Panchenko D. The structure of point perturbations of the Schrödinger operator in one-dimensional and two-dimensional quantum systems.

Thesis is devoted to the physical structure of point perturbations of the Schrödinger operator and their realizations in one-dimensional (layered) and two-dimensional (Abrikosov vortex) quantum systems. The classification of singular point perturbations of the one-dimensional Schrödinger operator for a spinless free particle is obtained on the basis of calibration transformations analysis. It is shown that all point perturbations can be divided into two classes: “electric” and “magnetic” related to the mass and charge of a particle correspondingly. The mass calibration is associated with the choice of the spatial scale x → λ x, and the charge calibration with the canonical transformation pˆ_x → pˆ_x − A. It is shown that so called point interactions X_4 and X_2 (in accordance with the Kurasov’s notation [Kurasov P. Distribution Theory for Discontinuous Test Functions and Differential Operators with Generalized Coefficients / P. Kurasov // Journal of Mathematical Analysis and Applications.––1996.––Vol. 201.––P. 297 – 323. https://doi.org/10.1006/jmaa.1996.0256.]) can be realized in layered systems where the effective mass of elementary excitation with qualitatively different character in the transition region takes place. Moreover, the X_2-coupling has an additional quantized magnetic flux, whereas the case of X_4 is purely of “electrostatic” nature. The existence of quantized magnetic flux in X_2-case is proven by explicit demonstration of the Zeeman-like splitting for states with the opposite projections of angular momentum. The point interactions ( X_1^{(r)} and X_4^{(r)} ) associated with the spin-flip interaction are also identified as one-dimensional analogs of the Rashba interaction. The considered simple spin-filtering devices (spin-resonator and spin-filter) allow us to establish that the case of boundary conditions with X_4^{(r)}-coupling is more effective than the one with X_1^{(r)}-coupling as spin-flip mechanism. Also, using the technique of distribution theory (in accordance with [Kurasov P. Distribution Theory for Discontinuous Test Functions and Differential Operators with Generalized Coefficients / P. Kurasov // Journal of Mathematical Analysis and Applications.––1996.––Vol. 201.––P. 297 – 323. https://doi.org/10.1006/jmaa.1996.0256.]), a regularized form of the Hamiltonian with spin-flip interaction is constructed, which establishes correspondences between boundary conditions and point perturbations associated with spin-flip. The physical nature of Schrödinger operator point perturbations in two-dimensional geometry is investigated on the example of the Abrikosov vortex. In particular, a relationship has been established between the Bogolyubov – de Gennes Hamiltonian for elementary excitations, localized near the Abrikosov vortex core, and the Aharonov – Bohm Hamiltonian, which describes a charged particle in the localized magnetic field. The equivalence of these Hamiltonians is demonstrated for the case of low-energy states in the quantum limit (T → 0). In this regard, a different approach to the interpretation of low-energy states near the Abrikosov vortex core in the quantum limit was proposed, based on self-adjoint extension of the Aharonov – Bohm-type Hamiltonian with a localized magnetic field, within which the inner structure of the vortex is determined by the parameter of the corresponding boundary condition. This allows us to describe the Kramer – Pesch anomaly of the order parameter slope d∆(r)/dr in the Abrikosov vortex core. Finally, the model of the electronic structure of the Abrikosov vortex is constructed and the corresponding spectrum in the quantum limit is calculated based on the analytical solution for the Aharonov – Bohm effect with singular perturbations.