Some mathematical aspects of the new mechanics

Hamza Fawzi, Omar Fawzi, Aram W. Harrow +1 more

Quantum information theory seeks to understand the fundamental limits set by quantum mechanics for information processing tasks. The mathematical aspects of quantum information rely on tools from various fields including mathematical optimization, high-dimensional convex geometry, operator algebras and representation theory. The goal of this meeting is to focus on the mathematical aspects connecting geometry, optimization and quantum information theory and develop new tools to solve some of the open problems at the intersection of these fields.

G.C. Stey

T. E. Phipps

Hervé Bergeron, Petr Siegl, A Youssef

In a recent short note (Bergeron et al 2010 Europhys. Lett. 92 60003), we have presented the good properties of a new family of semi-classical states for Pöschl–Teller potentials. These states are built from a supersymmetric quantum mechanics (SUSYQM) approach and the parameters of these 'coherent' states are points in the classical phase space. In this paper, we develop all the mathematical aspects that have been left out of the previous paper (proof of the resolution of unity, detailed calculations of the quantized version of classical observables and mathematical study of the resulting operators: problems of domains, self-adjointness or self-adjoint extensions). Some additional questions such as asymptotic behavior are also studied. Moreover, the framework is extended to a larger class of Pöschl–Teller potentials. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.

Abstract Our study of the infinite well family of potentials has provided us with an array of insights into the physical meaning and mathematical structure of wave mechan- ics. There are other quantum mechanical properties which do not appear in such systems, and in this chapter we study several model potentials which illustrate many new aspects, both formal and intuitive, which have wide applications.

J.G. Krüger, Dirk K. Callebaut