An upper bound for the number of eigenvalues of a non-self-adjoint Schröbinger operator

Abstract

Estimates for the total multiplicity of eigenvalues for Schrödinger operator are established in the case of compactly supported or exponentially decreasing complex-valued potential.

On the number of eigenvalues of Schrödinger operators with complex potentials

“…In dimensions one and three the resolvent kernel is explicit and this is important for the proofs in [31,32]. In contrast, our Theorem 1.2 is valid in arbitrary odd dimensions d ≥ 3, where the resolvent kernel is only given in terms of Bessel functions, which become increasingly more complicated as the dimension increases.…”

“…Despite this activity, there have been almost no results on the number of eigenvalues of Schrödinger operators with complex potentials. (The only exceptions are two papers [31,32] in one and three dimensions, whose relation to our work we discuss below.) To see that this is a subtle question we recall that, for instance, for the Schrödinger operator −d 2 /dx 2 + V on the half-line with a Dirichlet boundary condition at the origin, Bargmann's bound states that for real potentials V the number of eigenvalues can be bounded by |x||V (x)| dx.…”

Complete Wave Operators in Nonselfadjoint Kato Model of Smooth Perturbation Theory

“…In conclusion we remark that Kato sufficient condition which ensures similarity of operators L and T (see Section 1) is optimal in the sense that among the potentials V (x) for which first momentum exceeds the critical value 1 there exist (see [10]) such that σ d (L) = ∅. The estimates for the total number of eigenvalues and spectral singularities of Schroedinger operator with complex potential were obtained by the author in [13] and [18].…”

Spectral Properties of an Energy-Dependent Hamiltonian

“…Concerning the 3D non-selfadjoint case let us quote related papers by Kato [18,19], Mizohata and Mochizuki [26], Iwasaki [17], Mochizuki [27,28], Mochizuki and Motai [29], Ikebe [15], Ikebe and Saito [16], Saito [46]. Bounds on the discrete spectrum are proposed by Abramov, Aslanyan and Davies [1], Frank, Laptev, Lieb and Seiringer [11], Laptev and Safronov [23], Wang [49,50], Safronov [44], Stepin [47,48], Enblom [6], Frank [9,10,12]. For energy-dependent potentials let us mention among few studies concerning specific types of energy dependence, the case U 0 (x) = 0,U(x,ρ) = −iρW(x) studied by Nakazawa [32], and various works corresponding in fact to a damped equation in the space-time domain (see for instance Iwazaki [17], Mochizuki [28]).…”