Spectral shift function for a discretized continuum

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8121/ac4b8c ◽

2022 ◽

Author(s):

Olga Rubtsova ◽

Vladimir N Pomerantsev

Keyword(s):

Scattering Problem ◽

Spectral Shift ◽

Shift Function ◽

Spectral Shift Function ◽

Scattering Problems ◽

Scale Parameters ◽

Densities Of States ◽

Important Object ◽

The Continuum ◽

Square Integrable Basis

Abstract A spectral shift function (SSF) is an important object in the scattering theory which is related both to the spectral density and to the scattering matrix. In the paper, it is shown how to employ the SSF formalism to solve scattering problems when the continuum is discretized, e.g. when solving a scattering problem in a finite volume or in the representation of some finite square-integrable basis. A new algorithm is proposed for reconstructing integrated densities of states and the SSF using a union of discretized spectra corresponding to a set of Gaussian bases with the shifted scale parameters. The examples given show that knowledge of the discretized spectra of the total and asymptotic Hamiltonians is sufficient to find the scattering partial phase shifts at any required energy, as well as the resonances parameters.

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

Reviews in Mathematical Physics ◽

10.1142/s0129055x0700319x ◽

2007 ◽

Vol 19 (10) ◽

pp. 1071-1115 ◽

Cited By ~ 6

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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