# General solution of a specific eigenvalue problem.

Consider a Schroedinger-like equation with a generalized harmonic potential:

$$\left(\sum_i\mu_i^2\frac{\partial^2}{\partial x_i^2}-\sum_{ij}\Omega_{ij}x_ix_j+E\right)\Psi=0,$$ where indices run from 1 to $N$, $x_i$ are real coordinates, $\mu_i$ are positive real numbers, and $\Omega$ is a positive-definite symmetric real matrix.

Is the following statement valid:

Any eigenvalue of the problem is determined by a $N$-tuple of non-negative integer numbers $(n_1,n_2,\dots,n_N)$ as $$E_{n_1n_2\dots n_N}=\sum_{i=1}^N\left(1+2n_i\right)\omega_i,$$ where $\omega$'s are the square roots of the eigenvalues of the matrix $\tilde\Omega$: $$\tilde\Omega_{ij}=\mu_i\mu_j\Omega_{ij}$$?

If the general solution of the problem is known I would be thankful for corresponding hints and/or references.

• I made a small edit to your post and added a few tags. Hope this is OK. – Robert Lewis Jun 13 '17 at 18:34
• I think I was able to prove the statement, except for showing that $\tilde\Omega$ is positive-definite. I would be thankful for any hint concerning a proof of the latter property. – user Jun 14 '17 at 7:52
• The proof of the positive definiteness has appeared to be trivial. – user Jun 14 '17 at 13:10

## 1 Answer

This is just a bunch of N linear oscillators, a system readily diagonalizable, upon defining real coordinates $$y_i\equiv x_i/\mu_i$$, so you have $$\left(\sum_i \frac{\partial^2}{\partial y_i^2}-\sum_{ij}\tilde \Omega_{ij}y_iy_j+E\right)\Psi=0,$$ with $$\tilde \Omega$$ real, symmetric (symmetrize by the ys) and positive.

As such, it is orthogonally diagonalizable to its real positive eigenvalues' matrix $$\tilde { \tilde \Omega}_{ij}=\delta_{ij}\omega^2_i$$, with eigenvectors $$z_i$$, to wit $$\left(\sum_i ( \frac{\partial^2}{\partial z_i^2}- \omega_{i}^2 z_i^2)+E\right)\Psi=0,$$ N decoupled harmonic oscillators.