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Victor Ivrii

Publications and source records attributed to Victor Ivrii.

At least 37 records · Page 2Linked to original sources
arXiv2015

Spectral asymptotics for the semiclassical Dirichlet to Neumann operator

Let $M$ be a compact Riemannian manifold with smooth boundary, and let $R(λ)$ be the Dirichlet-to-Neumann operator at frequency $λ$. We obtain a leading asymptotic for the spectral counting function for $λ^{-1}R(λ)$ in an interval $[a_1, a_2)$ as $λ\to \infty$, under the assumption that the measure of periodic billiards on $T^*M$ is zero. The asymptotic takes the form \begin{equation*} N(λ; a_1,a_2) = \bigl(κ(a_2)-κ(a_1)\bigr)\mathsf{vol}'(\partial M) λ^{d-1}+o(λ^{d-1}), \end{equation*} where $κ(a)$ is given explicitly by \begin{equation*} κ(a) = \frac{ω_{d-1}}{(2π)^{d-1}} \biggl( -\frac{1}{2π} \int_{-1}^1 (1 - η^2)^{(d-1)/2} \frac{a}{a^2 + η^2} \, dη- \frac{1}{4} + H(a) (1+a^2)^{(d-1)/2} \biggr) \end{equation*} with the Heavyside function $H(a)$.

math.SP↗
arXiv2014

Asymptotics of the ground state energy of heavy atoms and molecules in combined magnetic field

We consider asymptotics of the ground state energy of heavy atoms and molecules in the self-generatedl magnetic field. Namely, we consider $$H=((D-A)\cdot\boldsymbolσ)^2-V$$ with $$V=\sum_{1\le m\le M} \frac{Z_m}{|x-y_m|}$$ and a corresponding Multiparticle Quantum Hamiltonian $$ \mathsf{H}=\sum_{1\le n\le N} H_{x_n} +\sum_{1\le n < n'\le N}|x_n-x_n'|^{-1} $$ on the Fock space $\wedge _{1\le n\le N} L^2(\mathbb{R}^3, \mathbb{C}^2)$. Here $A=A^0+A'$ where $A^0=\frac{1}{2}(-B x_2, B x_1,0)$ is an external magnetic field and $A'$ is self-generated magnetic field. Then the ground state energy is given by $$ \mathsf{E}(A)=\inf \operatorname{Spec}(\mathsf{H})+\frac{1}α\int |\nabla \times A'|^2\,dx $$ where the last term is the energy of magnetic field. Under assumptions $αZ\le κ^*$ (with a small constant $κ^*$) and $M=1$ (atomic case) we study the ground State Energy $$\mathsf{E}^*=\inf_{A'}\mathsf{E}(A).$$ We derive its asymptotics including Scott, and Schwinger and Dirac corrections (depending on $B\ll Z^3$). In the next versions we will consider also molecules and related topics: an excessive negative charge, ionization energy and excessive positive charge when atoms can still bind into molecules.

math.SP↗
arXiv2013

Asymptotics of the ground state energy of heavy molecules in self-generated magnetic field

We consider asymptotics of the ground state energy of heavy atoms and molecules in the self-generatedl magnetic field. Namely, we consider $$ H=((D-A)\cdot\boldsymbolσ)^2-V $$ with $$V=\sum_{1\le m\le M} \frac{Z_m}{|x-y_m|}$$ and a corresponding Multiparticle Quantum Hamiltonian $$ \mathsf{H}=\sum_{1\le n\le N} H_{x_n} +\sum_{1\le n < n'\le N}|x_n-x_{n'}|^{-1} $$ on the Fock space $\wedge _{1\le n\le N} L^2(\mathbb{R}^3, \mathbb{C}^2)$. Here $A$ is a self-generated magnetic fiels. Then the ground state energy is given by $$ \mathsf{E}(A)=\inf \operatorname{Spec}(\mathsf{H})+\frac{1}α\int |\nabla \times A|^2\,dx $$ where the last term is the energy of magnetic field. Under assumption $αZ\le κ^*$ (with a small constant $κ^*$) we study the ground State Energy $$ \mathsf{E}^*=\inf _{A}\mathsf{E}(A). $$ We derive its asymptotics including Scott, and Schwinger and Dirac corrections. We also consider related topics: an excessive negative charge, ionization energy and excessive positive charge when atoms can still bind into molecules.

math-ph↗
arXiv2011

Local trace asymptotics in the self-generated magnetic field

We consider a semiclassical asymptotics of local trace for the 3D-Schroedinger operator with self-generated magnetic field; it is given by Weyl expression with O(h^{-1}) error and under standard condition to Hamiltonian trajectories even o(h^{-1}). In comparison to v1,2 errors are corrected, new results are added and more details are provided. Misprint correction in comparison to v3

math.SP↗
arXiv2011

Global trace asymptotics in the self-generated magnetic field in the case of Coulomb-like singularities

We consider a semiclassical asymptotics of local trace for the 3D-Schroedinger operator with self-generated magnetic field in the case when electric potential has one or several Coulomb-like singularities; it is given by Weyl expression plus (magnetic) Scott correction term with O(h^{-4/3}) error provided distance between singularities is large enough. Significant improvement and simplification in comparison to v1.

math.SP↗
arXiv2011

Sharp Spectral Asymptotics for Operators with Irregular Coefficients. V. Multidimensional Schroedinger operator with a strong magnetic field. Non-Full-rank case

Sharp spectral asymptotics for multidimensional Schroedinger operators with the strong magnetic field are derived under rather weak smoothness conditions. I assume that magnetic intensity matrix has constant defect r>0 at each point. In comparison with version 1 of 5.5 year ago this version contains more results (we also study some degenerations), improvements and some minor corrections.

math.AP↗
arXiv2011

Sharp Spectral Asymptotics for Operators with Irregular Coefficients. IV. Multidimensional Schroedinger operator with a strong magnetic field. Full-rank case

With derive sharp spectral asymptotics (with the remainder estimate $O(μ^{-1}h^{1-d}+μ^{\frac{d} {2}-1}h^{1-\frac{d}{2}})$ for $d$-dimensional Schrödinger operator with a strong magnetic field; here $h$ and $μ$ are Plank and binding constants respectively and magnetic intensity matrix has full rank at each point. In comparison with version 1 of 4.5 year ago this version contains more results (we also study some degenerations), improvements and some minor corrections.

math.AP↗
arXiv2010

2D- and 3DMagnetic Schroedinger Operators: Short Loops, Pointwise Spectral Asymptotics and Asymptotics of Dirac Energy

We consider 2- and 3-dimensional Schrödinger or generalized Schrödinger-Pauli operators with the non-degenerating magnetic field in the open domain under certain non-degeneracy assumptions we derive pointwise spectral asymptotics. We also consider asymptotics of some related expressions (see below). For all asymptotics loops rather than periodic trajectories play important role.

math.SP↗
arXiv2010

Operators with Periodic Hamiltonian Flows in Domains with the Boundary

We consider operators in the domains with the boundaries and derive sharp spectral asymptotics (containing non-Weyl correction) in the case when Hamiltonian flow is periodic. Even if operator is scalar but not second order (or even second-order but there is an inner boundary with both refraction and reflection present) Hamiltonian flow is branching on the boundary and the notion of periodicity becomes more complicated.

math.AP↗
arXiv2010

Short Loops and Pointwise Spectral Asymptotics

We consider pointwise semiclassical spectral asymptotics i.e. asymptotics of $e(x,x,0)$ as $h\to +0$ where $e(x,y,τ)$ is the Schwartz kernel of the spectral projector and consider two cases when schort loops give contribution above $O(h^{1-d})$: (i) Schroedinger operator in dimensions $1,2$ as potential $V=0\implies \nabla V\ne 0$; (ii) Operators near boundaries.

math.AP↗
arXiv2010

Schroedinger Operator with Strong Magnetic Field near Boundary

We consider 2-dimensional Schroedinger operator with the non-degenerating magnetic field in the domain with the boundary and under certain non-degeneracy assumptions we derive spectral asymptotics with the remainder estimate better than $O(h^{-1})$, up to $O(μ^{-1}h^{-1})$ and the principal part $\asymp h^{-2}$ where $h\ll 1$ is Planck constant and $μ\gg 1$ is the intensity of the magnetic field; $μh \le 1$. We also consider generalized Schrödinger-Pauli operator in the same framework albeit with $μh\ge 1$ and derive spectral asymptotics with the remainder estimate up to O(1) and with the principal part $\asymp μh^{-1}$, or, under certain special circumstances with the principal part $\asymp μ^{1/2} h^{-1/2}$.

math.SP↗
arXiv2010

Schroedinger Operator with Strong Magnetic Field: Propagation of singularities and sharper asymptotics

We consider 2-dimensional Schrödinger operator with the non-degenerating magnetic field and we discuss spectral asymptotics with the remainder estimate $o(μ^{-1}h^{-1})$ or better. We also consider 3-dimensional Schrödinger operator with the non-degenerating magnetic field and we discuss spectral asymptotics with the remainder estimate $o(μ^{-1}h^{-1})$ or better. We also consider generalized Schrödinger-Pauli operator in the same framework albeit with $μh\ge 1$ and derive spectral asymptotics with the remainder estimate up to $O(h^{-1}|\log h|)$ and with the principal part $\asymp μh^{-2}$.

math.SP↗
arXiv2007

Sharp Spectral Asymptotics for Dirac Energy

I derive sharp semiclassical asymptotics of $\int |e_h(x,y,0)|^2ω(x,y) dx dy$ where $e_h(x,y,τ)$ is the Schwartz kernel of the spectral projector and $ω(x,y)$ is singular as $x=y$. I also consider asymptotics of more general expressions.

math.AP↗
arXiv2006

Sharp Spectral Asymptotics for four-dimensional Schroedinger operator with a strong magnetic field. II

I consider 4-dimensional Schrödinger operator with the generic non-degenerating magnetic field and for a generic potential I derive spectral asymptotics with the remainder estimate $O(μ^{-1}h^{-3})$ and the principal part $\asymp h^{-4}$ where $h\ll 1$ is Planck constant and $μ\gg 1$ is the intensity of the magnetic field. For general potentials remainder estimate $O(μ^{-1}h^{-3}+μ^2h^{-2})$ is achieved.

math.AP↗