:12)(cid:12)(cid:12) x + 1 λ (cid:12)(cid:12)(cid:12)(cid:12)! . ERTURBATION THEORY FOR THE THERMAL HAMILTONIAN: 1D CASE 3
In (1.9) kei and ker denote the irregular Kelvin functions of -th order (cf. [OMS, Chap. 55]or [OLBC, Sect. 10.61]) while the sign function is defined by sgn( x ) := x/ | x | if x = 0 and sgn(0) := 0 . It turns out that the operator defined by (1.8) is essentially self-adjoint on the domain(1.7), and this fact provides a rigorous definition for the thermal Hamiltonian. Definition 1.1 (1-D unperturbed thermal Hamiltonian) . The unperturbed thermal Hamiltonian ,still denoted with H T , is the self-adjoint operator on L ( R ) defined by (1.7) and (1.8) on thedomain D T := D T, k k HT obtained by the closure of D T, with respect to the graph norm k ψ k H T := k ψ k L + k H T ψ k L .In view of [DL, Theorem 1.1] we know that H T has a purely absolutely continuous spectrumgiven by(1.10) σ (cid:0) H T (cid:1) = σ a . c . (cid:0) H T (cid:1) = R independently of λ > .We are now in position to present the main results of this work. For that, let us recall thedefinition of the critical point (1.11) x c ≡ x c ( λ ) := − λ − which plays an important role in the singular behavior of the dynamics generated by the unper-turbed operator H T [DL]. The first result concerns the determination of a class of self-adjointperturbation of H T . Theorem 1.1 (Self-adjoint perturbations) . Let H T be the unperturbed thermal Hamiltonian de-scribed in Definition 1.1. Let V : R → R be a background potential such that V ( x ) = V ( x ) | x − x c | + V ( x ) | x − x c | with V ∈ L ( R ) and V ∈ L ∞ ( R ) . Then, the perturbed thermal Hamiltonian H T,V given by (1.3) , with potential W V given by (1.5) , is self-adjoint on the domain D T . Remark 1.1.
Although Theorem 1.1 stipulates that H T,V is self-adjoint for a large class of back-ground potentials V , from a physical point of view this result is not yet totally satisfactory. In fact,the standard model for the dynamics of a charged particle in a (semi-)metal is h V per := p + V per with V per a periodic background potential. However, every V per = 0 does not meet the conditionsof Theorem 1.1, and as a consequence the question of the self-adjointness of H T,V per remainsopen. This is not an irrelevant fact since H T,V per is the relevant model (tacitally) considered in[SS, VMT] for the derivation of thermal conductivity in condensed matter systems. It is alsoworth noting that Theorem 1.1 allows background potentials which are singular around the criticalpoint x c . ◭ The second main result describes a class of relatively compact perturbations. For that we willneed to introduce the family of resolvents R z ( H T ) := ( H T − z ) − , z ∈ C \ R . Theorem 1.2 (Relatively compact perturbations) . Let H T be the unperturbed thermal Hamilton-ian described in Definition 1.1. Let V : R → R be a background potential such that V ( x ) = V ( x ) | x − x c | G. DE NITTIS AND V. LENZ with V ∈ L ( R ) . Then, W V R z ( H T ) is Hilbert-Schmidt (hence compact) for every z ∈ C \ R ,where the potential W V is given by (1.5) . By combining Theorem 1.2 with the Weyl Theorem about the stability of the essential spectrum[RS4, Theorem XIII.14] one obtains the following result.
Corollary 1.1 (Essential spectrum) . Let V be a background potential as in Theorem 1.2. Then, theessential spectrum of the perturbed thermal Hamiltonian H T,V is σ ess ( H T,V ) = σ ess ( H T ) = R . The question of the existence of embedded eigenvalues is not answered by Corollary 1.1 and isleft open for future investigations.The final result concerns the scattering theory for the pair ( H T , H T,V ) . Let us recall the (formal)definition of the wave operators [RS3, Section XI.3](1.12) Ω ± ( V ) := s − lim t →∓∞ e i H T,V t e − i H T t , where the limits are meant in the strong operator topology. It is worth noting that in the definition(1.12) we have tacitly used the fact that the spectral projection on the absolutely continuous part ofthe spectrum of H T coincides with the identity in view of (1.10). The scattering matrix is definedby(1.13) S ( V ) := Ω − ( V ) ∗ Ω + ( V ) . Theorem 1.3 (Scattering theory) . Let V : R → R be a background potential such that bot V and | V | satisfy the conditions of Theorem 1.2. Then, the wave operators Ω ± ( V ) exist and thescattering matrix S ( V ) is unitary. Theorem 1.3 boils up the application of the celebrated
Kuroda-Birman Theorem [RS3, TheoremXI.9] which guarantees the existence and the completeness of the wave operators. In particular,the unitarity of S ( V ) is a consequence of the completeness of the wave operators.A special class of bounded background potentials that meet the conditions of Theorem 1.2 andTheorem 1.3 are described in Remark 3.2 and Remark 4.2, respectively.It is worth to end this introductory section with few words about the strategy used for the proofsof the main results described above. Instead of working with the “physical” operator H T we foundmore convenient to work with the unitarily equivalent (up to a scale factor) operator(1.14) T := λ − S λ H T S ∗ λ obtained from H T via the unitary shift(1.15) ( S λ ψ )( x ) := ψ (cid:18) x − λ (cid:19) = ψ ( x + x c ) , ψ ∈ L ( R ) . The advantage relies on the fact that T has a simpler expression with respect to H T . In fact, atleast formally, one has that T = pxp . It turns out that Theorem 1.1, Theorem 1.2 and Theorem 1.3are nothing more that the transposition via the conjugation by S λ of the equivalent results provedfor T in Proposition 3.1, Proposition 3.2 and Proposition 4.1, respectively. Anyway, the passagefrom the results concerning T to the related results concerning H T is described in some detail inRemark 3.1, Remark 3.2 and Remark 4.2. Structure of the paper. In Section 2 we recall some basic result for the operator T originallyobtained in [DL] and we provide new results about the regularity and the decay of the elements ERTURBATION THEORY FOR THE THERMAL HAMILTONIAN: 1D CASE 5 of the domain of T . Section 3 contains the results about the self-adjoint and relatively compactperturbations of the operator T . Finally, Section 4 provides the results about the scattering theory.
Acknowledgements.
GD’s research is supported by the grant
Fondecyt Regular - 1190204. Theauthors are indebted to Olivier Bourget, Claudio Fernandez, Marius Mantoiu and Serge Richardfor many stimulating discussions.2. A
NALYSIS OF THE DOMAIN
In this section we will provide some result about regularity and decay properties for for elementin the domain of the operator T . Such results can be inmediately transported to elements in thedomain of H T in view of the unitary mapping (1.14).2.1. Basic facts about the unperturbed operator.
We will start by recalling some important re-sult concerning the spectral theory of the operator T given by (1.14). All the information presentedhere are taken form [DL].An important role for the study of the operator T is played by the bounded operator B initiallydefined on elements ψ ∈ L ( R ) ∩ L ( R ) by the integral formula(2.1) ( Bψ )( x ) := ˆ R d y B ( x, y ) ψ ( y ) with kernel(2.2) B ( x, y ) := i sgn ( x ) − sgn( y )2 J (cid:16) p | xy | (cid:17) . where J denotes the Bessel function of the first kind [GR]. We will use the symbol B to denotethe unique linear, bounded extension the dense defined operator (2.1). It turns out that B is aunitary involution on L ( R ) , i.e. B = B ∗ = B − .Let x be the position operator on L ( R ) , acting as multiplication by x on its natural domain Q ( R ) := (cid:26) ψ ∈ L ( R ) (cid:12)(cid:12)(cid:12) ˆ R d x x | ψ ( x ) | < + ∞ (cid:27) . The involution B introduced above intertwines between the operator T and the position operators x . in fact it holds true that T = − BxB , and as a consequence the domain of T can be describeda D ( T ) = B [ Q ( R )] .ative.The resolvent of T admits an explicit integral expression when evaluated on element of thedense domain L ( R ) ∩ L ( R ) . Let z ∈ C \ R and consider the polar representation z = | z | e ± i φ with < φ < π . Let R z ( T ) := ( T − z ) − be the resolvent of T at z . If ψ ∈ L ( R ) ∩ L ( R ) ,then it holds true that(2.3) (cid:0) R z ( T ) ψ (cid:1) ( x ) = ˆ R d y (cid:0) sgn( x ) + sgn( y ) (cid:1) F z ( x, y ) ψ ( y ) Where(2.4) F z ( x, y ) := I (cid:18) p | z | min {| x | , | y |} e ± i h φ − π (cid:0) sgn( x )+1 (cid:1) i (cid:19) × K (cid:18) p | z | max {| x | , | y |} e ± i h φ − π (cid:0) sgn( x )+1 (cid:1) i (cid:19) G. DE NITTIS AND V. LENZ with I and K are the modified Bessel functions of the firstand second kind, respectively. In thespecial case z = ± i , by using [OLBC, eq. 10.61.1 & eq. 10.61.2], the formula above reduces to F ± i ( x, y ) := (cid:16) ber (cid:16) p min {| x | , | y |} (cid:17) ∓ i sgn( x )bei (cid:16) p min {| x | , | y |} (cid:17)(cid:17) × (cid:16) ker (cid:16) p max {| x | , | y |} (cid:17) ∓ i sgn( x )kei (cid:16) p max {| x | , | y |} (cid:17)(cid:17) , with ber, bei, ker and kei the Kelvin functions of -th order [OMS, Chapter 55] or [OLBC, Section10.61]. In accordance to [OLBC, Section 10.68] let us introduce the following notation M ( s ) := p ber( s ) + bei( s ) N ( s ) := p ker( s ) + kei( s ) , s > . This allows to rewrite the modulus of the kernel F ± i as follows(2.5) | F | ( x, y ) : = | F ± i ( x, y ) | = M (cid:16) p min {| x | , | y |} (cid:17) N (cid:16) p max {| x | , | y |} (cid:17) . From (2.5) one infers immediately that | F | is invariant under the reflections x
7→ − x and y
7→ − y and is independent on the sign in ± i . For the study of the integrability properties of (2.5) we willmake use of the inequalities(2.6) M ( s ) C M e √ s s ,N ( s ) C N e −√ s s , ∀ s > which can be deduced by the asymptotic expansions [OLBC, eq. 10.67.9 & eq. 10.67.13]. Theexact value of the positive constants C M and C N is not important for the purposes of this work .For small arguments ( s ∼ ) one has that the functions ber and bei are continuous around theorigin with ber(0) = 1 and bei(0) = 0 . Consequently, also M is continuous in and lim s → + M ( s ) = M (0) = 1 . On the other hand kei(0) = − π , but ker , and in turn N , have a logarithmic pole in . By usingthe series expansions [OLBC, eq. 10.65.5] one can prove that lim s → + (cid:0) N ( s ) + ln( s ) (cid:1) = ln(2) − γ where γ is the Euler-Mascheroni constant. For the next result, that will play a crucial role inSection 3.2, we need to introduce the positive semi-axis R + := (0 , + ∞ ) and the first quadrant R := R + × R + . Lemma 2.1.
Let w : R + → R + ∪ { } be a non negative function such that ˆ R + d x w ( x ) √ x < + ∞ and | F | the function defined by (2.5) . Then the integral I w := ¨ R d x d y w ( x ) | F | ( x, y ) < + ∞ , The best constant C M is fixed by the maximum of the function g M ( s ) := sM ( s ) e −√ s . A numerical inspectionwith Wolfram Mathematica (version 12.1) shows that one can choose C M > . π . A similar argumentalso provides C N > π . ERTURBATION THEORY FOR THE THERMAL HAMILTONIAN: 1D CASE 7 is finite.Proof.
Let us split the integral I w as follows I w = I w (Σ ) + I w (Σ ) where we used the notation I w (Σ i ) := ¨ Σ i d x d y w ( x ) | F | ( x, y ) , i = 1 , with Σ := { ( x, y ) ∈ R | x < y } and Σ := { ( x, y ) ∈ R | y < x } . We start with the integral I w (Σ ) . In view of the definition of | F | one has that I w (Σ ) := ¨ Σ d x d y w ( x ) M (cid:0) √ x (cid:1) N (2 √ y ) . From the inequalities (2.6) one infers that M (cid:0) √ x (cid:1) N (2 √ y ) C e √ x e − √ y √ xy , ∀ ( x, y ) ∈ Σ . with C := C M C N > . As a consequence one has that I w (Σ ) C ¨ Σ d x d y w ( x ) e √ x e − √ y √ xy = C ˆ R + d x w ( x ) e √ x √ x ˆ + ∞ x d y e − √ y √ y ! = C √ ˆ R + d x w ( x ) √ x < + ∞ where the first equality (second line) follows from the Tonelli’s theorem and the last inequality isguaranteed by hypothesis. The treatment of the integral I w (Σ ) is quite similar. Indeed, one hasthat I w (Σ ) = ¨ Σ d x d y w ( x ) N (cid:0) √ x (cid:1) M (2 √ y ) C ˆ R + d x w ( x ) e − √ x √ x ˆ x d y e √ y √ y ! = C √ ˆ R + d x w ( x ) 1 − e − √ x √ x C √ ˆ R + d x w ( x ) √ x < + ∞ . This concludes the proof. (cid:3)
Regularity and decay.
This section is devoted to the description of elements in D ( T ) . Letus start from a preliminary result. Lemma 2.2.
For every ψ ∈ D ( T ) it holds true that (2.7) k Bψ k L π k T ψ k L k ψ k L . As a consequence one has that B [ D ( T )] ⊂ L ( R ) . G. DE NITTIS AND V. LENZ
Proof.
Let v > be an arbitrary number and consider the identity k Bψ k L = ˆ + ∞−∞ d x | ( Bψ )( x ) | (cid:0) x + v (cid:1) (cid:0) x + v (cid:1) − . By the classical Cauchy-Bunyakovsky-Schwarz inequality one gets k Bψ k L (cid:18) ˆ + ∞−∞ d x | ( Bψ )( x ) | (cid:0) x + v (cid:1)(cid:19) (cid:18) ˆ + ∞−∞ d x (cid:0) x + v (cid:1) − (cid:19) . By recalling the equivalence between T and the position operator described in Section 2.1 one canrewrite the first integral as follows: ˆ + ∞−∞ d x | ( Bψ )( x ) | (cid:0) x + v (cid:1) = k BT ψ k L + v k Bψ k L . By using the fact that B is unitary and the known formula ˆ + ∞−∞ d x (cid:0) x + v (cid:1) − = πv one gets k Bψ k L πv (cid:0) k T ψ k L + v k ψ k L (cid:1) independently of v > . By minimizing with respect to v one obtains the bound (2.7). (cid:3) The following lemma makes use of the explicit form of the kernel of B . Lemma 2.3 (Boundedness) . For every ψ ∈ D ( T ) it holds true that (2.8) k ψ k L ∞ (2 π k T ψ k L k ψ k L ) √ π ( k T ψ k L + k ψ k L ) . As a consequence one has that D ( T ) ⊂ L ∞ ( R ) .Proof. Since B is an involution one has that ψ = B ( Bψ ) , and in view of Lemma 2.2 one infersthat Bψ ∈ L ( R ) ∩ L ( R ) . Therefore, the action of B on Bψ can be computed via the integralformula (2.1) and (2.2). This provides | ψ ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12) ˆ R d y B ( x, y ) ( Bψ )( y ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ R d y | B ( x, y ) | | ( Bψ )( y ) | ˆ R d y | ( Bψ )( y ) | = k Bψ k L where we used the bound | B ( x, y ) | . Since the inequality above holds for every x ∈ R onegets k ψ k L ∞ k Bψ k L . The rest of the proof follows from the inequality in Lemma 2.2. (cid:3) The next result describes the continuity properties of elements of D ( T ) . Lemma 2.4 (H ¨older continuity) . Let ψ ∈ D ( T ) and x, y ∈ R \ { } with sgn( x ) = sgn( y ) . Then,it holds true that (2.9) | ψ ( x ) − ψ ( y ) | G k | x − y | k k T ψ k kL k ψ k − kL for every k < , with G k a constant depending only on k . As a consequence the element of D ( T ) are α -H¨older continuous in R \ { } with α < . ERTURBATION THEORY FOR THE THERMAL HAMILTONIAN: 1D CASE 9
Proof.
By using the identity ψ = B ( Bψ ) and the integral expression of B as in the proof ofLemma 2.3 one obtains | ψ ( x ) − ψ ( y ) | = (cid:12)(cid:12)(cid:12)(cid:12) ˆ R d s [ B ( x, s ) − B ( y, s )] ( Bψ )( s ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ R d s (cid:12)(cid:12)(cid:12) J (cid:16) p | sx | (cid:17) − J (cid:16) p | sy | (cid:17)(cid:12)(cid:12)(cid:12) | ( Bψ )( s ) | where in the last inequality the hypothesis sgn( x ) = sgn( y ) has been used. In view of the integralrepresentation of the Bessel J (2.10) J ( z ) = 12 π ˆ π − π d τ e i z sin( τ ) one obtains(2.11) | ψ ( x ) − ψ ( y ) | π ˆ R d s ˆ π − π d τ g ( x,y ) ( s, τ ) | ( Bψ )( s ) | where g ( x,y ) ( s, τ ) := (cid:12)(cid:12)(cid:12) e i 2 √ | sx | sin( τ ) − e i 2 √ | sy | sin( τ ) (cid:12)(cid:12)(cid:12) . For fixed values of s , and using the classical estimate | − e i s | | s | one gets g ( x,y ) ( s, τ ) min n , (cid:12)(cid:12)(cid:12)p | sx | − p | sy | (cid:12)(cid:12)(cid:12) | sin( τ ) | o . Since the minimum between two numbers is dominated by any weighted geometric mean of thesame one obtains that g ( x,y ) ( s, τ ) − k (cid:16) (cid:12)(cid:12)(cid:12)p | sx | − p | sy | (cid:12)(cid:12)(cid:12) | sin( τ ) | (cid:17) k , ∀ k . After inserting the last inequality in (2.11) one gets(2.12) | ψ ( x ) − ψ ( y ) | C k ˆ R d s (cid:12)(cid:12)(cid:12)p | sx | − p | sy | (cid:12)(cid:12)(cid:12) k | ( Bψ )( s ) | where(2.13) C k := 1 π ˆ π − π d τ | sin( τ ) | k = 4 π ˆ π d τ sin( τ ) k = 2 √ π Γ (cid:0) k +12 (cid:1) Γ (cid:0) k +22 (cid:1) with Γ denoting the gamma function. Observing that (cid:12)(cid:12)(cid:12)p | sx | − p | sy | (cid:12)(cid:12)(cid:12) k = | s | k (cid:12)(cid:12)(cid:12)p | x | − p | y | (cid:12)(cid:12)(cid:12) k | s | k (cid:12)(cid:12) | x | − | y | (cid:12)(cid:12) k | s | k | x − y | k one obtains(2.14) | ψ ( x ) − ψ ( y ) | C k | x − y | k ˆ R d s | s | k | ( Bψ )( s ) | . By inserting inside the integral the identity ( s + v ) ( s + v ) − with v > , and using the sametrick employed in the proof of Lemma 2.2 one gets(2.15) | ψ ( x ) − ψ ( y ) | C k | x − y | k q k ( v ) (cid:0) k T ψ k L + v k ψ k L (cid:1) where q k ( v ) := ˆ R d s | s | k s + v = 1 v − k π cos (cid:0) kπ (cid:1) , k < . A minimization procedure on v provides min v> k T ψ k L + v k ψ k L v − k = I k k T ψ k kL k ψ k − kL with I k := 2(1 + k ) k (1 − k ) − k . Since inequality (2.15) holds for every v > , it holds true also when v coincides with the min-imizer. This provides the inequality (2.9) with the constant given by G k := C k I k π cos ( kπ ) . Thisconcludes the proof. (cid:3) The last result allows to control the behavior of the elements of the domain D ( T ) around thecritical point x = 0 and gives an upper bound for the decay rate at infinity . Lemma 2.5 (Global control of the behavior) . Let ψ ∈ D ( T ) . Then, it holds true that for somepostive constant K , (2.16) | ψ ( x ) | K | x | − k T ψ k L k ψ k L , ∀ x ∈ R \ { } . Proof.
Let us start by observing that from [OLBC, eq. 10.7.8] one infers that there exists a positiveconstant
M > such that | J ( z ) | M √ z , ∀ z > . The exact value of the constants M is not important for the purposes of this work . Using the sameargument in the proof of Lemma 2.4 one gets | ψ ( x ) | ˆ R d s (cid:12)(cid:12)(cid:12) J (cid:16) p | sx | (cid:17)(cid:12)(cid:12)(cid:12) | ( Bψ )( s ) | M | x | − ˆ R d s | s | − | ( Bψ )( s ) | . which shows that | ψ ( x ) | is dominated by | x | − . For the determination of the constant one caninserting inside the integral the identity ( s + v ) ( s + v ) − with v > and, with the sametrick employed in the proof of Lemma 2.2, one gets ˆ R d s | s | − | ( Bψ )( s ) | ˆ + ∞−∞ d s | s | − ( s + v ) ! (cid:0) k T ψ k L + v k ψ k L (cid:1) = 2 √ πv (cid:0) k T ψ k L + v k ψ k L (cid:1) . After minimizing the last inequality with respect v > , and grouping all constants in K > , onefinally obtains | ψ ( x ) | K | x | − k T ψ k L k ψ k L . This concludes the proof. (cid:3)
Remark 2.1 (Synopsis) . Let us summarize the main properties of elements in the domain D ( T ) obtained in this section. Lemma 2.3 provides:(i) D ( T ) ∈ L ( R ) ∩ L ∞ ( R ) . The best constant M is fixed by the maximum of the function f ( z ) := | J ( z ) |√ z . A numerical inspection with Wolfram Mathematica (version 12.1) shows that one can choose
M > q π . ERTURBATION THEORY FOR THE THERMAL HAMILTONIAN: 1D CASE 11
By combining Lemma 2.3 and Lemma 2.4 one infers that:(ii) Every ψ ∈ D ( T ) is at least H ¨older continuous in R \ { } of order α < . Moreover,the possible discontinuity in the critical point x = 0 is of the first kind meaning that bothlimits lim x → ± ψ ( x ) = L ± exist and are finite.Finally Lemma 2.5 tells us that:(iii) The global behavior of ψ ∈ D ( T ) in R \ { } is dominated by | x | − . In particular theelements of D ( T ) vanish at infinity.Let us point out that the existence of singular discontinuous elements in D ( T ) , as stipulated by(ii) is unavoidable. Indeed, we know from [DL] that smooth functions aren’t enough to provide acore for T , while a core is provided by S ( R ) + C [ κ ] . The function κ ( x ) ∝ sgn( x )kei(2 p | x | ) is smooth in R \ { } and has a discontinuity of first kind in x = 0 . The decay at infinity of κ us quite rapid. Indeed on has that κ ( x ) = o( e − √ | x | ) . Finally, it should be noted that Lemma2.5 provides a somehow weak information about the decay of elements of D ( T ) at infinity. Infact a decay of type | x | − it is not enough to guarantee that ψ ∈ L ( R ) . However, Lemma 2.5gives an important information that will be crucial in Proposition 3.1. Let ψ ∈ D ( T ) and define φ ( x ) := | x | ψ ( x ) . Then k φ k L ∞ ∞ . Moreover, from the proof of Lemma 2.5 one also infersthat(2.17) k φ k L ∞ Cv (cid:0) k T ψ k L + v k ψ k L (cid:1) with C > a suitable constant and for every v > . Let us conclude this remark by observingthat the properties (i), (ii) and (iii) remain valid for elements of the domain D T of the thermalHamiltonian H T with the only difference that the critical point has to be shifted from to x c aseffect of the translation S λ . ◭
3. P
ERTURBATIONS BY POTENTIAL
In this section we will study the perturbations of the operator T by multiplicative, real-valuedpotentials.3.1. Self-adjoint perturbations.
Let W : R → R be a real-valued function. With a slightabuse of notation we will denote with the same symbol also the multiplication operator defined on ψ ∈ L ( R ) by ( W ψ )( x ) := W ( x ) ψ ( x ) . This is a self-adjoint operator with natural domain givenby D ( W ) := { ψ ∈ L ( R ) | W ψ ∈ L ( R ) } . The next result provides condition on the potential W for the self-adjointness of T + W . Proposition 3.1 (Self-adjoint perturbations) . Let W : R → R be a real-valued function such that W ( x ) := | x | V ( x ) + V ( x ) , x ∈ R with V ∈ L ( R ) and V ∈ L ∞ ( R ) . Then, for every ǫ > there exists a positive constant B ǫ > such that (3.1) k W ψ k L ǫ k T ψ k L + B ǫ k ψ k L . As a consequence T + W defines a self-adjoint operator with domain D ( T ) .Proof. Let ψ ∈ D ( T ) . Then one has that(3.2) k W ψ k L k V φ k L + k V ψ k L k V k L k φ k L ∞ + k V k L ∞ k ψ k L where φ ( x ) := | x | ψ ( x ) with k φ k L ∞ < ∞ in view of Lemma 2.5. This proves that D ( T ) ⊆D ( W ) . By combining (3.2) with the inequality (2.17) (also deduced from Lemma 2.5) one obtains(3.3) k W ψ k L ( k V k L k φ k L ∞ + k V k L ∞ k ψ k L ) (cid:0) k V k L k φ k L ∞ + k V k L ∞ k ψ k L (cid:1) C k V k L v k T ψ k L + (cid:16) Cv + k V k L ∞ (cid:17) k ψ k L . Since (3.3) holds for every v > one can always find a v ǫ such that C k V k L v − ǫ = ǫ . Bysetting B ǫ := 2 Cv ǫ + k V k L ∞ one obtains the inequality (3.1). The latter implies that W isinfinitesimally small with respect to T in the sense of Kato (cf. [RS2, eq. X.19a & eq. X.19b]).Consequently T + W defined a self-adjoint operator with domain D ( T ) in view of the Kato-Rellichtheorem [RS2, Theorem X.12]. (cid:3) Remark 3.1 (Self-adjoint perturbation of the thermal Hamiltonian) . Let W a potential whichmeets the conditions of Proposition 3.1. Then T + W is a self-adjoint operator with domain D ( T ) . As a consequence H T + f W , with f W := λ S ∗ λ W S λ is a self-adjoint perturbation of the thermal Hamiltonian H T in view of the equivalence (1.14).The splitting of W stipulated in Proposition 3.1 translates to f W ( x ) = | x − x c | e V ( x ) + e V ( x ) where e V i := λS ∗ λ V i S λ and i = 1 , . Since the conjugation by S λ is a translation one obtains that e V ∈ L ( R ) and e V ∈ L ∞ ( R ) . The considerations above provide the main argument to deduceTheorem 1.1 from Proposition 3.1. ◭ Relatively compact perturbations.
The task of this section is to find suitable class of rela-tively compact (or T -compact) perturbations W : R → R of the free operator T . This means that D ( T ) ⊆ D ( W ) and the product W R ± i ( T ) must be a compact operator [RS4, Section XIII.4],with R z ( T ) := ( T − z ) − , z ∈ C \ R the resolvent of T evaluated at z . In the next result we will provide conditions on W : R → R such that the product W R ± i ( T ) is a Hilbert-Schmidt operator.
Proposition 3.2.
Let W : R → R be a real-valued function such that W ( x ) = | x | V ( x ) with V ∈ L ( R ) . Then W R ± i ( T ) are Hilbert-Schmidt operators (hence compact).Proof. The resolvents R ± i ( T ) are integral operators with explicit integral kernels given by (2.4)when evaluated on the dense domain L ( R ) ∩ L ( R ) . Let us introduce the symbol(3.4) K ± ( x, y ) := (cid:0) sgn( x ) + sgn( y ) (cid:1) W ( x ) F ± i ( x, y ) for the integral kernel of W R ± i ( T ) . Since the term sgn( x ) + sgn( y ) vanishes when x and y havedifferent signs, one has that ¨ R d x d y | K ± ( x, y ) | = 2 ( I + + I − ) ERTURBATION THEORY FOR THE THERMAL HAMILTONIAN: 1D CASE 13 where I ± : = ¨ R ± d x d y | W ( x ) | | F | ( x, y ) = ¨ R d x d y | W ( ± x ) | | F | ( x, y ) . In the definition of the integrals I ± we used the following notation for the domains of integration: R ± := { x ∈ R | ± x > } are the positive and negative semi-axis and R ± := R ± × R ± are the firstand third quadrants. The function | F | is defined by (2.5). In the second equality the invarianceproperty of | F | under the reflections x
7→ − x and y
7→ − y has been exploited. In view of thesplitting above, and after a comparison with the notation introduced in Lemma 2.1 one obtains that I ± = I w ± := ¨ R d x d y w ± ( x ) | F | ( x, y ) where the functions w ± : R + → R + ∪ { } are defined as follow: w + ( x ) := | W ( x ) | , w − ( x ) := | W ( − x ) | . Since ˆ R + d x w ± ( x ) √ x = ˆ R ± d x | V ( x ) | k V k L one can apply Lemma 2.1 to conclude that I ± < + ∞ . This implies that the kernels (3.4) aresquare-integrable, i.e. K ± ∈ L ( R ) . Therefore, the kernels K ± define two Hilbert-Schmidtoperators K ± on L ( R ) [RS1, Theorem VI.23]. Moreover, one has that K ± coincides with W R ± i ( T ) on the dense domain L ( R ) ∩ L ( R ) . This is enough to conclude that K ± = W R ± i ( T ) and this completes the proof. (cid:3) The hypotheses on W stipulated in Proposition 3.2 are stronger than the hypotheses in Propo-sition 3.1. Consequently it holds true that under the hypotheses of Proposition 3.2 T + W is aself-adjoint operator and W is a relatively compact perturbation of T . As a consequence of thecelebrated Weyl Theorem [RS4, Theorem XIII.14] one obtains the equality of the essential spectra,i.e. σ ess ( T + W ) = σ ess ( T ) = R . It is also worth noting that under the hypotheses of Proposition 3.2 it holds true that
W R z ( T ) is aHilbert-Schmidt operator for every z ∈ C \ R . This follows from the identity W R z ( T ) = W R ± i ( T ) [( T ∓ i ) R z ( T )] , the boundedness of the operator inside the square brackets and the fact that the class of Hilbert-Schmidt operators is a two-sided ideals inside the bounded operator.The next result provides a class of prototipe bounded potentials which meet the condition stip-ulated in Proposition 3.2. Corollary 3.1.
The potentials W r ( x ) := (cid:0) x (cid:1) − r , r > are relatively compact perturbations of the operator T and W r R z ( T ) are Hilbert-Schmidt opera-tors for every z ∈ C \ R . Proof.
An explicit computation provides ˆ R d x | W r ( x ) | | x | = 4Γ (cid:18) (cid:19) Γ (cid:0) r − (cid:1) Γ (2 r ) , r > where Γ denotes the Gamma function. Therefore the potentials W r , with r > , satisfy thehypotheses of Proposition 3.2. (cid:3) For the next result let us introduce the
Japanese brackets h x i : R → R + defined by h x i ( x ) := (cid:0) x (cid:1) . Corollary 3.2.
Let W be a real valued potential such that W h x i s ∈ L ∞ ( R ) for some s > . Then W is a relatively compact perturbation of the operator T and W R z ( T ) areHilbert-Schmidt operators for every z ∈ C \ R .Proof. Consider the identity
W R z ( T ) = (cid:0) W h x i s (cid:1)(cid:0) h x i − s R z ( T ) (cid:1) . Since h x i − s = W s one gets from Corollary 3.1 that h x i − s R z ( T ) is Hilbert-Schmidt whenever s > . To conclude the proof it is enough to observe that W h x i s is a bounded operator accordingto the hypothesis and that the class of the Hilbert-Schmidt operators is an ideal inside the boundedoperators. (cid:3) Remark 3.2 (Relatively compact perturbations of the thermal Hamiltonian) . In the spirit of Re-mark 3.1 one can translate the conditions for the relative compactness of a perturbation from theoperator T to the thermal Hamiltonian H T just by using the equivalence (1.14) implemented bythe translation S λ . As a result one gets that potentials of the type f W ( x ) = | x − x c | e V ( x ) with e V ∈ L ( R ) are relatively compact perturbations of H T of Hilbert-Schmidt type This is thekey observation to deduce Theorem 1.2 from Proposition 3.2. It is worth to translate the result ofCorollary 3.2 in terms of the background potential V which defines the perturbation of the thermalHamiltonian via the equation (1.5). By using the usual argument involving the shift operator S λ one infers that the condition(3.5) V h x − x c i s +1 ∈ L ∞ ( R ) , for some s > , implies that W V is a relatively compact perturbation of H T of Hilbert-Schmidttype. In the last equation we introduced the shifted Japanese brackets h x − x c i ( x ) := (cid:0) | x − x c | (cid:1) . Since the function h x − x c ih x i − is bounded and invertible one can reformulate the condition (3.5)as follows V h x i s ∈ L ∞ ( R ) , for some s > . ◭ ERTURBATION THEORY FOR THE THERMAL HAMILTONIAN: 1D CASE 15
4. S
CATTERING THEORY
In this section we will present a class of perturbations W of the operator T for which the waveoperators(4.1) Ω ± ( W ) := s − lim t →∓∞ e i ( T + W ) t e − i T t . exist and are complete. It is worth noticing that in the definition (4.1) we have tacitly used the factthat the spectrum of T is purely absolutely continuous, and consequently the spectral projection onthe absolutely continuous part of the spectrum of T coincides with the identity, i.e. P a.c. ( T ) = .The proof of the existence and completeness of Ω ± ( W ) rely on the results of Section 3.2 alongwith the celebrated Kuroda-Birman theorem [RS3, Theorem XI.9] which, in our specific case, canbe stated as follows:
Theorem 4.1 (Kuroda-Birman) . Let W : R → R be a potential such that T + W is self-adjoint.Consider the resolvents R − i ( T ) := ( T + i ) − , R − i ( T + W ) := ( T + W + i ) − . If the difference R − i ( T ) − R − i ( T + W ) is trace-class, then the wave operators Ω ± ( W ) existand are complete. We are now in position to present our main result concerning the scattering theory of the oper-ator T . Proposition 4.1 (Scattering theory) . Let W : R → R be a potential such that both W and W ′ := | W | satisfy the conditions of Proposition 3.2. Then, the wave operators Ω ± ( W ) exist andare complete.Proof. An iterated use of the resolvent identity provides R − i ( T ) − R − i ( T + W ) = R − i ( T ) W R − i ( T + W )= R − i ( T ) W (cid:2) R − i ( T ) − R − i ( T ) W R − i ( T + W ) (cid:3) = Z + Z where the two terms in the last line are given by Z := R − i ( T ) W R − i ( T ) , Z := − (cid:2) R − i ( T ) W (cid:3) R − i ( T + W ) . In view of the hypotheses and of Proposition 3.2 one has that R − i ( T ) W = (cid:0) W R i ( T ) (cid:1) ∗ is a Hilbert-Schmidt operator. As a consequence one has that its square, and consequently Z , aretrace-class. The operator Z can be rewritten as Z = (cid:0) W ′ R i ( T ) (cid:1) ∗ sgn( W ) W ′ R − i ( T ) where sgn( W ) denotes the bounded multiplicative operator which multiplies elements of L ( R ) by the sign of the function W . Again, from the hypotheses and of Proposition 3.2 one gets that W ′ R ± i ( T ) are Hilbert-Schmidt operators and consequently Z is trace-class. Since we provedthat the difference R − i ( T ) − R − i ( T + W ) is trace-class, the result follows from Theorem 4.1. (cid:3) Remark 4.1 (Scattering matrix) . The existence of the wave operators Ω ± ( W ) allows to define the scattering matrix (4.2) S ( W ) := Ω − ( W ) ∗ Ω + ( W ) . The completeness of the wave operators, along with P a.c. ( T ) = , ensures that S ( W ) is a unitaryoperator on L ( R ) . ◭ The next results provides a class of prototipe bounded potentials which meet the conditionstipulated in Proposition 4.1.
Corollary 4.1.
The wave operators Ω ± ( W r ) associated to the potentials W r ( x ) := (cid:0) x (cid:1) − r , r > exist and are complete.Proof. In view of Corollary 3.1 both W r and | W r | = W r meet the conditions of Proposition 3.2.Then, Proposition 4.1 applies. (cid:3) Corollary 4.2.
Let W be a real valued potential such that W h x i s ∈ L ∞ ( R ) for some s > . Then the related wave operators Ω ± ( W ) exist and are complete.Proof. By hypothesis one has that W = W ∞ W s where W ∞ := W h x i s ∈ L ∞ ( R ) and W s = h x i − s . If s > then W s meets the condition of Proposition 3.2 in view of Corollary (4.1). Sincethe multiplication by an essentially bounded function does not change the integrability propertiesof a given function, it follows that also W meets the condition of Proposition 3.2. Then, Proposi-tion 4.1 applies. (cid:3) Remark 4.2 (Scattering theory of the thermal Hamiltonian) . The passage from Proposition 4.1 toTheorem 1.3 is justified by the same argument already discussed in Remark 3.1 and Remark 3.2. Itis interesting to translate the content of Corollary 4.2 in terms of the background potential V whichdefines the perturbation of the thermal Hamiltonian via the equation (1.5). An analysis similar tothat in Remark 3.2 provides the following result: If if the background potential V : R → R meetsthe condition V h x i s ∈ L ∞ ( R ) , for some s > then the wave operators defined by (1.12) exist and are complete. ◭ R EFERENCES [DL] De Nittis, G.; Lenz, V.: SpectralTheoryoftheThermalHamiltonian:1DCase. E-print arXiv:2002.10382,(2020)[GR] Gradshteyn, I. S.; Ryzhik, I. M.:
Table of Integrals, Series, and Products . Seventh edition. Academic Press,San Diego, 2007[Kat] Kato, T.:
Perturbation Theory for Linear Operators . Springer-Verlag, Berlin-Heidelberg-New York, 1995[Lut] Luttinger, J. M.: TheoryofThermalTransportCoeffcients. Phys. Rev. , A1505-A1514 (1964)[OMS] Oldham, K.; Myland, J.; Spanier, J.:
An Atlas of Functions . 2nd edition. Springer, 2009[OLBC] Olver, F.; Lozier, D.; Boisvert, R.; Clark, C.:
NIST Handbook of Mathematical Functions . NIST & Cam-bridge University Press, 2010[RS1] Reed, M.; Simon, B.:
Functional Analysis (Methods of modern mathematical physics I) . Academic Press,New York-London, 1972[RS2] Reed, M.; Simon, B.:
Fourier Analysis. Self-adjointness (Methods of modern mathematical physics II) .Academic Press, New York-London, 1975[RS3] Reed, M.; Simon, B.:
Scattering Theory (Methods of modern mathematical physics III) . Academic Press,New York-London, 1975[RS4] Reed, M.; Simon, B.:
Analysis of Operators (Methods of modern mathematical physics IV) . AcademicPress, New York-London, 1978
ERTURBATION THEORY FOR THE THERMAL HAMILTONIAN: 1D CASE 17 [SS] Smrˇcka, L.; Stˇreda. P.: Transport coefficients in strong magnetic fields. J. Phys. C: Solid State Phys. ,2153-2161 (1977)[VMT] Vafek, O.; Melikyan, A.; Teˇsanovi´c, Z.: Quasiparticle Hall transport of d-wave superconductors in thevortexstate. Phys. Rev. B , 224508 (2001)[Yaf] Yafaev, D. R.: Mathematical scattering theory , volume 105 of
Translations of Mathematical Monographs .AMS, Providence, 1992(G. De Nittis) F
ACULTAD DE M ATEM ´ ATICAS & I
NSTITUTO DE F´ ISICA , P
ONTIFICIA U NIVERSIDAD C AT ´ OLICADE C HILE , S
ANTIAGO , C
HILE . Email address : [email protected] (V. Lenz) D EPARTAMENTO DE M ATEM ´ ATICAS , F
ACULTAD DE C IENCIAS , U
NIVERSIDAD DE C HILE , S
ANTIAGO ,C HILE
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