Quantum particle under dynamical confinement: from quantum fermi acceleration to high harmonic generation. - In: Physica scripta, ISSN 1402-4896, Bd. 99 (2024), 7, 075308, S. 1-13
Quantum dynamics of a particle confined in a box with time-dependent wall is revisited by considering some unexplored aspects of the problem. In particular, the case of dynamical confinement in a time-dependent box in the presence of purely time-varying external potential is treated by obtaining exact solution. Also, some external potentials approving separation of space and time variables in the Schrödinger equation with time-dependent boundary conditions are classified. Time-dependence of the average kinetic energy and average quantum force are analyzed. A model for optical high harmonic generation in the presence of dynamical confinement and external monochromatic time-dependent homogeneous electric field is proposed.
https://doi.org/10.1088/1402-4896/ad52c8
Perturbation and spectral theory for singular indefinite Sturm-Liouville operators. - In: Journal of differential equations, ISSN 1090-2732, Bd. 405 (2024), S. 151-178
https://doi.org/10.1016/j.jde.2024.05.043
PT-symmetric dynamical confinement: Fermi acceleration, quantum force, and Berry phase. - In: Physical review, ISSN 2469-9934, Bd. 109 (2024), 5, 053519
We consider a quantum particle under the dynamical confinement caused by PT-symmetric box with a moving wall. The latter is described in terms of the time-dependent Schrödinger equation obeying the time-dependent PT-symmetric boundary conditions. The class of the functions, describing time dependence of the wall's position and keeping the system as PT-symmetric is found. Physically observable characteristics, such as average kinetic energy and the average quantum force are calculated as a function of time. It is found that the behavior of the average kinetic energy as a function of time is completely different than that for of Hermitian counterpart of the model, while the average quantum force behaves similarly to that for Hermitian system. Also, geometric phase is calculated for the harmonically oscillating wall regime. The experimental realization of the proposed model is discussed.
https://doi.org/10.1103/PhysRevA.109.053519
Indefinite Sturm-Liouville operators in polar form. - In: Integral equations and operator theory, ISSN 1420-8989, Bd. 96 (2024), 2, S. 1-58
https://doi.org/10.1007/s00020-023-02746-3
Feedback rectifiable pairs and stabilization of switched linear systems. - In: Systems & control letters, ISSN 1872-7956, Bd. 186 (2024), 105755, S. 1-10
We address the feedback design problem for switched linear systems. In particular we aim to design a switched state-feedback such that the resulting closed-loop subsystems share the same eigenstructure. To this effect we formulate and analyse the feedback rectification problem for pairs of matrices. We present necessary and sufficient conditions for the feedback rectifiability of pairs for two subsystems and give a constructive procedure to design stabilizing state-feedback for a class of switched systems. In particular the proposed algorithm provides sets of eigenvalues and corresponding eigenvectors for the closed-loop subsystems that guarantee stability for arbitrary switching. Several examples illustrate the characteristics of the problem considered and the application of the proposed design procedure.
https://doi.org/10.1016/j.sysconle.2024.105755
The rank-one perturbation problem for linear relations. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (29 Seiten). - (Preprint ; M23,12)
We use the recently introduced Weyr characteristic of linear relations in Cn and its relation with the Kronecker canonical form of matrix pencils to describe their dimension. Then, this is applied to study one-dimensional perturbations of linear relations.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200305
On the essential spectrum of operator pencils. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (9 Seiten). - (Preprint ; M23,11)
For a closed densely defined linear operator A and a bounded linear operator B on a Banach space X whose essential spectrums are contained in disjoint sectors, we show that the essential spectrum of the associated operator pencil λA + B is contained in a sector of the complex plane whose boundaries are determined purely by the angles that define the two sectors, which contain the essential spectrums of A and B.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200291
Lower bounds for self-adjoint Sturm-Liouville operators. - In: Proceedings of the American Mathematical Society, ISSN 1088-6826, Bd. 151 (2023), 12, S. 5313-5323
https://doi.org/10.1090/proc/16523
Limit point and limit circle trichotomy for Sturm-Liouville problems with complex potentials. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (15 Seiten). - (Preprint ; M23,10)
The limit point and limit circle classification of real Sturm-Liouville problems by H. Weyl more than 100 years ago was extended by A.R. Sims around 60 years ago to the case when the coefficients are complex. Here the main result is a collection of various criteria which allow us to decide to which class of Sims' scheme a given Sturm-Liouville problem with complex coefficients belongs. This is subsequently applied to a second order differential equation defined on a ray in C which is motivated by the recent intensive research connected with PT-symmetric Hamiltonians.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200258
Functional models of symmetric and selfadjoint operators. - In: From complex analysis to operator theory: a panorama, (2023), S. 75-122
https://doi.org/10.1007/978-3-031-31139-0_7