Eigenvalue bounds for Schrödinger operators with complex potentials

Abstract

Abstract. We prove Lieb-Thirring inequalities for Schrödinger operators with a homogeneous magnetic field in two and three space dimensions. The inequalities bound sums of eigenvalues by a semi-classical approximation which depends on the strength of the magnetic field, and hence quantifies the diamagnetic behavior of the system. For a harmonic oscillator in a homogenous magnetic field, we obtain the sharp constants in the inequalities.

A remark on analytic Fredholm alternative

“…The following result, Theorem 3.1, is a cornerstone of the paper [5]. By {λ j } we always denote the eigenvalues of W = I + T of finite type, repeated accordingly to their algebraic multiplicity.…”

“…The proof of this result is based on the identification of the eigenvalues of finite type of W with the zeros of certain scalar analytic functions, known as the regularized determinants f (λ) := det p (I + T (λ)), see [7,9] for their definition and basic properties. The point is that the set of eigenvalues of finite type of W agrees with the zero set of f , and moreover, ν(λ 0 , W ) = µ f (λ 0 ), the multiplicity of zero of f at λ 0 (see [5,Lemma 3.2] for the rigorous proof). Thereby, the problem is reduced to the study of the zero distributions of certain analytic functions, the latter being a classical topic of complex analysis going back to Jensen [8] and Blaschke [1].…”

“…The goal of this note is to refine partially (for a certain range of parameters) a recent result of R. Frank [5,Theorem 3.1] on some quantitative aspects of the analytic Fredholm alternative. Precisely, the problem concerns the distribution of eigenvalues of finite type of an operator-valued function W (·) = I + T (·), analytic on a domain Ω of the complex plane.…”

“…A key ingredient of the proof in [5] is a result of [2, Theorem 0.2] on the Blaschketype conditions for zeros of analytic functions in the unit disk which can grow at the direction of certain (finite) subsets of the unit circle. In a recent manuscript [3] some new such conditions on zeros of analytic functions in the unit disk and on some other domains, including the complex plane with a cut along the positive semi-axis, are suggested.…”

Trace Class Conditions for Functions of Schrödinger Operators

“…We also note that the bound of the lemma in the case V ≡ 0 is due to [20] (see also [11] for the case d = 2).…”