### Book chapter

## Квазиклассическое приближение для разностных уравнений 2-ого порядка в неограниченной области

In this paper, we consider a linear homogeneous difference equation of the 2nd order with smooth coefficients. It is known that there are the several solutions to such equation. We have constructed explicit restrictions on the smoothness and growth of the coefficients of the equation, under which the solution retains the WKB form in an unbounded domain that does not contain pivot points and singularity points. The asymptotics of Laguerre polynomials are constructed for simultaneously large values of the argument and the order of the polynomials using proven asymptotic estimates as an application example.

We consider a model quantum Hamiltonian of a charge in a resonance electromagnetic trap. Using the operator averaging method, we obtain an effective quantum operator that asymptotically describes the anharmonic part of the Hamiltonian. We show that the operator becomes a second-order difference operator in a specially chosen quantum action-angle representation. Using the discrete WKB method for this difference equation, we obtain the semiclassical WKB asymptotics of the spectrum and stationary states of the charge.

Painlevé equations, holomorphic vector fields and normal forms, summability of WKB solutions, Gevrey order and summability of formal solutions for ordinary and partial differ- ential equations, • Stokes phenomena of formal solutions of non-linear PDEs, and the small divisors phenomenon, • summability of solutions of difference equations, • applications to integrable systems and mathematical physics.

In this work we construct and discuss special solutions of a homogeneous problem for the Laplace equation in a domain with the cone-shaped boundaries. The problem at hand is interpreted as that describing oscillatory linear wave movement of a uid under gravity in such a domain. These so- lutions are found in terms of the Mellin transform and by means of the reduction to some new functional-difference equations solved in an explicit form (in quadratures). The behavior of the so- lutions at far distances is studied by use of the saddle point technique. The corresponding eigenoscil- lations of a uid are then interpreted as eigenfunctions of the continuous spectrum.

In this paper, we consider the spectral problem for the magnetic Schrödinger operator on the 2-D plane (*x*1*, x*2) with the constant magnetic field normal to this plane and with the potential *V *having the form of a harmonic oscillator in the direction *x*1 and periodic with respect to variable *x*2. Such a potential can be used for modeling a long molecule. We assume that the magnetic field is quite large, this allows us to make the averaging and to reduce the original problem to a spectral problem for a 1-D Schrödinger operator with effective periodic potential. Then we use semiclassical analysis to construct the band spectrum of this reduced operator, as well as that of the original 2-D magnetic Schrödinger operator.

For the Dirac 2D-operator in a constant magnetic field with perturbing electric potential, we derive Hamiltonians describing the quantum quasiparticles (Larmor vortices) at Landau levels. The discrete spectrum of this Hall-effect quantum Hamiltonian can be computed to all orders of the semiclassical approximation by a deformed Planck-type quantization condition on the 2D-plane; the standard magnetic (symplectic) form on the plane is corrected by an “electric curvature” determined via derivatives of the electric field. The electric curvature does not depend on the magnitude of the electric field and vanishes for homogeneous fields (i.e., for the canonical Hall effect). This curvature can be treated as an effective magnetic charge of the inhomogeneous Hall 2D-nanosystem.