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11.
We prove a theorem on the completeness of the system of root functions of the Schrödinger operator L = ?d 2/dx 2 + p(x) on the half-line R+ with a potential p for which L appears to be maximal sectorial. An application of this theorem to the complex Airy operator L c = ?d 2/dx 2 + cx, c = const, implies the completeness of the system of eigenfunctions of L c for the case in which |arg c| < 2π/3.We use subtler methods to prove a theorem stating that the system of eigenfunctions of this special operator remains complete under the condition that |arg c| < 5π/6. 相似文献
12.
A. M. Savchuk A. A. Shkalikov 《Proceedings of the Steklov Institute of Mathematics》2013,283(1):181-196
We consider the inverse problem of recovering the potential for the Sturm-Liouville operator Ly = ?y″ + q(x)y on the interval [0, π] from the spectrum of the Dirichlet problem and norming constants (from the spectral function). For a fixed θ ≥ 0, with this problem we associate a map F: W 2 θ → l D θ , F(σ) = {s k } 1 ∞ , where W 2 θ = W 2 θ [0, π] is the Sobolev space, σ = ∫ q is a primitive of the potential q ∈ W 2 θ ? 1 , and l D θ is a specially constructed finite-dimensional extension of the weighted space l 2 θ ; this extension contains the regularized spectral data s = {s k } 1 ∞ for the problem of recovering the potential from the spectral function. The main result consists in proving both lower and upper uniform estimates for the norm of the difference ‖σ ? σ 1‖ θ in terms of the l D θ norm of the difference of the regularized spectral data ‖s ? s1‖ θ . The result is new even for the classical case q ∈ L 2, which corresponds to the case θ = 1. 相似文献
13.
14.
We study Schrödinger operatorsT+Q, whereT=?Δ is the Laplace operator andQ is the multiplication operator by a generalized function (distribution). We also consider generalizations for the case of the polyharmonic operatorT = (-δ) n 相似文献
15.
I. Z. Latypov A. A. Kalinkin A. V. Shkalikov A. A. Kalachev V. V. Samartsev 《Bulletin of the Russian Academy of Sciences: Physics》2009,73(12):1659-1664
The problem of generating a narrow-band biphoton field during spontaneous parametric down conversion of light in a quadratic
nonlinear medium in a cavity is discussed. It is shown that, in the case of double resonance (for signal and idle fields),
the form of the single-photon wave packet is biexponential and has a half-width determined by the photon life time in the
cavity. An experiment is carried out on the generation of narrow-band biphotons in a crystal of lithium iodine with I-type
matching in a frequency degenerate collinear regime. The second-order correlation function of the cavity-enhanced biphotons
is measured using the electric nanosecond time delay line. 相似文献
16.
A. A. Shkalikov 《Differential Equations》2009,45(4):580-590
We consider the operator function L(α, θ) = A(α) ? θR of two complex arguments, where A(α) is an analytic operator function defined in some neighborhood of a real point α 0 ∈ ? and ranging in the space of bounded operators in a Hilbert space ?. We assume that A(α) is a dissipative operator for real α in a small neighborhood of the point α 0 and R is a bounded positive operator; moreover, the point α 0 is a normal eigenvalue of the operator function L(α, θ 0) for some θ 0 ∈ ?, and the number θ 0 is a normal eigenvalue of the operator function L(α 0 θ). We obtain analogs and generalizations of well-known results for self-adjoint operator functions A(α) on the decomposition of α- and θ-eigenvalues in neighborhoods of the points α 0 and θ 0, respectively, and on the representation of the corresponding eigenfunctions by series. 相似文献
17.
In this paper, we deal with operators of the form
on the space ℝn. It is assumed that the principal part of L is a uniformly strongly elliptic operator and the coefficients c
α,β with |α| + |β| < 2m are distributions. We find sufficient conditions on these coefficients (in terms of generalized Sobolev spaces with negative
smoothness indices to which these coefficients belong) for the operator in question to be well defined in the sense of quadratic
forms.
Dedicated to the memory of B. M. Levitan
Paper supported by RFBR under grant no. 04-01-00712. 相似文献
18.
Selfadjoint linear pencils F–G are considered which have discrete spectrum and neither F nor G is definite. Several characterizations are given of a strongly definitizable property when F and G are bounded, and also when both operators are unbounded. The theory is applied to analysis of the stability of a linear second order initial-boundary value problem with boundary conditions dependent on the eigenvalue parameter.Research supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. 相似文献
19.
Turaikhanov D. A. Shkalikov A. V. Kalachev A. A. Imangulova I. R. Losevskii N. N. Razuvaeva E. V. Samagin S. A. Kotova S. P. 《Optics and Spectroscopy》2019,126(1):25-28
Optics and Spectroscopy - Specific features of the construction and recording of axially symmetric vortex fields using diffraction optical elements, which can be used for generating single-photon... 相似文献
20.
Mathematical Notes - 相似文献