The spectra of the quadratic Hamiltonians on the twodimensional Euclidean space are determined completely by using the theory of the metaplectic representation. In some cases, the corresponding heat kernels are studied in connection with the well-definedness of the Wiener integrations. A proof of the Lévy formula for the stochastic area and a relation between the real and complex Hermite polynomials are given in our framework.
Hiroyuki MATSUMOTO. Naomasa UEKI. "Applications of the theory of the metaplectic representation to quadratic Hamiltonians on the two-dimensional Euclidean space." J. Math. Soc. Japan 52 (2) 269 - 292, April, 2000. https://doi.org/10.2969/jmsj/05220269