Pseudo-Hermiticity and Theory of Singular Perturbations

Bijan Kumar Bagchi, C. Quesne

F. Cannata, M. V. Ioffe, D. N. Nishnianidze

F. Cannata, M. V. Ioffe, D. N. Nishnianidze

Hafida Moulla, N. Mebarki, Ilhem Leghrib

Abstract The non hermiticity of the Hamiltonian is shown to be generated from the q-deformation of the algebra. As an example the q-deformed harmonic oscillator is investigated within the leading order approximation in the pseudo Hermitian perturbation theory. A generalized form of the metric is constructed. The spectrum as well as eigenstates of the system are derived explicitly.. Some applications and illustrations in q-deformed quantum statistical mechanics are also discussed.

J. čada

Jacob L. Barnett

Pseudo-Hermitian operators generalize the concept of Hermiticity. This class of operators includes the quasi-Hermitian operators, which reformulate quantum theory while retaining real-valued measurement outcomes and unitary time evolution. This thesis is devoted to the study of locality in quasi-Hermitian theory, the symmetries and conserved quantities associated with non-Hermitian operators, and the perturbative features of pseudo-Hermitian matrices. In addition to the presented original research, scholars will appreciate the lengthy introduction to non-Hermitian physics. Local quasi-Hermitian observable algebras are examined. Expectation values of local quasi-Hermitian observables equal expectation values of local Hermitian observables. Thus, quasi-Hermitian theories do not increase the values of nonlocal games set by Hermitian theories. Furthermore, Bell's inequality violations in quasi-Hermitian theories never exceed the Tsirelson bound of Hermitian quantum theory. Exceptional points, which are branch points in the spectrum, are a perturbative feature unique to non-Hermitian operators. Cusp singularities of algebraic curves are related to higher-order exceptional points. To exemplify novelties of non-Hermiticity, one-dimensional lattice models with a pair of non-Hermitian defect potentials with balanced loss and gain, $Δ\pm i γ$, are explored. When the defects are nearest neighbour, the entire spectrum becomes complex when $γ$ is tuned past a second-order exceptional point. When the defects are at the edges of the chain and the hopping amplitudes are 2-periodic, as in the Su-Schrieffer-Heeger chain, the $\mathcal{PT}$-phase transition is dictated by the topological phase. Chiral symmetry and representation theory are used to derive large classes of pseudo-Hermitian operators with closed-form intertwining operators.