scholarly journals Analytic Scattering Theory for Jacobi Operators and Bernstein–Szegö Asymptotics of Orthogonal Polynomials

2018 ◽  
Vol 30 (08) ◽  
pp. 1840019 ◽  
Author(s):  
D. R. Yafaev
Keyword(s):  
Jacobi Matrix ◽  
Trace Class ◽  
Spectral Shift ◽  
Shift Function ◽  
Discrete Schrödinger Operator ◽  
Wave Operators ◽  
Jacobi Operators ◽  
Matrix Formula ◽  
Asymptotics Of Orthogonal Polynomials ◽  
Szego Asymptotics

We study semi-infinite Jacobi matrices [Formula: see text] corresponding to trace class perturbations [Formula: see text] of the “free” discrete Schrödinger operator [Formula: see text]. Our goal is to construct various spectral quantities of the operator [Formula: see text], such as the weight function, eigenfunctions of its continuous spectrum, the wave operators for the pair [Formula: see text], [Formula: see text], the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the orthonormal polynomials [Formula: see text] associated to the Jacobi matrix [Formula: see text] as [Formula: see text]. In particular, we consider the case of [Formula: see text] inside the spectrum [Formula: see text] of [Formula: see text] when this asymptotic has an oscillating character of the Bernstein–Szegö type and the case of [Formula: see text] at the end points [Formula: see text].

scholarly journals Trace singularities in obstacle scattering and the Poisson relation for the relative trace

2021 ◽  
Author(s):  
Yan-Long Fang ◽  
Alexander Strohmaier
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Trace Class ◽  
Spectral Shift ◽  
Dirichlet Boundary Conditions ◽  
Shift Function ◽  
Relative Trace Formula ◽  
Casimir Interactions ◽  
Wave Trace ◽  
Obstacle Scattering

AbstractWe consider the case of scattering by several obstacles in $${\mathbb {R}}^d$$ R d , $$d \ge 2$$ d ≥ 2 for the Laplace operator $$\Delta $$ Δ with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators $$\Delta _1$$ Δ 1 and $$\Delta _2$$ Δ 2 obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative operator $$g(\Delta ) - g(\Delta _1) - g(\Delta _2) + g(\Delta _0)$$ g ( Δ ) - g ( Δ 1 ) - g ( Δ 2 ) + g ( Δ 0 ) was introduced in Hanisch, Waters and one of the authors in (A relative trace formula for obstacle scattering. arXiv:2002.07291, 2020) and shown to be trace-class for a large class of functions g, including certain functions of polynomial growth. When g is sufficiently regular at zero and fast decaying at infinity then, by the Birman–Krein formula, this trace can be computed from the relative spectral shift function $$\xi _\mathrm {rel}(\lambda ) = -\frac{1}{\pi } {\text {Im}}(\Xi (\lambda ))$$ ξ rel ( λ ) = - 1 π Im ( Ξ ( λ ) ) , where $$\Xi (\lambda )$$ Ξ ( λ ) is holomorphic in the upper half-plane and fast decaying. In this paper we study the wave-trace contributions to the singularities of the Fourier transform of $$\xi _\mathrm {rel}$$ ξ rel . In particular we prove that $${\hat{\xi }}_\mathrm {rel}$$ ξ ^ rel is real-analytic near zero and we relate the decay of $$\Xi (\lambda )$$ Ξ ( λ ) along the imaginary axis to the first wave-trace invariant of the shortest bouncing ball orbit between the obstacles. The function $$\Xi (\lambda )$$ Ξ ( λ ) is important in the physics of quantum fields as it determines the Casimir interactions between the objects.

scholarly journals On the imaginary part of coupling resonance points

2019 ◽  
Vol 39 (5) ◽  
pp. 611-621 ◽  
Author(s):  
Nurulla Azamov ◽  
Tom Daniels
Keyword(s):  
Imaginary Part ◽  
Trace Class ◽  
Spectral Shift ◽  
Rate Of Change ◽  
Quantum Scattering ◽  
Shift Function ◽  
Spectral Shift Function ◽  
Resonance Point ◽  
Rank One ◽  
Scattering Phase

We prove for rank one perturbations that the imaginary part of a coupling resonance point is inversely proportional by a factor of \(-2\) to the rate of change of the scattering phase, as a function of the coupling variable, evaluated at the real part of the resonance point. This equality is analogous to the Breit-Wigner formula from quantum scattering theory. For more general relatively trace class perturbations, we also give a formula for the spectral shift function in terms of coupling resonance points, non-real and real.

scholarly journals The index formula and the spectral shift function for relatively trace class perturbations

2011 ◽  
Vol 227 (1) ◽  
pp. 319-420 ◽  
Author(s):  
Fritz Gesztesy ◽  
Yuri Latushkin ◽  
Konstantin A. Makarov ◽  
Fedor Sukochev ◽  
Yuri Tomilov
Keyword(s):  
Trace Class ◽  
Spectral Shift ◽  
Index Formula ◽  
Shift Function ◽  
Spectral Shift Function

Resonance index and singular μ-invariant

Analysis ◽  
2020 ◽  
Vol 40 (3) ◽  
pp. 151-161
Author(s):  
Nurulla Azamov ◽  
Tom Daniels
Keyword(s):  
Essential Spectrum ◽  
Coupling Parameter ◽  
Adjoint Operator ◽  
Direct Proof ◽  
Trace Class ◽  
Spectral Shift ◽  
Shift Function ◽  
Spectral Shift Function ◽  
Poles And Zeros ◽  
Open Question

AbstractGiven a self-adjoint operator and a relatively trace class perturbation, one can associate the singular spectral shift function – an integer-valued function on the real line which measures the flow of singular spectrum, not only at points outside of the essential spectrum, where it coincides with the classical notion of spectral flow, but at points within the essential spectrum too. The singular spectral shift function coincides with both the total resonance index and the singular μ-invariant. In this paper we give a direct proof of the equality of the total resonance index and singular μ-invariant assuming only the limiting absorption principle and no condition of trace class type – a context in which the existence of the singular spectral shift function is an open question. The proof is based on an application of the argument principle to the poles and zeros of the analytic continuation of the scattering matrix considered as a function of the coupling parameter.

scholarly journals Kreĭn's trace formula and the spectral shift function

2001 ◽  
Vol 25 (4) ◽  
pp. 239-252 ◽  
Author(s):  
Khristo N. Boyadzhiev
Keyword(s):  
Integral Representation ◽  
Harmonic Functions ◽  
Trace Class ◽  
Spectral Shift ◽  
Shift Function ◽  
Large Set ◽  
Spectral Shift Function ◽  
Selfadjoint Operators ◽  
Upper Half Plane ◽  
Admissible Functions

LetA,Bbe two selfadjoint operators whose differenceB−Ais trace class. Kreĭn proved the existence of a certain functionξ∈L1(ℝ)such thattr[f(B)−f(A)]=∫ℝf′(x)ξ(x)dxfor a large set of functionsf. We give here a new proof of this result and discuss the class of admissible functions. Our proof is based on the integral representation of harmonic functions on the upper half plane and also uses the Baker-Campbell-Hausdorff formula.

scholarly journals RESONANCES AND SPECTRAL SHIFT FUNCTION FOR THE SEMI-CLASSICAL DIRAC OPERATOR

2007 ◽  
Vol 19 (10) ◽  
pp. 1071-1115 ◽  
Author(s):  
ABDALLAH KHOCHMAN
Keyword(s):  
Dirac Operator ◽  
Upper Bound ◽  
Selfadjoint Operator ◽  
Energy Levels ◽  
Critical Energy ◽  
Spectral Shift ◽  
Approximation Formula ◽  
Shift Function ◽  
Spectral Shift Function ◽  
Electro Magnetic

We consider the selfadjoint operator H = H0+ V, where H0is the free semi-classical Dirac operator on ℝ3. We suppose that the smooth matrix-valued potential V = O(〈x〉-δ), δ > 0, has an analytic continuation in a complex sector outside a compact. We define the resonances as the eigenvalues of the non-selfadjoint operator obtained from the Dirac operator H by complex distortions of ℝ3. We establish an upper bound O(h-3) for the number of resonances in any compact domain. For δ > 3, a representation of the derivative of the spectral shift function ξ(λ,h) related to the semi-classical resonances of H and a local trace formula are obtained. In particular, if V is an electro-magnetic potential, we deduce a Weyl-type asymptotics of the spectral shift function. As a by-product, we obtain an upper bound O(h-2) for the number of resonances close to non-critical energy levels in domains of width h and a Breit–Wigner approximation formula for the derivative of the spectral shift function.

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