Non‐self‐adjoint operators in quantum physics: ideas, people, and trends

Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

“…In this sense, our present study may be perceived as originating from the recent brief account of the recent history of quantum theory [19] in which the possible respective weakening of Postulates {A.2} and {A.3} was still presented as strictly independent.…”

“…Thus, in the former setting of Postulate {A.2}, the self-adjoint nature of the Hamiltonian in the conventional Schrödinger Equation (1) was emphasized, whenever needed, by the replacement of the ambiguous symbol H(t) by its lower-case version h(t) (cf., e.g., Equation (1.5.1) in [19]). Then, a transition was considered to certain generalized, non-self-adjoint Hamiltonian-resembling operators G(t) controlling the evolution of wave kets (for the rigorous definitions and for further details, the readers may consult also review [20]).…”

Parity-Time Symmetry and the Toy Models of Gain-Loss Dynamics near the Real Kato’s Exceptional Points

“…Although, as we already indicated, the trick itself was already well known in mathematics [8,16] as well as in physics [5][6][7]17], the key innovation was that the operator Θ playing the role of an inner-product metric has been shown obtainable, in closed form, for all of the tridiagonal toy-model matrices D (N) (t) of the family [20].…”

“…By the, up to now, more or less purely formal change of representation (4), the main problem of the phenomenological acceptability of the family of quantum toy models of [13] becomes transferred from the description of the spectrum of D (N) (t) (which has been shown trivial in [19]) to the construction of the mapping Ω(t) (which is, usually, enormously difficult in practice [7]). In this sense, a decisive merit of working with toy models of [13] may be seen in the feasibility of such a construction, i.e., of the construction of the related metric Θ(t) and/or of the mappings Ω(t) themselves.…”

“…In parallel, their study inspired also generalizations of mathematically similar theories in several other branches of physics including even the classical mechanics and, first of all, classical optics (also [6]; updated historical and contextual remarks and also many related fresh as well as older references may be found, e.g., in the physics-reviewing chapter [7] and/or in the more mathematics-oriented rest of monograph [8]). …”