arXiv subjects

Sergey Dobrokhotov

Publications and source records attributed to Sergey Dobrokhotov.

6 recordsLinked to original sources

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↗

arXiv2013

The semiclassical Maupertuis-Jacobi correspondence for quasi-periodic Hamiltonian flows: stable and unstable spectra

We investigate semi-classical properties of Maupertuis-Jacobi correspondence in 2-D for families of Hamiltonians $(H_λ(x,ξ), {\cal H}_λ(x,ξ))$, when ${\cal H}_λ(x,ξ)$ is the perturbation of completely integrable Hamiltonian $\widetilde{\cal H}$ veriying some isoenergetic non-degeneracy conditions. Assuming the Weyl $h$-PDO $H^w_λ$ has only discrete spectrum near $E$, and the energy surface $\{\widetilde{\cal H}={\cal E}\}$ is separated by some pairwise disjoint lagrangian tori, we show that most of eigenvalues for $\hat H_λ$ near $E$ are asymptotically degenerate as $h\to0$. This applies in particular for the determination of trapped modes by an island, in the linear theory of water-waves. We also consider quasi-modes localized near rational tori. Finally, we discuss breaking of Maupertuis-Jacobi correspondence on the equator of Katok sphere.

math-ph↗

arXiv2012

The semi-classical Maupertuis-Jacobi correspondence: stable and unstable spectra

We investigate semi-classical properties of Maupertuis-Jacobi correspondence for families of 2-D Hamiltonians $(H_λ(x,ξ), {\cal H}_λ(x,ξ))$, when ${\cal H}_λ(x,ξ)$ is the perturbation of a completely integrable Hamiltonian $\widetilde{\cal H}$ veriying some isoenergetic non-degeneracy conditions. Assuming $\hat H_λ$ has only discrete spectrum near $E$, and the energy surface $\{\widetilde{\cal H}_0={\cal E}\}$ is separated by some pairwise disjoint Lagrangian tori, we show that most of eigenvalues for $\hat H_λ$ near $E$ are asymptotically degenerate as $h\to0$. This applies in particular for the determination of trapped modes by an island, in the linear theory of water-waves. We also consider quasi-modes localized near rational tori. Finally, we discuss breaking of Maupertuis-Jacobi correspondence on the equator of Katok sphere.

math-ph↗

arXiv2008

The Semi Classical Maupertuis-Jacobi Correspondance for Quasi-Periodic Hamiltonian Flows

We extend to the semi-classical setting the Maupertuis-Jacobi correspondance for a pair of hamiltonians $(H(x,hD_x), {\cal H}(x,hD_x)$. If ${\cal H}(p,x)$ is completely integrable, or has merely has invariant diohantine torus $Λ$ in energy surface ${\cal E}$, then we can construct a family of quasi-modes for $H(x,hD_x)$ at the corresponding energy $E$. This applies in particular to the theory of water-waves in shallow water, and determines trapped modes by an island, from the knowledge of Liouville metrics.

math-ph↗

arXiv2005

Generalized Maslov canonical operator and tsunami asymptotics over nonuniform bottom. I

We suggest a new asymptotic representation for the solutions to the 2-D wave equation with variable velocity with localized initial data. This representation is a generalization of the Maslov canonical operator and gives the formulas for the relationship between initial localized perturbations and wave profiles near the wave fronts including the neighborhood of backtracking (focal or turning) and selfintersection points. We apply these formulas to the problem of a propagation of tsunami waves in the frame of so-called piston model. Finally we suggest the fast asymptotically-numerical algorithm for simulation of tsunami wave over nonuniform bottom. In this first part we present the final formulas and some geometrical construction. The proofs concerning analytical calculations will be done in the second part.

math-ph↗

arXiv2004

The Spectral Asymptotics of the Two-Dimensional Schrödinger operator with a Strong Magnetic Field

We consider the spectral problem for the two-dimensional Schrödinger operator for a charged particle in strong uniform magnetic and periodic electric fields. The related classical problem is analyzed first by means of the Krylov-Bogoljubov-Alfven and Neishtadt averaging methods. It allows us to show ``almost integrability'' of the the original two-dimensional classical Hamilton system, and to reduce it to a one-dimensional one on the phase space which is a two-dimensional torus. Using the topological methods for integrable Hamiltonian system and elementary facts from the Morse theory, we give a general classification of the classical motion. According this classification the classical motion is separated into different regimes with different topological characteristics (like rotation numbers and Maslov indices). Using these regimes, the semiclassical approximation, the Bohr-Sommerfeld rule and the correspondence principle, we give a general asymptotic description of the (band) spectrum of the original Schrödinger operator and, in particular, estimation for the number of subbands in each Landau band. From this point of view the regimes, are the classical preimages of ``spectral series'' of the Schrödinger operator. We also discuss the relationship between this spectrum and the spectrum of one-dimensional difference operators.

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