Geodesic
Spectral theory in $p$-adic quantum mechanics, and representation theory
Izvestiya. Mathematics , Tome 36 (1991) no. 2, pp. 281-309

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Spectral theory is discussed for a harmonic oscillator in $p$-adic quantum mechanics. The problem of the decomposition into irreducible representations of the restriction of a projective representation of the symplectic group to a compact abelian subgroup is solved, the dimensions of the invariant subspaces are calculated, and the eigenfunctions are analyzed. Spectral problems are considered for $p$-adic pseudodifferential operators of Schrödinger type. A complete orthonormal system of eigenfunctions is constructed for the $p$-adic analog of the differentiation operator. 
@article{IM2_1991_36_2_a3,
     author = {V. S. Vladimirov and I. V. Volovich and E. I. Zelenov},
     title = {Spectral theory in $p$-adic quantum mechanics, and representation theory},
     journal = {Izvestiya. Mathematics },
     pages = {281--309},
     publisher = {mathdoc},
     volume = {36},
     number = {2},
     year = {1991},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1991_36_2_a3/}
}
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V. S. Vladimirov; I. V. Volovich; E. I. Zelenov. Spectral theory in $p$-adic quantum mechanics, and representation theory. Izvestiya. Mathematics , Tome 36 (1991) no. 2, pp. 281-309. http://geodesic.mathdoc.fr/item/IM2_1991_36_2_a3/