Information on the thesis

The work is devoted to the study of mathematical models that arose from quantum mechanics. The main aim is to build so-called exactly solvable models, which not only provide a qualitative description of the actual process, but can also be relatively easily solved to obtain quantitative characteristics such as spectra or scattering data. The object of research is differential operators with singularly perturbed coefficients. The main tool is asymptotic methods for differential equations, which can be used to prove convergence of families of perturbed operators in the uniform or strong resolvent topologies. The domains of the limit operators can be described in terms of boundary conditions or coupling conditions on sets, where the perturbations are localized. These operators are the solvable models that best approximate an actual physical process in the class of   point interactions or interactions on submanifolds.

A mathematical theory of the one-dimensional hydrogen atom is constructed. The problem has given rise to many scientific discussions and many publications in the physical and mathematical literature. We have studied Schrödinger operators with Coulomb-type potentials, i.e., potentials with power singularities. To construct a Hamiltonian of the hydrogen atom, i.e., a self-adjoint operator corresponding to the energy of system, it is necessary to explicitly indicate coupling conditions at the point of singularity. We have considered   more general problem for regularizations of Coulomb-type potentials and we have proved the convergence of the corresponding operators in the norm resolvent topology. The consequence of these results is a complete mathematical description of the one-dimensional model of hydrogen atom.

The Thesis contains a complete mathematical solution to the well-known problem of δʹ-potential in quantum mechanics. The problem consists in constructing one-dimensional solvable models for the localized dipole.    The norm resolvent convergence of Schrödinger operators with δʹ-like potentials has been established for the shapes of perturbation with compact supports. The solvable models have been constructed that can be used to describe the dipole interactions. These results have been extended to the shapes of the Faddeev-Marchenko class. Also, we have described the interaction of δ and δʹ potentials. We have studied the families of Schrödinger operators, which can be interpreted as a regularization of Hamiltonians with pseudopotentials aδʹ+bδ. The regularized potentials contain two parameters associated with the localization rate of the δ-like and
δʹ-like sequences, respectively. The convergence of such operators has been obtained in the norm resolvent topology and it has been shown that the solvable models depend not only on the potential shapes, but also on the ratio of the localization rates of each term in the perturbations.

We have generalized the problem of δʹ-potential to the two-dimensional case. The spectral properties of Schrödinger operators with the dipole type potentials concentrated in a vicinity of a closed curve have been studied. Solvable models with interactions on the curve have been obtained by construction of the asymptotics of eigenvalues and eigenfunctions. The coupling conditions on the curve depend in nontrivial way on the geometry of curve and the spectral characteristics of localized potentials.

Since the product of the first derivative of Dirac’s function and a smooth function is always a linear combination of two functionals δ and δʹ, the δʹ-potential can be formally treated as a perturbation by an operator of rank two. We have established convergence in the uniform resolvent topology of Schrödinger operators with singular local perturbations of rank two. For such families of operators, several qualitatively different cases of the limit behavior were obtained and a wide class of solvable models with point interactions was constructed.

We have also established conditions for the existence of negative eigenvalues and conditions when these eigenvalues are absorbed by the lower bound of continuous spectrum as a coupling constant goes to zero.  The threshold behavior of negative eigenvalues has been studied for families of Schrödinger operators with potentials that depend on a coupling constant in the nonlinear way. Two-term asymptotics of threshold eigenvalues have been constructed.

The spectral properties of Sturm-Liouville operators and boundary value problems for the elliptic operators with singular perturbation of the weight function have been studied. These asymptotic results give non-trivial examples of self-adjoint operators acting in variable Hilbert spaces for which the behavior of spectrum and eigenspaces is described by a non-self-adjoint operator with nontrivial Jordan cells.

The thesis is a mathematical research in fields of the asymptotic analysis of differential operators and the spectral theory of linear operators. The obtained results are new and have direct application in nonrelativistic quantum mechanics. Most of them concern one-dimensional physical models, which are not a simplification of 3-dimensional ones, but play the role of primary mathematical models in various branches of modern physics. The models of one-dimensional physics have become relevant in recent decades, because only now can they be implemented experimentally.  From the mathematical viewpoint, some problems of one-dimensional physics are much more complicated than the corresponding problems in higher dimensions. In particular, this applies to the problem of hydrogen atom. The mathematics of one-dimensional models also underlies the modern theory of quantum graphs, which describes the processes in quantum waveguides.

Key words: Schrödinger operator, singular potential, δʹ-potential, Coulomb potential, one-dimensional hydrogen atom, scattering problem, asymptotics of spectrum, solvable model, point interaction, finite rank perturbation, magnetic potential, zero-energy resonance, half-bound state, Sturm-Liouville operator, concentrated masses.