On the imaginary part of coupling resonance points

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.

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

Advances in Mathematics ◽

10.1016/j.aim.2011.01.022 ◽

2011 ◽

Vol 227 (1) ◽

pp. 319-420 ◽

Cited By ~ 15

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

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Resonance index and singular μ-invariant

Analysis ◽

10.1515/anly-2019-0053 ◽

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.

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Kreĭn's trace formula and the spectral shift function

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171201004318 ◽

2001 ◽

Vol 25 (4) ◽

pp. 239-252 ◽

Cited By ~ 1

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.

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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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