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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2010 • Question 3

Question

Math 6420 • PDE Fall 2010 • Question 3

  1. Find all the eigenvalues and eigenfunctions for the operator \[ (Lu)(x)=\int _0^1 xyu(y)\,dy. \]
  2. Do these eigenfunctions form a complete set? If so, how do you know (state a relevant theorem) and on what Hilbert space, and if not, why not?

Raw LaTeX

\begin{enumerate}
\item[(a)] Find all the eigenvalues and eigenfunctions for the operator
\[
  (Lu)(x)=\int_0^1 xyu(y)\,dy.
\]

\item[(b)] Do these eigenfunctions form a complete set? If so, how do you know (state a
relevant theorem) and on what Hilbert space, and if not, why not?
\end{enumerate}

Topic Tags

Eigenvalues, Spectral Radius, and Gershgorin Theorem Hilbert Spaces and Orthogonality Linear Programming and Optimization Spectral Theory of Operators