991 Energy-dependent point interaction: Self-adjointness F.A.B. Coutinho, Y. Nogami, Lauro Tomio, and F.M. Toyama Abstract: Recently, we constructed an energy-dependent point interaction (EDPI) in its most general form in one-dimensional quantum mechanics. In this paper, we show that stationary solutions of the Schrödinger equation with the EDPI form a complete set. Then any nonstationary solution of the time-dependent Schrödinger equation can be expressed as a linear combination of stationary solutions. This, however, does not necessarily mean that the EDPI is self-adjoint and the time-development of the nonstationary state is unitary. The EDPI is self-adjoint provided that the stationary solutions are all orthogonal to one another. We illustrate situations in which this orthogonality condition is not satisfied. PACS Nos.: 03.65.−w, 03.65.Nk, 03.65.Ge Résumé : Nous avons récemment construit une interaction ponctuelle à dépendance en énergie (EDPI) sous sa forme la plus générale pour un système quantique à une dimension. Nous montrons ici que les solutions de l’équation de Schrödinger avec la EDPI forment un ensemble complet. Dans ce cas, toute solution non stationnaire dépendante du temps de l’équation de Schrödinger peut être exprimée comme une combinaison linéaire des solutions stationnaires. Cependant, ceci ne signifie pas que la EDPI est self-adjointe et que le développement dans le temps des états non stationnaires est unitaire. La EDPI est self-adjoint si les états stationnaires sont tous orthogonaux l’un à l’autre. Nous montrons quelques cas où cette condition n’est pas satisfaite. [Traduit par la Rédaction] 1. Introduction Recently, we constructed an energy-dependent point interaction (EDPI) in its most general form for a one-particle system in one-dimensional quantum mechanics [1]. The EDPI is represented by a boundary condition that depends on the energy of the system under consideration. When we say “energy- dependent”, it is understood that we are applying the boundary condition to a stationary state with a specified energy. The EDPI leads to a unitary S -matrix for the transmission–reflection problem. The boundary condition can also be applied to a nonstationary state provided that the nonstationary state can be expressed as a linear combination of stationary states. This, however, is based on the assumption that Received 26 June 2006. Accepted 28 September 2006. Published on the NRC Research Press Web site at http://cjp.nrc.ca/ on 29 November 2006. F.A.B. Coutinho. Faculdade de Medicina, Universidade de São Paulo, 01246-903, SP, Brazil. Y. Nogami. 1 Department of Physics andAstronomy, McMaster University, Hamilton, ON L8S 4M1, Canada. L. Tomio. Instituto de FísicaTeórica, Universidade Estadual Paulista, 01405-900, São Paulo, SP, Brazil. F.M. Toyama. Department of Information and Communication Sciences, Kyoto Sangyo University, Kyoto 603-8555, Japan. 1 Corresponding author (e-mail: nogami@mcmaster.ca). Can. J. Phys. 84: 991–1005 (2006) doi: 10.1139/P06-086 © 2006 NRC Canada