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Spectral asymptotics for two-dimensional Schrödinger operators with strong magnetic fields. (English) Zbl 1269.81047

Summary: In this paper, we study the perturbed quadratic Hamiltonian in two-dimensional case, \(P(b,\omega)=(D_x-by)^2+D^2_y+\omega^2x^2-\sqrt{b^2+\omega^2}+V(x,y)\). Here, b is the strong constant magnetic field, \(\omega\neq 0\) is a fixed constant, and the potential \(V\) vanishes at infinity. For \(f\in C^\infty_0 ((-\infty,0)\mathbb R)\) and \(b\) large enough, we give a full asymptotic expansion in powers of \(b^{-1}\) of the trace of \(f(P(b,\omega))\). Moreover, we also obtain a Weyl formula with optimal remainder estimate of the counting function of eigenvalues of \(P(b,\omega)\) as \(b\to\infty\)

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35C20 Asymptotic expansions of solutions to PDEs
35J10 Schrödinger operator, Schrödinger equation
35P20 Asymptotic distributions of eigenvalues in context of PDEs
47F05 General theory of partial differential operators
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