On-diagonal singularities of the Green functions for Schroedinger operators

Abstract

We investigate the behavior of the Green functions of Schroedinger operators near the diagonal. The only non-trivial cases, where the on-diagonal singularities are non-zero and do not depend on the spectral parameter, are two and three dimensions. In the case of two dimensions, we show that the singularity is independent of both the scalar and the gauge potentials. In dimension three, we obtain conditions for preserving the singularity under perturbations by non-regular potentials. Some examples illustrating dependence of the singularity on general scalar and gauge potentials are presented.

17 pages

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Cited by in corpus (6)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• Boundary relations and generalized resolvents of symmetric operators
• Resolvents of self-adjoint extensions with mixed boundary conditions
• Continuity properties of integral kernels associated with Schroedinger operators on manifolds
• Approximation by point potentials in a magnetic field
• Bounds on the heat kernel of the Schroedinger operator in a random electromagnetic field