Grigory Tashchiyan

Bio: Grigory Tashchiyan is an academic researcher from Saint Petersburg State University. The author has contributed to research in topics: Lipschitz continuity & Magnetic field. The author has an hindex of 5, co-authored 6 publications receiving 100 citations.

On the spectral properties of the perturbed Landau Hamiltonian.

Abstract: For the Schrodinger and Pauli operators with constant magnetic field it is investigated how the spectrum is perturbed if a perturbation by a compactly supported magnetic field is performed

On the Spectral Properties of the Perturbed Landau Hamiltonian

Abstract: The Landau Hamiltonian governing the behavior of a quantum particle in dimension 2 in a constant magnetic field is perturbed by a compactly supported magnetic field and a similar electric field. We describe how the spectral subspaces change and how the Landau levels split under this perturbation.

Eigenvalue Asymptotics for Potential Type Operators on Lipschitz Surfaces

Abstract: For potential-type integral operators on a Lipschitz surface, an asymptotic formula for eigenvalues is proved. The reasoning is based upon the study of the rate of operator convergence as smooth surfaces approximate the Lipschitz surface.

On the Spectral Properties of the Landau Hamiltonian Perturbed by a moderately Decaying Magnetic Field

Abstract: The Landau Hamiltonian, describing the behavior of a quantum particle in dimension 2 in a constant magnetic field, is perturbed by a magnetic field with power-like decay at infinity and a similar electric potential. We describe how the spectral subspaces change and how the Landau levels split under this perturbation.

Eigenvalue asymptotics for potential type operators on Lipschitz surfaces of codimension greater than 1

Abstract: For potential type integral operators on a Lipschitz submanifold the asymptotic formula for eigenvalues is proved. The reasoning is based upon the study of the rate of operator convergence as smooth surfaces approximate the Lipschitz one.

100 years of Weyl’s law

Abstract: We discuss the asymptotics of the eigenvalue counting function for partial differential operators and related expressions paying the most attention to the sharp asymptotics. We consider Weyl asymptotics, asymptotics with Weyl principal parts and correction terms and asymptotics with non-Weyl principal parts. Semiclassical microlocal analysis, propagation of singularities and related dynamics play crucial role. We start from the general theory, then consider Schrodinger and Dirac operators with the strong magnetic field and, finally, applications to the asymptotics of the ground state energy of heavy atoms and molecules with or without a magnetic field.

100 years of Weyl's law

Abstract: We discuss the asymptotics of the eigenvalue counting function for partial differential operators and related expressions paying the most attention to the sharp asymptotics. We consider Weyl asymptotics, asymptotics with Weyl principal parts and correction terms and asymptotics with non-Weyl principal parts. Semiclassical microlocal analysis, propagation of singularities and related dynamics play crucial role. We start from the general theory, then consider Schrodinger and Dirac operators with the strong magnetic field and, finally, applications to the asymptotics of the ground state energy of heavy atoms and molecules with or without a magnetic field.

Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field

Abstract: In this paper, we study the Cauchy problem for the Landau Hamiltonian wave equation, with time-dependent irregular (distributional) electromagnetic field and similarly irregular velocity. For such equations, we describe the notion of a ‘very weak solution’ adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifier of the coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or distributional-type solutions under conditions when such solutions also exist.

Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed

Abstract: Given a Hilbert space $${\mathcal{H}}$$ , we investigate the well-posedness of the Cauchy problem for the wave equation for operators with a discrete non-negative spectrum acting on $${\mathcal{H}}$$ . We consider the cases when the time-dependent propagation speed is regular, Holder, and distributional. We also consider cases when it is strictly positive (strictly hyperbolic case) and when it is non-negative (weakly hyperbolic case). When the propagation speed is a distribution, we introduce the notion of “very weak solutions” to the Cauchy problem. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique “very weak solution” in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the harmonic and anharmonic oscillators, the Landau Hamiltonian on $${\mathbb{R}^n}$$ , uniformly elliptic operators of different orders on domains, Hormander’s sums of squares on compact Lie groups and compact manifolds, operators on manifolds with boundary, and many others.

Eigenvalues of the Neumann–Poincaré operator in dimension 3: Weyl’s law and geometry

Abstract: Asymptotic properties of the eigenvalues of the Neumann-Poincare (NP) operator in three dimensions are treated. The region Omega subset of R-3 is bounded by a compact surface Gamma = partial derivative Omega, with certain smoothness conditions imposed. The NP operator K-Gamma, called often 'the direct value of the double layer potential', acting in L-2(Gamma), is defined by K-Gamma[psi](X) := 1/4 pi integral(Gamma) /vertical bar x - y vertical bar(3)psi(y) dS(y), where dS(y) is the surface element and n(y) is the outer unit normal on F. The firstnamed author proved in [27] that the singular numbers s(j) (K-Gamma) of K-Gamma and the ordered moduli of its eigenvalues lambda(j) (K-Gamma) satisfy the Weyl law s(j)(K(Gamma)) similar to vertical bar lambda(j)(K-Gamma)vertical bar similar to {3W(Gamma) - 2 pi chi(Gamma)/128 pi}(1/2) j(-1/2), under the condition that Gamma belongs to the class C-2,C-alpha with alpha > 0, where W(Gamma) and chi(Gamma) denote, respectively, the Willmore energy and the Euler characteristic of the boundary surface Gamma. Although the NP operator is not selfadjoint (and therefore no general relationships between eigenvalues and singular numbers exists), the ordered moduli of the eigenvalues of K-Gamma satisfy the same asymptotic relation. The main purpose here is to investigate the asymptotic behavior of positive and negative eigenvalues separately under the condition of infinite smoothness of the boundary Gamma. These formulas are used, in particular, to obtain certain answers to the long-standing problem of the existence or finiteness of negative eigenvalues of K-Gamma. A more sophisticated estimate allows us to give a natural extension of the Weyl law for the case of a smooth boundary.