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The Jacobi operator and its Donoghue $m$-functions

Fritz Gesztesy, Mateusz Piorkowski, Jonathan Stanfill

Abstract

In this paper we construct Donoghue $m$-functions for the Jacobi differential operator in $L^2\big((-1,1); (1-x)^α (1+x)^β dx\big)$, associated to the differential expression \begin{align*} \begin{split} τ_{α,β} = - (1-x)^{-α} (1+x)^{-β}(d/dx) \big((1-x)^{α+ 1}(1+x)^{β+ 1}\big) (d/dx),& \\ x \in (-1,1), \; α, β\in \mathbb{R}, \end{split} \end{align*} whenever at least one endpoint, $x=\pm 1$, is in the limit circle case. In doing so, we provide a full treatment of the Jacobi operator's $m$-functions corresponding to coupled boundary conditions whenever both endpoints are in the limit circle case, a topic not covered in the literature.

The Jacobi operator and its Donoghue $m$-functions

Abstract

In this paper we construct Donoghue -functions for the Jacobi differential operator in , associated to the differential expression \begin{align*} \begin{split} τ_{α,β} = - (1-x)^{-α} (1+x)^{-β}(d/dx) \big((1-x)^{α+ 1}(1+x)^{β+ 1}\big) (d/dx),& \\ x \in (-1,1), \; α, β\in \mathbb{R}, \end{split} \end{align*} whenever at least one endpoint, , is in the limit circle case. In doing so, we provide a full treatment of the Jacobi operator's -functions corresponding to coupled boundary conditions whenever both endpoints are in the limit circle case, a topic not covered in the literature.

Paper Structure

This paper contains 10 sections, 8 theorems, 106 equations, 1 table.

Table of Contents

  1. Introduction
  2. Some Background
  3. Donoghue $m$-functions: Two Limit Circle Endpoints
  4. Donoghue $m$-functions: One Limit Circle Endpoint
  5. The Jacobi Operator and its Donoghue $m$-functions
  6. The Regular and Limit Circle Case $\alpha, \beta \in (-1,1)$
  7. Precisely One Interval Endpoint in the Limit Point Case
  8. The Hypergeometric and Jacobi Differential Equations
  9. Connection Formulas
  10. Behavior of $y_{j,\alpha,\beta,\mp 1}(z,x)$, $j=1,2$, near $x=\pm 1$

Key Result

Theorem 2.2

${}$ Assume Hypothesis h2.1. Then the following alternative holds: Either $(i)$ for every $z\in{\mathbb{C}}$, all solutions $u$ of $(\tau-z)u=0$ are in $L^2((a,b);rdx)$ near $b$$($resp., near $a$$)$, or, $(ii)$ for every $z\in{\mathbb{C}}$, there exists at least one solution $u$ of $(\tau-z)u=0$ whi

Theorems & Definitions (14)

  • Theorem 2.2: Weyl's Alternative
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Definition 2.6
  • Theorem 2.7
  • Definition 2.8
  • Theorem 2.9: GLN20
  • Theorem 2.10
  • Theorem 3.2
  • ...and 4 more