Eigenvalue Asymptotics for a Schrödinger Operator with Non-Constant Magnetic Field Along One Direction

Abstract

We consider the discrete spectrum of the two-dimensional Hamiltonian H = H0 + V , where H0 is a Schrödinger operator with a non-constant magnetic field B that depends only on one of the spatial variables, and V is an electric potential that decays at infinity. We study the accumulation rate of the eigenvalues of H in the gaps of its essential spectrum. First, under certain general conditions on B and V , we introduce effective Hamiltonians that govern the main asymptotic term of the eigenvalue counting function. Further, we use the effective Hamiltonians to find the asymptotic behavior of the eigenvalues in the case where the potential V is a power-like decaying function and in the case where it is a compactly supported function, showing a semiclassical behavior of the eigenvalues in the first case and a non-semiclassical behavior in the second one. We also provide a criterion for the finiteness of the number of eigenvalues in the gaps of the essential spectrum of H.

Threshold singularities of the spectral shift function for a half-plane magnetic Hamiltonian

“…Let W ≥ 0 be in S m 0 and supp W ⊂ [K 0 , ∞) × R for some K 0 ∈ R. Let r > 0 and r ± > 0 such that r − < r < r + . Then for any ν > 0n + (r + λ, W j ) + o(λ − 1 m λ −ν ) Q j (W ) 1 (̺j (λ−ǫ λ ),∞) (·) |E j (·) − E j − λ| −1/2 ≤ n + (r − λ, W j ) + o(ǫ − 1 m λ λ −ν ), λ ↓ 0,(4 26). …”

Magnetic fields and boundary conditions in spectral and asymptotic analysis

“…In this section we are concerned about extensions of the results from [130,131,102], counting the number of eigenvalues in the gap of the bands of the spectrum of the Iwatsuka model B1. In [102], an effective operator of the same kind as in [25] appears:…”

“…On the contrary, if V is compactly supported, then, as in [25], the counting function of eigenvalues is not related to a volume in phase space, see [102,Corollary 2.3] for the precise result. In particular, even a compactly supported potential can give rise to an infinite number of eigenvalues.…”

Spectrum of the Iwatsuka Hamiltonian at thresholds

“…However, discrete eigenvalues may appear. Counting the number of such eigenvalues has been considered for b constant and non-constant (for the Landau case, b " constant, see for instance [27,28,26,29], and for Iwatsuka [31,11,23]). In this article we study an extension of this problem to the continuous spectrum of H, namely, we describe the properties of the spectral shift function (SSF) for the operator pair pH, H 0 q.…”

“…The approach we use here to study the SSF comes somehow from [27], where the discrete spectrum of the Landau Hamiltonian was described. This approach was also used to analyze the discrete spectrum of the Iwatsuka Hamiltonian in [23]. However, in the aforementioned articles it was not necessary to make a detailed analysis of the corresponding band functions; they are constants in the Landau case and some variational estimates are enough for the Iwatsuka case.…”