Classical and quantum mechanical systems of Toda-lattice type. III: Joint eigenfunctions of the quantized systems
In the first paper of the series to which the present one belongs [the authors, ibid. 83, 355-386 (1982; Zbl 0503.22013); part II, see ibid. 94, 177-217 (1984; Zbl 0592.58028)] it was shown that the Hamiltonian \(H=\Delta -V\) of a quantized one-dimensional system of Toda lattice type is part of a family \((L_ i)_{1\leq i\leq \ell}\) of mutually commuting differential operators on \(C^{\infty}({\mathbb{R}}^{\ell})\). The top-order symbols of these operators \(L_ i\) are a set of functionally independent polynomials that generate the polynomial invariants for the Weyl group W of the finite root system of rank \(\ell\) associated with the potential V. The main purpose of this paper is the study of the spaces of joint eigenfunctions for the family \((L_ i)\). It is proven that all these eigenspaces have dimension \(| W|\), the order of W. If we have a periodic Toda lattice with n particles, then this dimension is n! and the Hamiltonian has only bound states. The joint eigenfunctions are shown to be holomorphic functions on \({\mathbb{C}}^{\ell}\) which can be obtained (in principle) by integrating a system of ordinary differential equations. Moreover, explicit expansions for the joint eigenfunctions are given.
scientific article; zbMATH DE number 3998112
scientific article