Related papers: Estimates, asymptotics and trace formulas for peri…

In this article we extend B. Simon's construction and results for leading order eigenvalue asymptotics to $n$-dimensional Schr\"odinger operators with non-confining potentials given by: $H^\alpha_n=-\Delta +\prod\limits_{i=1}^n…

In this paper we discuss a constructive approach to check whether a constant Hamiltonian is Yang-Baxter integrable. We then apply our method to long-range interactions and find the Lax operator and $R$-matrix of the two-loop SU(2) sector in…

This paper demonstrates the power of the calculus developed in the two previous parts of the series for all real forms of the almost Hermitian symmetric structures on smooth manifolds, including e.g. conformal Riemannian and almost…

We develop physically admissible lattice models in the harmonic approximation which define by Hamilton's variational principle fractional Laplacian matrices of the forms of power law matrix functions on the n -dimensional periodic and…

This is a review of two of the fundamental tools for analysis of soliton equations: i) the algebraic ones based on Kac-Moody algebras, their central extensions and their dual algebras which underlie the Hamiltonian structures of the NLEE;…

In this paper, we study the sub-Riemannian problem associated with contact structures on connected, simply connected, solvable, non-nilpotent, regular three-dimensional Lie groups. For these groups, the vertical component of the Hamiltonian…

For Toeplitz operators on bounded symmetric domains of arbitrary rank, we define a Hilbert quotient module corresponding to partitions of length $1$ and prove that it belongs to the Macaev class ${\mathcal{L}}^{n,\infty}$. We next obtain an…

We show that to each symmetric elliptic operator of the form \[ \mathcal{A} = - \sum \partial_k \, a_{kl} \, \partial_l + c \] on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ one can associate a self-adjoint Dirichlet-to-Neumann…

The homogenization of elliptic divergence-type fourth-order operators with periodic coefficients is studied in a (periodic) domain. The aim is to find an operator with constant coefficients and represent the equation through a perturbation…

On L 2 (R), we consider the Schr\"odinger operator (1.1) H \k{o} = -- $\partial$ 2 $\partial$x 2 + v(x) -- \k{o}x, where v is a real analytic 1-periodic function and \k{o} is a positive constant. This operator is a model to study a Bloch…

In this article we obtain asymptotic formulas for the Bloch eigenvalues of the operator generated by a system of Schrodinger equations with periodic PT-symmetric complex-valued coefficients. Then using these formulas we classify the…

This thesis is devoted to asymptotic norm estimates for oscillatory integral operators acting on the L^2 space of functions of one real variable. The operators in question have compact support and an oscillatory kernel of the form exp(i…

A family of discrete Schroedinger operators is investigated through scattering theory. The continuous spectrum of these operators exhibit changes of multiplicity, and some of these operators possess resonances at thresholds. It is shown…

We prove an approximation result showing how operators of the type $-\Delta -\gamma \delta (x-\Gamma)$ in $L^2(\mathbb{R}^2)$, where $\Gamma$ is a graph, can be modeled in the strong resolvent sense by point-interaction Hamiltonians with an…

We consider motion of a material point placed in a constant homogeneous magnetic field restricted to the sphere $S^{n-1}$. We provide a Lax representation of the equations of motion and prove complete integrability of those systems for any…

We consider rotating wave solutions of the nonlinear wave equation \[ \left\{ \begin{aligned} \partial_{t}^2 v - \Delta v + m v & = |v|^{p-2} v \quad && \text{in $\mathbb{R} \times \textbf{B}$} \\ v & = 0 && \text{on $\mathbb{R} \times…

A high-frequency asymptotics of the symbol of the Dirichlet-to-Neumann map, treated as a periodic pseudodifferential operator, in 2D diffraction problems is discussed. Numerical results support a conjecture on a universal limit shape of the…

We consider the defocussing NLS equation with small periodic initial condition. A new approach to study the Hamiltonian as a function of action variables is demonstrated. The problems for the NLS equation is reformulated as the problem of…

This article deals with the spectral analysis of the semiclassical Neumann magnetic Laplacian on a smooth bounded domain in dimension three. When the magnetic field is constant and in the semiclassical limit, we establish a four-term…

We study several classes of non-Hermitian Hamiltonian systems, which can be expressed in terms of bilinear combinations of Euclidean Lie algebraic generators. The classes are distinguished by different versions of antilinear (PT)-symmetries…