# Eigenvalues & Eigenvector of infinite dimensional tridiagonal matrix

I am trying to find the eigenvalues of an infinite dimensional tridiagonal hermitian matrix. I need this because I want to calculate the evolution of the superposition coefficients of a harmonic oscillator perturbed under some potential.

I know that for a finite dimensional tridiagonal hermitian matrix you get sines. But how does it generalize to an infinite dimensional one?

thanks

• Can you be more explicit about the matrix? Is it infinite in one or both directions? If you write $(Ax)_n = a_n x_{n-1} + b_n x_n + c_n x_{n+1}$ you may be able to solve it with a Z transform, if the form of the entries is simple enough. – arkeet Oct 15 '16 at 2:00
• Thanks with the idea with the z transform. I can see roughly how it will work, but not completely. could you please expand on that idea? The matrix is basically square and has infinite number of rows and columns. – onephys Oct 16 '16 at 10:13