共查询到20条相似文献,搜索用时 11 毫秒
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We study non-self-adjoint second-order differential operators with a constant delay. We establish properties of the spectral characteristics and investigate the inverse problem of recovering operators from their spectra. The uniqueness theorem is proved for this inverse problem. 相似文献
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We consider Sturm–Liouville operators in the half axis generated by shifts of the potential and prove that Lebesgue measure is equivalent to a measure defined as an average of spectral measures which correspond to these operators. This is then used to obtain results on stability of spectral types under change of parameters such as boundary conditions, local perturbations, and shifts. In particular if for a set of shifts of positive measure the corresponding operators have α-singular, singular continuous and (or) point spectrum in a fixed interval, then this set of shifts has to be unbounded. Moreover, there are large sets of boundary conditions and local perturbations for which the corresponding operators enjoy the same property. 相似文献
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Natalia P. Bondarenko 《Applicable analysis》2018,97(7):1209-1222
The matrix Sturm–Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval is studied. We consider inverse problems by the Weyl matrix and by the spectral data for this equation. Constructive solutions, based on the method of spectral mappings, are obtained for these inverse problems. 相似文献
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Mathematical Notes - 相似文献
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We consider (in general noncoercive) mixed problems in a bounded domain D in ? n for a second-order elliptic partial differential operator A(x, ?). It is assumed that the operator is written in divergent form in D, the boundary operator B(x, ?) is the restriction of a linear combination of the function and its derivatives to ?D and the boundary of D is a Lipschitz surface. We separate a closed set Y ? ?D and control the growth of solutions near Y. We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, where the weight is a power of the distance to the singular set Y. Finally, we prove the completeness of the root functions associated with L.The article consists of two parts. The first part published in the present paper, is devoted to exposing the theory of the special weighted Sobolev–Slobodetskii? spaces in Lipschitz domains. We obtain theorems on the properties of these spaces; namely, theorems on the interpolation of these spaces, embedding theorems, and theorems about traces. We also study the properties of the weighted spaces defined by some (in general) noncoercive forms. 相似文献
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We consider a (generally, noncoercive) mixed boundary value problem in a bounded domain D of Rn for a second order elliptic differential operator A(x, ?). The differential operator is assumed to be of divergent form in D and the boundary operator B(x, ?) is of Robin type on ?D. The boundary of D is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset Y ? ?D and control the growth of solutions near Y. We prove that the pair (A, B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set Y. Moreover, we prove the completeness of root functions related to L. 相似文献
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This paper deals with the eigenvalue problems for the Sturm–Liouville operators generated by the differential expression with singular coefficients q(x) in the sense of distributions. We obtain the inequalities among the eigenvalues corresponding to different self-adjoint boundary conditions. The continuity region, the differentiability and the monotonicity of the nth eigenvalue corresponding to the separated boundary conditions are given. Oscillation properties of the eigenfunctions of all the coupled Sturm–Liouville problems are characterized. The main results of this paper can also be applied to solve a class of transmission problems. 相似文献
L(y)=−(p(x)y′)′+q(x)y
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The aim of this note is to prove the following theorem. Let
where P(ix) is a nonnegative homogeneous elliptic polynomial on R
d
and V is a nonnegative polynomial potential. Then for every 1 < p < ∞ and every α > 0 there exist constants C
1, C
2 > 0 such that
and
for f in the Schwartz class . We take advantage of the Christ inversion theorem for singular integral operators with a small amount of smoothness on
nilpotent Lie groups, the maximal subelliptic L
2-estimates for the generators of stable semi-groups of measures, and the principle of transference of Coifman–Weiss.
In memory of Tadek Pytlik, our teacher and friend.
Research supported by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis,
Nonlinear Analysis and Probability” MTKD-CT-2004-013389 and by Polish funds for science in years 2005–2008 (research project
1P03A03029). 相似文献
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The paper addresses nonlinear inverse Sturm–Liouville-type problems with constant delay. Since many processes in the real world possess nonlocal nature, operators with delay as well as other classes of nonlocal operators are continuously finding numerous applications in the natural sciences and engineering. However, in spite of a large number of works devoted to inverse problems for operators with delay, the existing results do not give a comprehensive picture for all values of the delay parameter. Namely, for small delays, even such a basic question as the unique solvability of the inverse problem has been remaining open for many years. Since the problems with delay approximate the classical Sturm–Liouville problems as soon as the delay parameter tends to zero, many researchers expected the unique solvability as in the classical case. Here we give, however, a negative answer to this long-term basic question by constructing infinite families of iso-bispectral potentials. For this purpose, we develop a unified general approach that simultaneously covers various types of boundary conditions and allows one to significantly shorten the related proofs. 相似文献
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Alberto Gil C. P. Ramos 《Advances in Computational Mathematics》2018,44(2):395-421
The current paper concerns the uniform and high-order discretization of the novel approach to the computation of Sturm–Liouville problems via Fer streamers, put forth in Ramos and Iserles (Numer. Math. 131(3), 541—565 2015). In particular, the discretization schemes are shown to enjoy large step sizes uniform over the entire eigenvalue range and tight error estimates uniform for every eigenvalue. They are made explicit for global orders 4,7,10. In addition, the present paper provides total error estimates that quantify the interplay between the truncation and the discretization in the approach by Fer streamers. 相似文献
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《Journal of Computational and Applied Mathematics》2001,132(2):443-459
The numerical solution of the Sturm–Liouville problem can be achieved using shooting to obtain an eigenvalue approximation as a solution of a suitable nonlinear equation and then computing the corresponding eigenfunction. In this paper we use the shooting method both for eigenvalues and eigenfunctions. In integrating the corresponding initial value problems we resort to the boundary value method. The technique proposed seems to be well suited to supplying a general formula for the global discretization error of the eigenfunctions depending on the discretization errors arising from the numerical integration of the initial value problems. A technique to estimate the eigenvalue errors is also suggested, and seems to be particularly effective for the higher-index eigenvalues. Numerical experiments on some classical Sturm–Liouville problems are presented. 相似文献
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This article is concerned with the oscillatory properties of the eigenfunctions of a class of singular Sturm—Liouville problems—(p y)+q y=w y on (a, b), where the weight functionw vanishes on a subinterval of positive measure, or where the weight functionw changes sign on (a, b).This work was supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38. 相似文献
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A. V. Menovshchikov 《Siberian Mathematical Journal》2016,57(5):849-859
We obtain necessary and sufficient conditions for a homeomorphism of domains in a Euclidean space to generate a bounded embedding operator of the Orlicz–Sobolev spaces defined by a special class of N-functions. 相似文献
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Classical Sturm–Liouville problems of q-difference variables are extended for symmetric discrete functions such that the corresponding solutions preserve the orthogonality property. Some illustrative examples are given in this sense. 相似文献