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31.
Mathematical Notes - Asymptotic formulas as x→∞ are obtained for a fundamental system of solutions to equations of the form $$l\left( y \right): = {\left( { - 1} \right)^n}{\left(... 相似文献
32.
Doklady Mathematics - The connection between the number of internal zeros of nontrivial solutions to fourth-order self-adjoint boundary value problems and the inertia index of these problems is... 相似文献
33.
A. E. Mihailov I. Z. Latypov A. A. Kalinkin A. V. Shkalikov A. A. Kalachev V. V. Samartsev 《Bulletin of the Russian Academy of Sciences: Physics》2011,75(12):1658-1663
Narrow-band orthogonally polarized collinear frequency-degenerate biphotons are generated in the process of the spontaneous
parametric down-conversion of light in a nonlinear BBO crystal placed in an optical resonator. Quantum polarization tomography
and polarization transformation of the generated biphoton states are performed. The results from our experiment agree well
with theoretical calculations. 相似文献
34.
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36.
A. M. Savchuk A. A. Shkalikov 《Proceedings of the Steklov Institute of Mathematics》2008,260(1):218-237
Let L D be the Sturm-Liouville operator generated by the differential expression L y = ?y″ + q(x)y on the finite interval [0, π] and by the Dirichlet boundary conditions. We assume that the potential q belongs to the Sobolev space W 2 ? [0, π] with some ? ≥ ?1. It is well known that one can uniquely recover the potential q from the spectrum and the norming constants of the operator L D. In this paper, we construct special spaces of sequences ? 2 θ in which the regularized spectral data {s k } ?∞ ∞ of the operator L D are placed. We prove the following main theorem: the map F q = {s k } from W 2 ? to ? 2 θ is weakly nonlinear (i.e., it is a compact perturbation of a linear map). A similar result is obtained for the operator L DN generated by the same differential expression and the Dirichlet-Neumann boundary conditions. These results serve as a basis for solving the problem of uniform stability of recovering a potential. Note that this problem has not been considered in the literature. The uniform stability results are formulated here, but their proof will be presented elsewhere. 相似文献
37.
The dynamics of the eigenvalues of a family of Sturm–Liouville operators with complex integrable PT-symmetric potential on a finite interval is studied. In the model case of the complex Airy operator, a criterion for the similarity of operators in the family to self-adjoint and normal operators is stated and the exceptional parameter values corresponding to multiple eigenvalues are analytically calculated. 相似文献
38.
A. A. Shkalikov 《Functional Analysis and Its Applications》2007,41(2):154-167
We prove that a dissipative operator in the Krein space has a maximal nonnegative invariant subspace provided that the operator admits matrix representation with respect to the canonical decomposition of the space and the upper right operator in this representation is compact relative to the lower right operator. Under the additional assumption that the upper and lower left operators are bounded (the so-called Langer condition), this result was proved (in increasing order of generality) by Pontryagin, Krein, Langer, and Azizov. We relax the Langer condition essentially and prove under the new assumptions that a maximal dissipative operator in the Krein space has a maximal nonnegative invariant subspace such that the spectrum of its restriction to this subspace lies in the left half-plane. Sufficient conditions are found for this restriction to be the generator of a holomorphic semigroup or a C 0-semigroup. 相似文献
39.
The paper deals with the Sturm-Liouville operator $$ Ly = - y'' + q(x)y, x \in [0,1], $$ generated in the space L 2 = L 2[0, 1] by periodic or antiperiodic boundary conditions. Several theorems on the Riesz basis property of the root functions of the operator L are proved. One of the main results is the following. Let q belong to the Sobolev spaceW 1 p [0, 1] for some integer p ≥ 0 and satisfy the conditions q (k)(0) = q (k)(1) = 0 for 0 ≤ k ≤ s ? 1, where s ≤ p. Let the functions Q and S be defined by the equalities $$ Q(x) = \int_0^x {q(t)dt, S(x) = Q^2 (x)} $$ and let q n , Q n , and S n be the Fourier coefficients of q, Q, and S with respect to the trigonometric system $ \{ e^{2\pi inx} \} _{ - \infty }^\infty $ . Assume that the sequence q 2n ? S 2n + 2Q 0 Q 2n decreases not faster than the powers n ?s?2. Then the system of eigenfunctions and associated functions of the operator L generated by periodic boundary conditions forms a Riesz basis in the space L 2[0, 1] (provided that the eigenfunctions are normalized) if and only if the condition $$ q_{2n} - s_{2n} + 2Q_0 Q_{2n} \asymp q_{ - 2n} - s_{2n} + 2Q_0 Q_{ - 2n} , n > 1, $$ holds. 相似文献
40.
The Sturm-Liouville problem is considered with a singular potential q(x) representing the derivative of a real function from the space L 2[0, 1] in the distributional sense. Two approaches are developed for the study of oscillation properties of eigenfunctions of this problem. The first approach is based on generalization of methods of the Sturm theory. The second one is based on development of variational principles.
相似文献
$\begin{array}{*{20}c} { - y'' + q(x)y = \lambda y,} \\ {y(0) = y(1) = 0} \\ \end{array} $