Mathematics Qualifying Exam Question Bank

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Question

Math 6410 • ODE

Fall 1995 • Question 12

Question

Math 6410 • ODE Fall 1995 • Question 12

Let \(S\) be the selfadjoint realization in the Hilbert space \(\mathcal {H}=L_2(0,\infty )\) for the singular Sturm-Liouville operator \(\mathcal {L}=-u''\) on \(0<x<\infty \) with boundary condition \(u'(0)=0\). Assume that \(S\) is defined by the domain \[ D(S)=\{u\in \mathcal {H}:u''\in \mathcal {H}\text { and }u'(0)=0\} \] and the relation \(Su=-u''\) for all \(u\in D(S)\).

Show that the Green’s function \(G\) for the operator \(S\) is given by the equation below, where \(\zeta =re^{i\theta }\) and \(\sqrt {\zeta }=\sqrt {r}e^{i\theta /2}\) on \(0<\theta <2\pi \): \[ G(x,y,\zeta )= \begin {cases} i e^{i\sqrt {\zeta }y}\cos (\sqrt {\zeta }x)/\sqrt {\zeta }, & 0\le x\le y<\infty ,\\ i e^{i\sqrt {\zeta }x}\cos (\sqrt {\zeta }y)/\sqrt {\zeta }, & 0\le y\le x<\infty . \end {cases} \]
Prove that the operator \(S\) has a pure continuous spectrum consisting of all non-negative real numbers.

Let \(S\) be the selfadjoint realization in the Hilbert space
\(\calH=L_2(0,\infty)\) for the singular Sturm-Liouville operator \(\calL=-u''\) on
\(0<x<\infty\) with boundary condition \(u'(0)=0\). Assume that \(S\) is defined by the
domain
\[
D(S)=\{u\in\calH:u''\in\calH\text{ and }u'(0)=0\}
\]
and the relation \(Su=-u''\) for all \(u\in D(S)\).

\begin{enumerate}
\item[(a)] Show that the Green's function \(G\) for the operator \(S\) is given by the
equation below, where \(\zeta=re^{i\theta}\) and
\(\sqrt{\zeta}=\sqrt{r}e^{i\theta/2}\) on \(0<\theta<2\pi\):
\[
G(x,y,\zeta)=
\begin{cases}
i e^{i\sqrt{\zeta}y}\cos(\sqrt{\zeta}x)/\sqrt{\zeta}, & 0\le x\le y<\infty,\\
i e^{i\sqrt{\zeta}x}\cos(\sqrt{\zeta}y)/\sqrt{\zeta}, & 0\le y\le x<\infty.
\end{cases}
\]

\item[(b)] Prove that the operator \(S\) has a pure continuous spectrum consisting of
all non-negative real numbers.
\end{enumerate}

Topic Tags

Adjoint Operators and Riesz Representation Boundary Value Problems and Sturm-Liouville Theory Fundamental Solutions Hilbert Spaces and Orthogonality