Duong, Anh Tuan Spectral asymptotics for two-dimensional Schrödinger operators with strong magnetic fields. (English) Zbl 1269.81047 Methods Appl. Anal. 19, No. 1, 77-98 (2012). 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 \(b\to\infty\) Cited in 1 Document 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 Keywords:Schrödinger operators; strong magnetic fields; asymptotic trace formula; eigenvalues distribution PDFBibTeX XMLCite \textit{A. T. Duong}, Methods Appl. Anal. 19, No. 1, 77--98 (2012; Zbl 1269.81047) Full Text: DOI