Cauchy Problem of the non-self-adjoint Gauss-Laguerre semigroups and uniform bounds of generalized Laguerre polynomials

TL;DR

This paper develops a new spectral expansion approach for non-self-adjoint Gauss-Laguerre semigroups, providing regularity results and uniform bounds for generalized Laguerre polynomials, advancing understanding of non-self-adjoint Markov processes.

Contribution

It introduces a novel eigenvalue expansion method for non-self-adjoint Laguerre-type semigroups using intertwining relations and non-harmonic analysis techniques.

Findings

Established intertwining relations between semigroups.

Derived uniform asymptotic bounds for generalized Laguerre polynomials.

Provided regularity properties for semigroups and heat kernels.

Abstract

We propose a new approach to construct the eigenvalue expansion in a weighted Hilbert space of the solution to the Cauchy problem associated to Gauss-Laguerre invariant Markov semigroups that we introduce. Their generators turn out to be natural non-self-adjoint and non-local generalizations of the Laguerre differential operator. Our methods rely on intertwining relations that we establish between these semigroups and the classical Laguerre semigroup and combine with techniques based on non-harmonic analysis. As a by-product we also provide regularity properties for the semigroups as well as for their heat kernels. The biorthogonal sequences that appear in their eigenvalue expansion can be expressed in terms of sequences of polynomials, and they generalize the Laguerre polynomials. By means of a delicate saddle point method, we derive uniform asymptotic bounds that allow us to get an upper bound for their norms in weighted Hilbert spaces. We believe that this work opens a way to construct spectral expansions for more general non-self-adjoint Markov semigroups.