Finite deficiency indices and uniform remainder in Weyl's law

TL;DR

This paper proves that for symmetric operators with finite deficiency indices, the spectral counting functions of different self-adjoint extensions differ by a bounded amount, with applications to quantum graphs, pseudo-laplacians, and conical surfaces.

Contribution

It establishes a uniform bound on the difference of spectral counting functions for self-adjoint extensions with finite deficiency indices, extending spectral theory results.

Findings

Bounded difference in spectral counting functions for extensions

Application to quantum graphs and surfaces with singularities

Extension of classical spectral theory results

Abstract

We give a proof that in settings where Von Neumann deficiency indices are finite the spectral counting functions of two different self-adjoint extensions of the same symmetric operator differ by a uniformly bounded term (see also Birman-Solomjak's 'Spectral Theory of Self-adjoint operators in Hilbert Space') >. We apply this result to quantum graphs, pseudo-laplacians and surfaces with conical singularities.