High energy asymptotics of the magnetic spectral shift function

Abstract

We consider the three-dimensional Schrödinger operators H0 and H where H0=(i∇+A)2−b, A is a magnetic potential generating a constant magnetic field of strength b>0, and H=H0+V where V∈L1(R3;R) satisfies certain regularity conditions. Then the spectral shift function ξ(E;H,H0) for the pair of operators H, H0 is well-defined for energies E≠2qb, q∈Z+. We study the asymptotic behavior of ξ(E;H,H0) as E→∞, E∈Or, r∈(0,b), where Or≔{E∈(0,∞)|dist(E,2bZ+)>r}. We obtain a Weyl-type formula limE→∞,E∈OrE−1/2ξ(E;H,H0)=(1/4π2) ∫ R3V(x)dx.