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AbstractIn a previous paper (Regular and Chaotic Dynamics7 (2002), 351–391, Ref. [1]), we obtained various results concerning reflectionless Hilbert space transforms arising from a general Cauchy system. Here we extend these results, proving in particular an isometry property conjectured in Ref. [1]. Crucial input for the proof comes from previous work on a special class of relativistic Calogero-Moser systems. Specifically, we exploit results on action-angle maps for the pertinent systems and their relation to the 2D Toda soliton tau-functions. The reflectionless transforms may be viewed as eigenfunction transforms for an algebra of higher-order analytic difference operators.

AbstractIn the previous two parts of this series of papers, we introduced and studied a large class of analytic difference operators admitting reflectionless eigenfunctions, focusing on algebraic and function-theoretic features in the first part, and on connections with solitons in the second one. In this third part we study our difference operators from a quantum mechanical viewpoint. We show in particular that for an arbitrary difference operator Afrom a certain subclass, the reflectionless A-eigenfunctions can be used to construct an unbounded self-adjoint reflectionless operator Âon L2(ℝ, dx), whose action on a suitable core coincides with that of A.

We consider a one-parameter family of Hamilton functions yielding the Newton equation of the harmonic oscillator, x¨+ω2x=0. The parameter may be viewed as the speed of light c, the nonrelativistic limit c→∞ yielding the usual Hamiltonian. For c<∞, the classical Hamiltonians are the product of a function of x and a function of p. In the quantum case, with a suitable ordering, we explicitly find the spectrum and the eigenfunctions of the Hamiltonian.

We show how various known results concerning the Barnes multiple zeta and gamma functions can be obtained as specializations of simple features shared by a quite extensive class of functions. The pertinent functions involve Laplace transforms, and their asymptotics is obtained by exploiting this. We also demonstrate how Barnes' multiple zeta and gamma functions fit into a recently developed theory of minimal solutions to first order analytic difference equations. Both of these new approaches to the Barnes functions give rise to novel integral representations. Copyright 2000 Academic Press.