arXiv subjects

Vladimir Nazaikinskii

Publications and source records attributed to Vladimir Nazaikinskii.

4 recordsLinked to original sources

arXiv2020

Partial spectral flow and the Aharonov-Bohm effect in graphene

We study the Aharonov-Bohm effect in an openended tube made of a graphene sheet whose dimensions are much larger than the interatomic distance in graphene. An external magnetic field vanishes on and in the vicinity of the graphene sheet and its flux through the tube is adiabatically switched on. It is shown that, in the process, the energy levels of the tight-binding Hamiltonian of pi-electrons unavoidably cross the Fermi level, which results in the creation of electron-hole pairs. The number of pairs is proven to be equal to the number of magnetic flux quanta of the external field. The proof is based on the new notion of partial spectral flow, which generalizes the ordinary spectral flow already having well-known applications (such as the Kopnin forces in superconductors and superfluids) in condensed matter physics.

math-ph↗

arXiv2018

Semi-classical Green functions

Let $H(x,p)\sim H_0(x,p)+hH_1(x,p)+\cdots$ be a semi-classical Hamiltonian on $T^*{\bf R}^n$, and $Σ_E=\{H_0(x,p)=E\}$ a non critical energy surface. Consider $f_h$ a semi-classical distribution (the "source") microlocalized on a Lagrangian manifold $Λ$ which intersects cleanly the flow-out $Λ_+$ of the Hamilton vector field $X_{H_0}$ in $Σ_E$. Using Maslov canonical operator, we look for a semi-classical distribution $u_h$ satisfying the limiting absorption principle and $H^w(x,hD_x)u_h=f_h$ (semi-classical Green kernel). In this report, we elaborate (still at an early stage) on some results announced in [Doklady Akad. Nauk, Vol. 76, No1, p.1-5, 2017] and provide some examples, in particular from the theory of wave beams.

math-ph↗

arXiv2017

Asymptotics of Green function for the linear waves equations in a domain with a non-uniform bottom

We consider the linear problem for water-waves created by sources on the bottom and the free surface in a 3-D basin having slowly varying profile $z=-D(x)$. The fluid verifies Euler-Poisson equations. These (non-linear) equations have been given a Hamiltonian form by Zakharov, involving canonical variables $(ξ(x,t),η(x,t))$ describing the dynamics of the free surface; variables $(ξ,η)$ are related by the free surface Dirichlet-to-Neumann (DtN) operator. For a single variable $x\in{\bf R}$ and constant depth, DtN operator was explicitely computed in terms of a convergent series. Here we neglect quadratic terms in Zakharov equations, and consider the linear response to a disturbance of $D(x)$ harmonic in time when the wave-lenght is small compared to the depth of the basin. We solve the Green function problem for a matrix-valued DtN operator, at the bottom and the free-surface.

math-ph↗

arXiv2016

The Complex Lagrangian Germ and the Canonical Operator

We give a manifestly invariant definition of the Lagrangian complex germ with the minimal degree of accuracy required to define the canonical operator. The equivalence with the traditional definition is proved, and the canonical operator is constructed in new terms. A new form of the quantization condition is given, in which the volume form is assumed to be defined on the universal covering of the Lagrangian manifold rather than on the manifold itself. This allows one to solve a wider class of eigenvalue problems.

math-ph↗