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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1992 • Question 12

Question

Math 6410 • ODE Fall 1992 • Question 12

The singular Sturm-Liouville operator \(Lu=-u''\) on \(-\infty <x<\infty \) is known to have a unique selfadjoint realization \(T\) in the Hilbert space \(H=L_2(-\infty ,\infty )\). It is defined by \(D(T)=H\cap \{u:u''\in H\}\) and \(Lu=-u''\) for all \(u\in D(T)\). (Do not prove this). Calculate the Green’s function (resolvent kernel) of \(T\).

Raw LaTeX

The singular Sturm-Liouville operator \(Lu=-u''\) on
\(-\infty<x<\infty\) is known to have a unique selfadjoint realization \(T\) in the
Hilbert space \(H=L_2(-\infty,\infty)\). It is defined by
\(D(T)=H\cap\{u:u''\in H\}\) and \(Lu=-u''\) for all \(u\in D(T)\). (Do not prove this).
Calculate the Green's function (resolvent kernel) of \(T\).

Topic Tags

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