The peak model for finite rank supersingular perturbations

TL;DR

This paper extends the peak model for finite rank supersingular perturbations of self-adjoint operators by removing the restriction on the Gram matrix, clarifying the origin of the Krein Q-function.

Contribution

It generalizes the peak model to include non-diagonal Gram matrices, providing new insights into the Krein Q-function's origin.

Findings

Removed the diagonal restriction on the Gram matrix in the peak model.

Connected the Krein Q-function to the Gram matrix's structure.

Enhanced understanding of supersingular perturbations in operator theory.

Abstract

In its original form the peak model for rank one supersingular perturbations of class $\mathfrak{H}_{-4}$ or higher of a nonnegative self-adjoint operator requires that the Gram matrix of the model should be diagonal. Here we remove the restriction on the Gram matrix. In particular we explain the origin of the Krein $Q$-function associated with the Gram matrix.