High energy asymptotics of the magnetic spectral shift function

2004 ◽  
Vol 45 (9) ◽  
pp. 3453-3461 ◽  
Author(s):  
Vincent Bruneau ◽  
Georgi D. Raikov
Keyword(s):  
High Energy ◽  
Spectral Shift ◽  
Shift Function ◽  
Spectral Shift Function

Spectral asymptotics for the Schrödinger operator with a non-decaying potential

2022 ◽  
pp. 1-31
Author(s):  
Mouez Dimassi ◽  
Setsuro Fujiié
Keyword(s):  
Asymptotic Expansion ◽  
High Energy ◽  
Spectral Shift ◽  
Degree Zero ◽  
Shift Function ◽  
Spectral Shift Function ◽  
Energy Limit ◽  
Semiclassical Asymptotics ◽  
High Energy Limit ◽  
Homogeneous Potentials

We study Schrödinger operators H ( h ) = − h 2 Δ + V ( x ) acting in L 2 ( R n ) for non-decaying potentials V. We give a full asymptotic expansion of the spectral shift function for a pair of such operators in the high energy limit. In particular for asymptotically homogeneous potentials W at infinity of degree zero, we also study the semiclassical asymptotics to give a Weyl formula of the spectral shift function above the threshold max W and Mourre estimates in the range of W except at its critical values.

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.

Singular spectral shift function for resolvent comparable operators

2019 ◽  
Vol 292 (9) ◽  
pp. 1911-1930
Author(s):  
Nurulla Azamov ◽  
Tom Daniels
Keyword(s):  
Spectral Shift ◽  
Shift Function ◽  
Spectral Shift Function
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