Mathematical aspects of quantum mechanics with non-self-adjoint operators

Pavel Exner, Seba, Petr

Natália Bebiano, João da Providência, João P. da Providência

Abstract Non-self-adjoint operators with a real discrete spectrum exhibit, in general, a quite different behavior from the one of self-adjoint operators, with the same spectrum. This fact has serious mathematical and physical implications, namely in the rigorous formulation of quantum mechanics (QM), where self-adjointness plays a central role, in particular in the spectral analysis of the involved operators. In this framework, a non-self-adjoint 2D-harmonic oscillator is considered. Its eigenvalues and eigenfunctions, as well as those of its adjoint, are explicitly determined. Although the eigensystems are complete they may not form bases, a drawback for the mathematical formulation of the QM of the system. The existence of a metric operator defining an inner product which renders the Hamiltonian self-adjoint, is discussed.

Andrea Cintio, Alessandro Michelangeli

César R. de Oliveira

The spectral theory of linear operators plays a key role in the mathematical formulation of quantum theory. Furthermore, such a rigorous mathematical foundation leads to a more profound insight into the nature of quantum mechanics. This textbook provides a concise and comprehensible introduction to the spectral theory of (unbounded) self-adjoint operators and its application in quantum dynamics. The book places emphasis on the symbiotic relationship of these two domains by (1) presenting the basic mathematics of nonrelativistic quantum mechanics of one particle, i.e., developing the spectral theory of self-adjoint operators in infinite-dimensional Hilbert spaces from the beginning, and (2) giving an overview of many of the basic functional aspects of quantum theory, from its physical principles to the mathematical models. The book is intended for graduate (or advanced undergraduate) students and researchers interested in mathematical physics. It starts with linear operator theory, spectral questions and self-adjointness, and ends with the effect of spectral type on the large time behaviour of quantum systems. Many examples and exercises are included that focus on quantum mechanics.

Reinhold A. Bertlmann, Nicolai Friis

Abstract In this chapter we present the mathematical framework of describing wave functions as vectors in a Hilbert space and observables as Hermitian operators on such a vector space. We describe the mathematical formulation of these ideas for finite and infinite-dimensional cases via the Dirac notation, and discuss the example of the Hilbert space of photon polarization. Important classes of linear operators on Hilbert spaces are discussed, including projectors, Hermitian and self-adjoint operators, as well as unitary operators. As an important centre piece for the mathematical language of quantum mechanics we present the spectral theorem. We discuss the subtleties of this framework in infinite dimensions, giving a brief overview of the distributional aspects of quantum mechanics and the conceptually important problem of finding self-adjoint extensions via the Weyl-von Neumann theorem