arXiv subjects

Quinn Wicks

Publications and source records attributed to Quinn Wicks.

3 recordsLinked to original sources

arXiv2026

A New Characterization of the Domains of Integral Powers of the Self-Adjoint Friedrichs-Legendre Operator

Let $A$ be the self-adjoint operator in $L^{2}(-1,1)$, generated by the second-order classical Legendre differential equation% \[ \ell\lbrack y](t)=-\left( (1-t^{2})y^{\prime}(t)\right) ^{\prime}+ky(t)=\lambda y(t)\quad(t\in(-1,1)), \] which has the Legendre polynomials $\{P_{m}\}_{m=0}^{\infty}$ as a complete sequence of eigenfunctions; here $k$ is a fixed, non-negative real number. This is the Friedrichs extension of the minimal operator associated with $\ell[\cdot]$ in $L^2(-1,1)$. For each $n \in \mathbb{N}$, we show that $\mathcal{D}(A^{n})$ is characterized by \textit{one} integrability condition instead of $2n$ boundary conditions as dictated by the classical Glazman-Krein-Naimark theory. We also prove that if $f\in\mathcal{D}(A^{n})$ then $f^{(n)}\in L^{2}(-1,1).$ This smoothness result extends known results when $n=1$ and $n=2.$ Furthermore, this result is optimal in the sense that there exists $g\in\mathcal{D}(A^{n})$ with $g^{(n+1)}\notin L^{2}(-1,1)$.

math.FA↗

arXiv2015

Glazman-Krein-Naimark Theory, Left-Definite Theory and the Square of the Legendre Polynomials Differential Operator

As an application of a general left-definite spectral theory, Everitt, Littlejohn and Wellman, in 2002, developed the left-definite theory associated with the classical Legendre self-adjoint second-order differential operator $A$ in $L^{2}(-1,1)$ which has the Legendre polynomials $\{P_{n}% \}_{n=0}^{\infty}$ as eigenfunctions. As a consequence, they explicitly determined the domain $\mathcal{D}(A^{2})$ of the self-adjoint operator $A^{2}.$ However, this domain, in their characterization, does not contain boundary conditions. In fact, this is a general feature of the left-definite approach developed by Littlejohn and Wellman. Yet, the square of the second-order Legendre expression is in the limit-4 case at each end point $x=\pm1$ in $L^{2}(-1,1)$ so $\mathcal{D}(A^{2})$ should exhibit four boundary conditions. In this paper, we show that this domain can, in fact, be expressed using four separated boundary conditions using the classical GKN (Glazman-Krein-Naimark) theory. In addition, we determine a new characterization of $\mathcal{D}(A^{2})$ that involves four \textit{non-GKN} boundary conditions. These new boundary conditions are surprisingly simple - and natural - and are equivalent to the boundary conditions obtained from the GKN theory.

math.SP↗

arXiv2014

A Spectral Study of the Second-Order Exceptional $X_1$-Jacobi Differential Expression and a Related Non-classical Jacobi Differential Expression

The exceptional $X_{1}$-Jacobi differential expression is a second-order ordinary differential expression with rational coefficients; it was discovered by Gómez-Ullate, Kamran and Milson in 2009. In their work, they showed that there is a sequence of polynomial eigenfunctions $\left\{\widehat{P} _{n}^{(α,β)}\right\}_{n=1}^{\infty}$ called the exceptional $X_{1}$-Jacobi polynomials. There is no exceptional $X_{1}$-Jacobi polynomial of degree zero. These polynomials form a complete orthogonal set in the weighted Hilbert space $L^{2}((-1,1);\widehat{w}_{α,β}),$ where $\widehat{w}_{α,β}$ is a positive rational weight function related to the classical Jacobi weight. Among other conditions placed on the parameters $α$ and $β,$ it is required that $α,β>0.$ In this paper, we develop the spectral theory of this expression in $L^{2}((-1,1);\widehat{w}_{α,β})$. We also consider the spectral analysis of the `extreme' non-exceptional case, namely when $α=0$. In this case, the polynomial solutions are the non-classical Jacobi polynomials $\left\{ P_{n}^{(-2,β)}\right\} _{n=2}^{\infty}.$ We study the corresponding Jacobi differential expression in several Hilbert spaces, including their natural $L^{2}$ setting and a certain Sobolev space $S$ where the full sequence $\left\{ P_{n}^{(-2,β)}\right\} _{n=0}^{\infty}$ is studied and a careful spectral analysis of the Jacobi expression is carried out.

math.CA↗