Generalized tight p-frames and spectral bounds for Laplace-like operators

TL;DR

This paper establishes sharp spectral bounds for Laplace-like operators using generalized tight p-frames, leveraging domain symmetries to avoid complex variational test functions, especially for fractional Laplacians.

Contribution

It introduces a generalization of tight p-frames to derive geometric spectral bounds for Laplace-like operators on symmetric domains.

Findings

Sharp upper bounds for eigenvalues of Laplace-like operators.

Maximal spectral functionals on original symmetric domains.

Generalized tight p-frames derived from conjugation of matrices.

Abstract

We prove sharp upper bounds for sums of eigenvalues (and other spectral functionals) of Laplace-like operators, including bi-Laplacian and fractional Laplacian. We show that among linear images of a highly symmetric domain, our spectral functionals are maximal on the original domain. We exploit the symmetries of the domain, and the operator, avoiding necessity of finding good test functions for variational problems. This is especially important for fractional Laplacian, since exact solutions are not even known on intervals, making it hard to find good test functions. To achieve our goals we generalize tight $p$-fusion frames, to extract the best possible geometric results for domains with isometry groups admitting tight $p$-frames. Any such group generates a tight $p$-fusion frame via conjugation of arbitrary projection matrix. We show that generalized tight $p$-frames can also be obtained by conjugation of arbitrary rectangular matrix, with frame constant depending on the singular values of the matrix.