Geodesic
Eigenfunctions of quadratic Hamiltonians in the Wigner representation
Teoretičeskaâ i matematičeskaâ fizika, Tome 60 (1984) no. 3, pp. 413-422

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Exact solutions of the Schrödinger equation in the Wigner representation are obtained for an arbitrary time-dependent $N$-dimensional quadratic Hamiltonian. It is shown that a complete system of solutions can always be chosen in the form of products of $N$ Laguerre polynomials having arguments that are quadratic integrals of the motion of the corresponding classical problem. The generating function found for the transition probabilities between the Foek states is a multidimensional generalization of Husimi's well-known expression for an oscillator with variable frequency. The motion of a charged particle in a uniform time-dependent electromagnetic field is considered in detail as an example. 
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     author = {\`E. A. Akhundova and V. V. Dodonov and V. I. Man'ko},
     title = {Eigenfunctions of quadratic {Hamiltonians} in the {Wigner} representation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     publisher = {mathdoc},
     volume = {60},
     number = {3},
     year = {1984},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1984_60_3_a7/}
}
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È. A. Akhundova; V. V. Dodonov; V. I. Man'ko. Eigenfunctions of quadratic Hamiltonians in the Wigner representation. Teoretičeskaâ i matematičeskaâ fizika, Tome 60 (1984) no. 3, pp. 413-422. http://geodesic.mathdoc.fr/item/TMF_1984_60_3_a7/