arXiv subjects

Victor Ivrii

Publications and source records attributed to Victor Ivrii.

41 records · Page 3Linked to original sources

arXiv2006

Sharp spectral asymptotics for four-dimensional Schrödinger operator with a strong degenerating magnetic field

I continue analysis of the Schrödinger operator with the strong degenerating magnetic field, started in \cite{IRO6}. Now I consider 4-dimensional case, assuming that magnetic field is generic degenerated and under certain conditions I derive spectral asymptotics with the principal part $\asymp h^{-4}$ and the remainder estimate $O(μ^{-1/2}h^{-3})$ where $μ\gg 1$ is the intensity of the field and $h\ll 1$ is the Plank constant; $μh\le 1$. These asymptotics can contain correction terms of magnitude $μ^{5/4}h^{-3/2}$ corresponding to the short periodic trajectories.

math.AP↗

arXiv2006

Sharp Spectral Asymptotics for Magnetic Schrödinger Operator with Irregular Potential

Survey: In this paper I consider sharp spectral asymptotics for multidimensional magnetic Schrödinger operator with irregular coefficients with respect to two parameters -- semiclassical parameter $h$ and coupling parameter $μ$. There are few principally different cases, depending on dimension, rank of magnetic intensity matrix, relation between $h$ and $μ$ and some extra assumptions.

math.AP↗

arXiv2006

Sharp Spectral Asymptotics for two-dimensional Schrödinger operator with a strong degenerating magnetic field

I consider two-dimensional Schrödinger operator with degenerating magnetic field and in the generic situation I derive spectral asymptotics as $h\to +0$ and $μ\to +\infty$ where $h$ and $μ$ are Planck and coupling parameters respectively. The remainder estimate is $O(μ^{-{\frac 1 2}}h^{-1})$ which is between $O(μ^{-1}h^{-1})$ valid as magnetic field non-degenerates and $O(h^{-1})$ valid as magnetic field is identically 0. As $μ$ is close to its maximal reasonable value $O(h^{-2})$ the principal part contains correction terms associated with short periodic trajectories of the corresponding classical dynamics.

math.AP↗

arXiv2006

Sharp Spectral Asymptotics for 2-dimensional Schrödinger operator with a strong but degenerating magnetic field. II

I consider the same operator as in part I assuming however that $μ\ge Ch^{-1}$ and $V$ is replaced by $(2l+1)μh F+W$ with $l\in \bZ^+$. Under some non-degeneracy conditions I recover remainder estimates up to $O \bigl(μ^{-{\frac 1 ν}}h^{-1} +1\bigr)$ but now case $μ\ge Ch^{-ν}$ is no more forbidden and the principal part is of magnitude $μh^{-1}$.

math.AP↗

arXiv2006

Sharp Spectral Asymptotics for 2-dimensional Schrödinger operator with a strong magnetic field. Note about forgotten generic case

I consider magnetic Schrödinger operator in dimension $d=2$ assuming that coefficients are smooth and magnetic field is non-degenerating. Then I extend the remainder estimate $O(μ^{-1}h^{-1}+1)$ derived in \cite{Ivr1} for the case when $V/F$ has no stationary points to the case when it has non-degenerating stationary points. If some of them are saddles and $μ^3h\ge 2$ then asymptotics contains correction terms of magnitude $μ^{-1}h^{-1}|\log μ^3 h|$.

math.AP↗