# Degeneracy and the Hamiltonian

How many linearly independent eigenfunctions can be associated with one degenerate eigenvalue of the Hamiltonian operator? (Is there a limit since it contains a 2nd order differential operator?) Thanks.

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Do you mean a Hamiltonian in general, or, as your question seems to imply, a Hamiltonian of the form $H = P^2/2m + V(X)$? –  Gerben Oct 17 '11 at 12:00

If the Hamiltonian is the zero operator, it's the entire Hilbert space.

A free charged particle in a uniform background magnetic field will have Landau levels with infinite degeneracy.

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Why would the Hamiltonian be the zero operator? –  tetra Oct 16 '11 at 20:31
Why wouldn't it? All you asked was if there's a limit to the number of eigenfunctions that can be associated with a given degenerate eigenvalue. There is not, as the clear example of a constant hamiltonian indicates. –  wsc Oct 16 '11 at 22:31