We determine all the potentials V(x) for the Schrόdinger equation (dl + Y(x))φ = k 2 φ such that some family of eigenfunctions φ satisfies a differential equation in the spectral parameter k of the formFor each such V{x) we determine the algebra of all possible operators B and the corresponding functions Θ(x). Table of Contents 0. Introduction 177 1. (adL) m + 1 ((9) = 0 180 2. K(oo) is Finite 187 3. The Rational KdV Potentials 192 4. The Even Case 203 5. The Even Potentials Work Too 213 6. F(oo) = oo is the Airy Case 218 7. Some Illustrative Examples 222
We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation.We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth-and-death processes using orthogonal polynomials on the real line.In perfect analogy with the classical case, the study of QRWs on the set of nonnegative integers can be handled using scalar-valued (Laurent) polynomials and a scalar-valued measure on the circle. In the case of classical or quantum random walks on the integers, one needs to allow for matrix-valued versions of these notions.We show how our tools yield results in the well-known case of the Hadamard walk, but we go beyond this translation-invariant model to analyze examples that are hard to analyze using other methods. More precisely, we consider QRWs on the set of nonnegative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes.The presentation here aims at being self-contained, but we refrain from trying to give an introduction to quantum random walks, a subject well surveyed in the literature we quote. For two excellent reviews, see [1,19]. See also the recent notes [20].
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