Trace asymptotics formula for the Schrödinger operators with constant magnetic fields

2013/02/17 by Mouez Dimassi, Dimassi, Mouez, Anh Tuan Duong +1
Mathematics · Physics and Astronomy · #35C20 #35J10 #35P20 #47F05 #81Q10 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.MP #math.SP #msc:35C20 #msc:35J10 #msc:35P20 #msc:47F05 #msc:81Q10

paper · pdf · doi:10.48550/arxiv.1302.4074

21 pages

arxiv created 2013/07/03 · arxiv updated 2013/07/04

Abstract

In this paper, we consider the 2D- Schrödinger operator with constant magnetic field H(V)=(D_x-By)²+D_y²+V_h(x,y), where V tends to zero at infinity and h is a small positive parameter. We will be concerned with two cases: the semi-classical limit regime V_h(x,y)=V(h x,h y), and the large coupling constant limit case V_h(x,y)=h^-δ V(x,y). We obtain a complete asymptotic expansion in powers of h² of \rm tr(Φ(H(V),h)), where Φ(⋅,h)∈ C^∞₀(\mathbb R;\mathbb R). We also give a Weyl type asymptotics formula with optimal remainder estimate of the counting function of eigenvalues of H(V).

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