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.