A more or less elementary exercise in (or, one could say, a
rediscovery of) the Bogoliubov transformation.

MR2402967 Xu, Shi-Min; Xu, Xing-Lei; Li, Hong-Qi Research on the
dynamic problems of 3D cross coupling quantum harmonic oscillator
by virtue of intermediate representation $\vert x\rangle\sb
{\lambda,\nu}$. Internat. J. Theoret. Phys. 47 (2008), no. 6,
1654--1662. 81R05 (81S05)