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We consider a magnetic Schrödinger operator H = (−ih∇− ~ A) with the Dirichlet boundary conditions in an open set Ω ⊂ R, where h > 0 is a small parameter. We suppose that the minimal value b0 of the module | ~ B| of the vector magnetic field ~ B is strictly positive, and there exists a unique minimum point of | ~ B|, which is non-degenerate. The main result of the paper is upper estimates for the low-lying eigenvalues of the operator H in the semiclassical limit. We also prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting. 1. Preliminaries and main results 1.1. Main assumptions. We would like to analyze the asymptotic behavior, in the semiclassical regime, of the low-lying eigenvalues of the Dirichlet realization of the magnetic Schrödinger operator in an open set Ω in R: H = (hDX1 −A1(X)) + (hDX2 −A2(X)) + (hDX3 −A3(X)) , where ~ A = (A1, A2, A3) ∈ C(Ω̄,R) is a magnetic potential and h > 0 is a small parameter. We will denote the coordinates in R as X = (X1, X2, X3) = (x, y, z) . Let ~ B = rot ~ A = (B1, B2, B3) be the corresponding vector magnetic field: B1 = ∂A3 ∂y − ∂A2 ∂z , B2 = ∂A1 ∂z − ∂A3 ∂x , B3 = ∂A2 ∂x − ∂A1 ∂y . Put b0 = min{| ~ B(X)| : X ∈ Ω}. We assume that there exist a (connected) bounded domain Ω1 ⊂⊂ Ω and a constant ǫ0 > 0 such that (1.1) | ~ B(X)| ≥ b0 + ǫ0, x 6∈ Ω1 . We also assume that: (1.2) b0 > 0 , and that there exists a unique minimum X0 ∈ Ω such that | ~ B(X0)| = b0, which is non-degenerate: in some neighborhood of X0 (1.3) C|X −X0| ≤ | ~ B(X)| − b0 ≤ C|X −X0| . 2000 Mathematics Subject Classification. 35P20, 35J10, 47F05, 81Q10.

Eigenvalue estimates for a three-dimensional magnetic Schrödinger operator