Discrete and Embedded Eigenvalues for One-Dimensional Schr?dinger Operators

摘要

I present an example of a discrete Schr?dinger operator that shows that it is possible to have embedded singular spectrum and, at the same time, discrete eigenvalues that approach the edges of the essential spectrum (much) faster than exponentially. This settles a conjecture of Simon (in the negative). The potential is of von Neumann-Wigner type, with careful navigation around a previously identified borderline situation.