Russian Journal of Mathematical Physics最新文献

We study the inverse spectral problem method for one class of band matrices which can be applied to the integration (via Lax formalism) of Bogoyavlensky lattice

We prove that every locally bounded endomorphism (pi) of a connected reductive Lie group taking the center of the group to the center is continuous if and only if the restriction (pi|_Z) of (pi) to the center (Z) of (G) is continuous with respect to the same topology.

Singleton approximations of sets in asymmetric spaces are studied. Estimate of the Chebyshev radius of a set depending on the behavior of the MAX-distance function and if the MAX - projection operator has not too many points of discontinuity.

We consider semiclassical operators equal to linear combinations of quantized canonical transformations with pseudodifferential operators as coefficients and study semiclassical asymptotics of the solutions of the corresponding equations. Under the assumption that the group of canonical transformations is finite, we reduce our problem to a similar problem for a matrix semiclassical pseudodifferential operator. The latter problem can be treated by standard methods.

It is proved that a metric graph with the minimal growth of the number of possible end positions of a random walk is the union of several paths outgoing from one vertex.

We consider a Schrödinger operator of a charge motion near a surface in a strong homogeneous external magnetic field in the resonant case when the cyclotron frequency is equal to the frequency of the quadratic confinement potential holding the charge near the surface. The small curvature of the surface causes a significant energy splitting of the Landau levels in this case. The corresponding drift of the charge oscillations centers induces a geometric current on the surface. Using the quantum averaging over the cyclotron and confinement oscillations we reduce the (3)-dimensional magnetic Schrödinger operator to an effective (2)-dimensional operator, which determines the small splittings of Landau levels and the quantization of corresponding current on the surface. We obtain an explicit form of the second order correction induced by the surface curvature to the lowest Landau level near a nondegenerate stationary point of the surface and the corresponding closed curves of the geometric current on the surface.

It is shown that, under a natural constraint, a set of generalized rational fractions in an atomless (L_1)-space is a Chebyshev set with continuous metric projection only if this set is convex. Hence this set is not a uniqueness set in (L_1), and therefore, some (xin L_1) has at least two nearest points in this set. As a result, it is shown that the set of classical algebraic fractions ( mathscr{R} _{n,m}) (consisting of ratios of algebraic polynomials of degree (le n), (le m), respectively) is not a Chebyshev set in ( L_1[a,b]), and therefore, there exists a function (xin L_1[a,b]) with at least two nearest points in ( mathscr{R} _{n,m}). This result solves one long-standing problem in rational approximation.

We introduce a natural class of multicomponent local Poisson structures (mathcal P_k + mathcal P_1), where (mathcal P_1) is a local Poisson bracket of order one and (mathcal P_k) is a homogeneous Poisson bracket of odd order (k) under the assumption that (mathcal P_k) has Darboux coordinates (Darboux–Poisson bracket) and is nondegenerate. For such brackets, we obtain general formulas in arbitrary coordinates, find normal forms (related to Frobenius triples), and provide the description of the Casimirs, using a purely algebraic procedure. In the two-component case, we completely classify such brackets up to a point transformation. From the description of Casimirs, we derive new Harry Dym (HD) hierarchies and new Hunter–Saxton (HS) equations for arbitrary number of components. In the two-component case, our HS equation differs from the well-known HS2 equation.

Efficient formulas for the Maslov canonical operator in a neighborhood of a generic caustic cusp were constructed in [1] based on the Lagrangian equivalence of a generic (A_3) singularity and the standard (A_3) singularity. For the constructive implementation of these formulas, one needs a closed-form expression for the diffeomorphism specifying this equivalence. Such an expression is obtained in the present paper. By way of example, we apply the resulting formulas to the asymptotic solution of a problem on the scattering of a plane wave in an inhomogeneous medium.

As is known, for a sufficiently small defect of a (not necessarily bounded) quasirepresentation of an amenable group in a reflexive Banach space (E) with dense set of bounded orbits, there is an extension of this quasirepresentation for which there is a close ordinary representation of the group in the space of this extension. In the present note it is proved that, if the original quasirepresentation (pi) of an amenable group (G) in a reflexive Banach space (E) is a pseudorepresentation, then an ordinary representation of (G), in the vector subspace (L) of (E) formed by vectors with bounded orbits and equipped with a natural Banach norm, which is close to (pi|_L) (this ordinary representation exists if the defect of (pi) is sufficiently small) is equivalent to (pi|_L).