On eigenfunction expansions of differential equations with degenerating weight

TL;DR

This paper develops a comprehensive framework for eigenfunction expansions of differential equations with degenerating weights, characterizing spectral properties and boundary conditions, and providing explicit methods for eigenfunction calculation and convergence analysis.

Contribution

It introduces a novel parameterization of boundary extensions using Nevanlinna functions and extends eigenfunction expansion theory to equations with degenerating weights and $ ext{l}$-dependent boundary conditions.

Findings

Eigenvalues form an infinite, real set without finite accumulation points.

Eigenfunction expansions converge in the $L^2$ space.

Explicit methods for calculating eigenfunctions are provided.

Abstract

Let $A$ be a symmetric operator. By using the method of boundary triplets we parameterize in terms of a Nevanlinna parameter $\tau$ all exit space extensions $\wt A=\wt A^*$ of $A$ with the discrete spectrum $\s(\wt A)$ and characterize the Shtraus family of $\wt A$ in terms of abstract boundary conditions. Next we apply these results to the eigenvalue problem for the $2r$-th order differential equation $l[y]= \l \D(x)y$ on an interval $[a,b), \; -\infty <a<b\leq \infty,$ subject to $\l$-depending separated boundary conditions with entire operator-functions $C_0(\l) $ and $C_1(\l)$, which form a Nevanlinna pair $(C_0,C_1)$. The weight $\D(x)$ is nonnegative and may vanish on some intervals $(\a,\b)\subset \cI$. We show that in the case when the minimal operator of the equation has the discrete spectrum (in particular, in the case of the quasiregular equation) the set of eigenvalues of the eigenvalue problem is an infinite subset of $\bR$ without finite limit points and each function $y\in\LI$ admits the eigenfunction expansion $y(x)=\sum_{k=1}^\infty y_k (x)$ converging in $\LI$. Moreover, we give an explicit method for calculation of eigenfunctions $y_k$ in this expansion and specify boundary conditions on $y$ implying the uniform convergence of the eigenfunction expansion of y. These results develop the known ones obtained for the case of the positive weight $\D$ and the more restrictive class of $\l$-depending boundary conditions.