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.

  • $\begingroup$ I made a small edit to your post and added a few tags. Hope this is OK. $\endgroup$ – Robert Lewis Jun 13 '17 at 18:34
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    $\begingroup$ 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. $\endgroup$ – user Jun 14 '17 at 7:52
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    $\begingroup$ The proof of the positive definiteness has appeared to be trivial. $\endgroup$ – user Jun 14 '17 at 13:10

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.


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