Applications of Functional Analysis in Quantum Physics

Dear Colleagues,

Functional analytic methods have been an essential mathematical tool for quantum mechanics since the very first years of the development of quantum theory, starting with the seminal work by von Neumann. Since then, an enormous effort has been made in order to give order and mathematical rigor to quantum physics. Great developments in the theory of operators in Hilbert spaces have helped very much in the understanding of quantum theory through a large variety of models.

We may mention a variety of very successful applications of functional analysis to quantum mechanics, mostly through the theory of operators on Hilbert spaces. In fact, self-adjoint operators and their spectral decomposition have played an important role, since states and observables are traditionally described by self-adjoint operators. This has driven impressive developments like the Kato perturbation theory, the formal scattering theory, the theory of extensions of symmetric non-self-adjoint operators or the renormalization techniques often necessary to properly define point potentials. In addition, we have the outstanding field of group and algebra representations as operators on Hilbert spaces. The above represent just a selection of the applications to date.

The objective of this Special Issue is to foster the extension of investigation in this field. In addition to the traditional fields of research mentioned above, we welcome contributions of algebras of operators in quantum mechanics, as well as quantum field theory, Banach spaces, locally convex spaces, Gelfand triplets, and theory of operators on all these structures with possible applications in quantum mechanics and quantum field theory. Additionally, rigorous mathematical developments of PT symmetries and non-Hermitian quantum mechanics would be most welcome. Finally, other rigorous developments not mentioned here would also be considered, provided that they could be included in this field. This Special Issue will accept high-quality papers including original research results with illustrative applications, as well as survey articles of exceptional merit.

Prof. Dr. Manuel Gadella
Guest Editor

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