In a series of recent papers [cf., e.g., V. S. Vladimirov and I. V. Volovich, Commun. Math. Phys. 123, 659-676 (1989; Zbl 0688.22004)], the authors have proposed (and largely elaborated) possible formulations of a quantum mechanics over the field \({\mathbb{Q}}_ p\) of p- adic numbers.
One of their approaches is based upon the study of triples \((L_ 2({\mathbb{Q}}_ p),W(z),U(t))\), where \(L_ 2({\mathbb{Q}}_ p)\) denotes the Hilbert space of quadratically summable complex functions on \({\mathbb{Q}}_ p\), W(z) a family of unitary operators on it, parametrized by the symplectic phase space \({\mathbb{Q}}_ p\times {\mathbb{Q}}_ p\), and U(t) an evolution operator corresponding to a unitary representation of a certain additive subgroup \(G_ p\) of \({\mathbb{Q}}_ p\) in \(L_ 2({\mathbb{Q}}_ p).\)
In an earlier paper [op. cit.], the authors had constructed a particular representation corresponding to U(t), which can be understood as being a model for the p-adic harmonic oscillator.
In the present paper, they continue this approach by studying the spectral properties of the p-adic harmonic oscillator via representation theory. More precisely, they look at the compact abelian subgroup \(G_ p\) of \({\mathbb{Q}}_ p\), which corresponds to the harmonic oscillator U(t) via a representation in \(L_ 2({\mathbb{Q}}_ p)\), and give a complete decomposition of that unitary representation of \(G_ p\) into irreducible ones. This is achieved by describing the characters of \(G_ p\), computing the dimensions of the invariant subspaces, and by explicitly computing the eigenfunctions of the corresponding evolution operator U(t). Concludingly, at the end of the paper, the authors discuss the analogous spectral problem for some p-adic pseudo-differential operators of Schrödinger type. Also in these cases, they obtain complete systems of orthonormal eigenfunctions in \(L_ 2({\mathbb{Q}}_ p).\)
As for a closely related, but different approach to this kind of problem in p-adic quantum mechanics, one should compare this work to a recent paper of Ph. Ruelle, E. Thiran, D. Verstegen, J. Weyers [J. Math. Phys. 30, No.12, 2854-2874 (1989; see the following review Zbl 0709.22011)].