Counting negative eigenvalues of one-dimensional Schrodinger operators with singular potentials

Abstract

In this paper, we extend the well known estimates for the number of negative eigenvalues of one-dimensional Schrodinger operators with potentials that are absolutely continuous with respect to the Lebesgue measure to the case of strongly singular potentials.

Eigenvalue estimates for magnetic Schrodinger operators in a waveguide

“…dx, where c is a constant that depends on the dimension n. For n = 1, the analogue of the CLR inquality exists for potentials that are monotone on R + or R − (see, e.g., [5]), otherwise, the operator H V has at least one negative eigenvalue, (see, e.g., [15] and the references therein). In the critical case n = 2, the CLR inequality fails, because if norms and Orlciz norms of the potential ( see, e.g.…”