2009/01/14 by Abdallah Khochman, Khochman, Abdallah
Mathematics · #35J10 #35P25 #47F05 #81Q10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.AP #math.SP #msc:35J10 #msc:35P25 #msc:47F05 #msc:81Q10
paper · pdf · doi:10.48550/arxiv.0901.1980
18 pages, 1 figure
arxiv created 2009/01/14 · openalex publication_date 2009/01/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the 3D Schrödinger operator H0 with constant magnetic field and subject to an electric potential v0 depending only on the variable along the magnetic field x3. The operator H0 has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb H0 by smooth scalar potentials V=O((x1,x2)>-\de_⊥x3>-\de_∥), \de_⊥>2, \de_∥>1. We assume also that V and v0 have an analytic continuation, in the magnetic field direction, in a complex sector outside a compact set. We define the resonances of H=H0+V as the eigenvalues of the non-selfadjoint operator obtained from H by analytic distortions of \Rx3. We study their distribution near any fixed real eigenvalue of H0, 2bq+\la for q∈\N. In a ring centered at 2bq+\la with radiuses (r,2r), we establish an upper bound, as r tends to 0, of the number of resonances. This upper bound depends on the decay of V at infinity only in the directions (x1,x2). Finally, we deduce a representation of the derivative of the spectral shift function (SSF) for the operator pair (H0,H) in terms of resonances. This representation justifies the Breit-Wigner approximation and implies a local trace formula.