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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2010 • Question 2

Question

Math 6420 • PDE Spring 2010 • Question 2

Consider linear operator \[ H=-\frac {d^2}{dx^2}, \] acting on \(\psi (x)\in L^2[0,\pi ]\) with periodic boundary conditions \(\psi (0)=\psi (\pi )=0\). Find the eigenvalues and eigenfunctions of \(H\). Show that the eigenvectors of \(H\) corresponding to distinct eigenvalues are orthogonal. Are the eigenvectors of \(H\) complete, and if so, in what sense?

(b) Solve the equation \[ H\psi (x)=x^2, \] representing the solution as an expansion in eigenfunctions of \(H\).

Raw LaTeX

Consider linear operator
\[
  H=-\frac{d^2}{dx^2},
\]
acting on \(\psi(x)\in L^2[0,\pi]\) with periodic boundary conditions
\(\psi(0)=\psi(\pi)=0\). Find the eigenvalues and eigenfunctions of \(H\). Show that the
eigenvectors of \(H\) corresponding to distinct eigenvalues are orthogonal. Are the eigenvectors
of \(H\) complete, and if so, in what sense?

(b) Solve the equation
\[
  H\psi(x)=x^2,
\]
representing the solution as an expansion in eigenfunctions of \(H\).

Topic Tags

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