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Victor Ivrii

Publications and source records attributed to Victor Ivrii.

At least 19 recordsLinked to original sources
arXiv2023

Pointwise Spectral Asymptotics near Singularity

We establish semiclassical asymptotics and estimates for the $e_h(x,x;\tau)$ where $e_h(x,y,\tau)$ is the Schwartz kernel of the spectral projector for a second order elliptic operator inside domain with power singularity in the origin. While such asymptotics for its trace $\mathsf{N}_h(\tau)= \int e_h(x,x,\tau)\,dx$ are well-known, the poinwise asymptotics are much less explored. Our main tools: microlocal methods, improved successive approximations and geometric optics methods.

math.SP↗
arXiv2022

Pointwise Spectral Asymptotics out of the Diagonal near Degeneration

We establish uniform (with respect to $x$, $y$) semiclassical asymptotics and estimates for the Schwartz kernel $e_h(x,y;τ)$ of spectral projector for a second order elliptic operator inside domain under microhyperbolicity (but not $ξ$-microhyperbolicity) assumption. While such asymptotics for its restriction to the diagonal $e_h(x,x,τ)$ and, especially, for its trace $\mathsf{N}_h(τ)= \int e_h(x,x,τ)\,dx$ are well-known, the out-of-diagonal asymptotics are much less explored, especially uniform ones. Our main tools: microlocal methods, improved successive approximations and geometric optics methods. Our results would also lead to classical asymptotics of $e_h(x,y,τ)$ for fixed $h$ (say, $h=1$) and $τ\to \infty$.

math.AP↗
arXiv2021

Pointwise Spectral Asymptotics out of the Diagonal near Boundary

We establish semiclassical asymptotics and estimates for the Schwartz kernel $e_h(x,y;τ)$ of spectral projector for a second order elliptic operator on the manifold with a boundary. While such asymptotics for its restriction to the diagonal $e_h(x,x,τ)$ and, especially, for its trace $\mathsf{N}_h(τ)= \int e_h(x,x,τ)\,dx$ are well-known, the out-of-diagonal asymptotics are much less explored. Our main tools: microlocal methods, improved successive approximations and geometric optics methods. Our results would also lead to classical asymptotics of $e_h(x,y,τ)$ for fixed $h$ (say, $h=1$) and $τ\to \infty$.

math.SP↗
arXiv2020

Upper Estimates for Electronic Density in Heavy Atoms and Molecules

We derive an upper estimate for electronic density $ρ_Ψ(x)$ in heavy atoms and molecules. While not sharp, on the distances $\gtrsim Z^{-1}$ from the nuclei it is still better than the known estimate $CZ^3$ ($Z$ is the total charge of the nuclei, $Z\asymp N$ the total number of electrons).

math-ph↗
arXiv2020

Strong Scott Conjecture

In heavy atoms and molecules, on the distances $a \ll Z^{-1/3}$ from one of the nuclei (with a charge $Z_m$) we prove that $ρ_Ψ(x)$ is approximated in $L^p$-norm, by the electronic density for a single atom in the model with no interactions between electrons. We cover also the relativistic case.

math-ph↗
arXiv2020

Thomas-Fermi approximation to electronic density

In heavy atoms and molecules, on the distances $a \gg Z^{-1}$ from all of the nuclei (with a charge $Z_m$) we prove that $ρ_Ψ(x)$ is approximated in $L^p$-norm, by the Thomas-Fermi density.

math.SP↗
arXiv2019

Bethe-Sommerfeld conjecture in semiclassical settings

Under certain assumptions (including $d\ge 2)$ we prove that the spectrum of a scalar operator in $\mathscr{L}^2(\mathbb{R}^d)$ \begin{equation*} A_\varepsilon (x,hD)= A^0(hD) + \varepsilon B(x,hD), \end{equation*} covers interval $(τ-ε,τ+ε)$, where $A^0$ is an elliptic operator and $B(x,hD)$ is a periodic perturbation, $\varepsilon=O(h^\varkappa)$, $\varkappa>0$. Further, we consider generalizations.

math.SP↗
arXiv2018

Complete Differentiable Semiclassical Spectral Asymptotics

For an operator $A:= A_h= A^0(hD) + V(x,hD)$ with a "potential" $V$ decaying as $|x|\to \infty$ we establish under certain assumptions the complete and differentiable with respect to $τ$ asymptotics of $e_h(x,x,τ)$ where $e_h(x,y,τ)$ is the Schwartz kernel of the spectral projector.

math.SP↗
arXiv2018

Complete Semiclassical Spectral Asymptotics for Periodic and Almost Periodic Perturbations of Constant Operators

Under certain assumptions we derive a complete semiclassical asymptotics of the spectral function $e_{h,\varepsilon}(x,x,λ)$ for a scalar operator \begin{equation*} A_\varepsilon (x,hD)= A^0(hD) + \varepsilon B(x,hD), \end{equation*} where $A^0$ is an elliptic operator and $B(x,hD)$ is a periodic or almost periodic perturbation. In particular, a complete semiclassical asymptotics of the integrated density of states also holds. Further, we consider generalizations.

math.SP↗
arXiv2018

Spectral asymptotics for Dirichlet to Neumann operator

We consider eigenvalues of the Dirichlet-to-Neumann operator for Laplacian in the domain (or manifold) with edges and establish the asymptotics of the eigenvalue counting function \begin{equation*} \mathsf{N}(λ)= κ_0λ^d +O(λ^{d-1})\qquad \text{as}\ \ λ\to+\infty, \end{equation*} where $d$ is dimension of the boundary. Further, in certain cases we establish two-term asymptotics \begin{equation*} \mathsf{N}(λ)= κ_0λ^d+κ_1λ^{d-1}+o(λ^{d-1})\qquad \text{as}\ \ λ\to+\infty. \end{equation*} We also establish improved asymptotics for Riesz means.

math.SP↗
arXiv2017

Spectral Asymptotics for Fractional Laplacians

In this article we consider fractional Laplacians which seem to be of interest to probability theory. This is a rather new class of operators for us but our methods works (with a twist, as usual). Our main goal is to derive a two-term asymptotics as one-term asymptotics is easily obtained by R.~Seeley's method.

math.SP↗
arXiv2017

Asymptotics of the ground state energy in the relativistic settings

The purpose of this paper is to derive sharp asymptotics of the ground state energy for the heavy atoms and molecules in the relativistic settings, and, in particular, to derive relativistic Scott correction term and also Dirac, Schwinger and relativistic correction terms. Also we will prove that Thomas-Fermi density approximates the actual density of the ground state, which opens the way to estimate the excessive negative and positive charges and the ionization energy.

math.SP↗
arXiv2017

Asymptotics of the ground state energy in the relativistic settings and with self-generated magnetic field

The purpose of this paper is to derive sharp asymptotics of the ground state energy for the heavy atoms and molecules in the relativistic settings, with the self-generated magnetic field, and, in particular, to derive relativistic Scott correction term and also Dirac, Schwinger and relativistic correction terms. Also we will prove that Thomas-Fermi density approximates the actual density of the ground state, which opens the way to estimate the excessive negative and positive charges and the ionization energy.

math.SP↗
arXiv2017

Spectral Asymptotics for Magnetic Schroedinger Operator

In this article we obtain eigenvalue asymptotics for 2D and 3D-Schroedinger, Schroedinger-Pauli and Dirac operators in the situations in which the role of the magnetic field is important. These operators are essentially different and there is a significant difference between 2D and 3D-operators.

math.AP↗
arXiv2017

Asymptotics of the ground state energy of heavy molecules and related topics. II

We consider asymptotics of the ground state energy of heavy atoms and molecules in the strong external magnetic field and derive it including Schwinger and Dirac corrections (if magnetic field is not too strong). We also consider related topics: an excessive negative charge, ionization energy and excessive positive charge when atoms can still bind into molecules.

math-ph↗
arXiv2017

100 years of Weyl's law

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 Schrödinger 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.

math.SP↗