Cubic String Boundary Value Problems and Cauchy Biorthogonal Polynomials

TL;DR

This paper explores cubic string boundary value problems linked to Cauchy biorthogonal polynomials, establishing a Fourier transform framework that connects these polynomials with orthogonal systems in L^2 spaces, relevant to integrable PDEs and random matrix models.

Contribution

It introduces a generalized class of cubic string boundary value problems with discrete inhomogeneity sources and develops a Fourier transform approach to relate biorthogonal polynomials to orthogonal systems.

Findings

Established a Parseval type identity for these boundary value problems

Connected Cauchy biorthogonal polynomials with orthogonal systems in L^2_g

Extended the cubic string problem beyond the original formulation

Abstract

Cauchy Biorthogonal Polynomials appear in the study of special solutions to the dispersive nonlinear partial differential equation called the Degasperis-Procesi (DP) equation, as well as in certain two-matrix random matrix models. Another context in which such biorthogonal polynomials play a role is the cubic string; a third order ODE boundary value problem -f'''=zg f which is a generalization of the inhomogeneous string problem studied by M.G. Krein. A general class of such boundary value problems going beyond the original cubic string problem associated with the DP equation is discussed under the assumption that the source of inhomogeneity g is a discrete measure. It is shown that by a suitable choice of a generalized Fourier transform associated to these boundary value problems one can establish a Parseval type identity which aligns Cauchy biorthogonal polynomials with certain natural orthogonal systems on L^2_g.