Multidimensional Spectral Problem Research Articles

On the solvability of a mixed problem for a fractional partial differential equation with delayed time argument and Laplace operators with nonlocal boundary conditions in Sobolev classes

In this paper, we study a problem with initial functions and boundary conditions for partial differential equations of fractional order with Laplace operators. The boundary conditions of the problem are nonlocal, and the solution is supposed to belong to one of Sobolev classes. The solution of the initial boundary value problem is constructed as the sum of a series of multidimensional spectral problem’s eigenfunctions. The eigenvalues of the spectral problem are found and the corresponding system of eigenfunctions is constructed. It is shown that this system is complete and forms a Riesz basis in the subspaces of Sobolev spaces. Basing on the completeness of the eigenfunctions’ system, the uniqueness theorem for the solution of the problem is proved. The existence of a regular solution of the initial boundary value problem is proved in Sobolev subspaces.

О РАЗРЕШИМОСТИ СМЕШАННОЙ ЗАДАЧИ С ПОМОЩЬЮ ОПЕРАТОРА ЛАПЛАСА С НЕЛОКАЛЬНЫМИ ГРАНИЧНЫМИ УСЛОВИЯМИ

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

Initial-Boundary Value Problem for the Beam Vibration Equation in the Multidimensional Case

In the multidimensional case, we study the problem with initial and boundary conditions for the equation of vibrations of a beam with one end clamped and the other hinged. An existence and uniqueness theorem is proved for the posed problem in Sobolev classes. A solution of the problem under consideration is constructed as the sum of a series in the system of eigenfunctions of a multidimensional spectral problem for which the eigenvalues are determined as the roots of a transcendental equation and the system of eigenfunctions is constructed. It is shown that this system of eigenfunctions is complete and forms a Riesz basis in Sobolev spaces. Based on the completeness of the system of eigenfunctions, a theorem about the uniqueness of a solution to the posed initial-boundary value problem is stated.

On a class of spectral problems on the half-line and their applications to multi-dimensional problems

A survey of estimates on the number N_-(\mathbf M_{\alpha G}) of negative eigenvalues (bound states) of the Sturm–Liouville operator \mathbf M_{\alpha G}u=-u''-\alpha G on the half-line, as depending on the properties of the function G and the value of the coupling parameter \alpha>0 . The central result is Theorem 5.1, giving a sharp sufficient condition for the semi-classical behavior N_-(\mathbf M_{\alpha G})=O(\alpha^{1/2}) , and the necessary and sufficient conditions for a “super-classical” growth rate N_-(\mathbf M_{\alpha G})=O(\alpha^q) with any given q>1/2 . Similar results for the problem on the whole \mathbb R are also presented. Applications to the multi-dimensional spectral problems are discussed.

Isotropic Tori, Complex Germ and Maslov Index, Normal Forms and Quasimodes of Multidimensional Spectral Problems

More than twenty years ago V. P. Maslov posed the question under what conditions it is possible to assign to invariant isotropic lower-dimensional tori of Hamiltonian systems sequences of asymptotic eigenvalues and eigenfunctions (spectral series) of the corresponding quantum mechanical and wave operators. In the present paper this question is answered in terms of the quadratic approximation to the theory of normal forms. We also discuss the quantization conditions for isotropic tori and their relation to topological, geometric, and dynamical characteristics (Maslov indices, rotation (winding) numbers, eigenvalues of dynamical flows, etc.).

On the uniqueness of solutions of inverse multidimensional spectral problems in dual asymptotic models

The solution of an elliptic equation in an unbounded region is approximated by the solution of an ordinary differential equation. To obtain a new inverse problem we prove a uniqueness theorem and exhibit a solution algorithm. Bibliography: 14 titles.

Semiclassical maslov asymptotics with complex phases. I. General approach

A method of constructing semiclassical asymptotics with complex phases is presented for multidimensional spectral problems (scalar, vector, and with operator-valued symbol) corresponding to both classically integrable and classically nonintegrable Hamiltonian systems. In the first case, the systems admit families of invariant Lagrangian tori (of complete dimension equal to the dimensionn of the configuration space) whose quantization in accordance with the Bohr—Sommerfeld rule with allowance for the Maslov index gives the semiclassical series in the region of large quantum numbers. In the nonintegrable case, families of Lagrangian tori with complete dimension do not exist. However, in the region of regular (nonchaotic) motion, such systems do have invariant Lagrangian tori of dimensionk<n (incomplete dimension). The construction method associates the families of such tori with spectral series covering the region of “intermediate” quantum numbers. The construction includes, in particular, new quantization conditions of Bohr—Sommerfeld type in which other characteristics of the tori appear instead of the Maslov index. Applications and also generalizations of the theory to Lie groups will be presented in subsequent publications of the series.

Recursion and group structures of the integrable equations in 1+1 and 1+2 dimensions. Bilocal approach

A bilocal approach to the construction of the general integrable equations and their general Backlund transformations connected with the one- and two-dimensional spectral problems is discussed. Three different, but equivalent methods of calculation are considered. The principal role of the bilocal adjoint representation of the spectral problem in such constructions, in particular in the calculation of the bilocal 'recursion' operator, is emphasised. Multi-dimensional spectral problems are discussed briefly.

On the adjoint representation for spectral problems and its relation with the Akns-method, gauge transformations and Riemann problem

It is shown that the eigenfunctions of the recursion operators, gauge transformations and Riemann problem are connected with the different irreducible components of the adjoint representations for the one-, two- and multidimensional spectral problems.