scholarly journals SPECTRAL SHIFT FUNCTION OF THE SCHRÖDINGER OPERATOR IN THE LARGE COUPLING CONSTANT LIMIT. II. POSITIVE PERTURBATIONS

2002 ◽  
Vol 27 (7-8) ◽  
pp. 1373-1405 ◽  
Author(s):  
A. Pushnitski ◽  
M. Ruzhansky
Keyword(s):  
Schrödinger Operator ◽  
Spectral Shift ◽  
Shift Function ◽  
Schrodinger Operator ◽  
Spectral Shift Function ◽  
Coupling Constant ◽  
Large Coupling ◽  
Large Coupling Constant

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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