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1.
This paper is devoted to the calculation of the deficiency index of a differential operator. In particular, we present sufficient conditions under which the operator with homogeneous boundary condition at zero is self-adjoint. 相似文献
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An uncertainty principle for the Sturm--Liouville operator
$$
L=\frac{d^2}{dt^2}+a(t)\frac{d}{dt}
$$
is established, as generalization of an inequality for Jacobi expansions proved in our
previous paper, which implies the uncertainty principle for ultraspherical expansions
by M. Rösler and M. Voit. The properties of the orthogonal set of eigenfunctions
of the operator L and the so-called conjugate orthogonal set are unified by introducing
the differential–difference operators, which are essential in our study. As
consequences, an uncertainty principle for Laguerre, Hermite, and generalized
Hermite expansions is obtained, respectively. 相似文献
3.
Michel Duhoux 《Mathematische Nachrichten》2001,225(1):19-38
We consider a Sturm – Liouville operator Lu = —(r(t)u′)′ + p (t)u , where r is a (strictly) positive continuous function on ]a, b [ and p is locally integrable on ]a, b[. Let r1(t) = (1/r) ds andchoose any c ∈]a, b[. We are interested in the eigenvalue problem Lu = λm(t)u, u (a) = u (b) = 0,and the corresponding maximal and anti .maximal principles, in the situation when 1/r ∈ L1 (a, c),1 /r ∉ L1 (c, b), pr1 ∉ L1 (a, c) and pr1 ∉ L1(c, b). 相似文献
4.
Henk De Snoo 《Mathematische Nachrichten》1996,182(1):99-126
In this note we consider regular Sturm-Liouville equations with a floating singularity of a special type: the coefficient of the second order derivative contains the eigenvalue parameter. We determine the form of the boundary conditions which make the problem selfadjoint after linearizing. In general the boundary conditions for the linearized system give rise to boundary conditions which involve the eigenvalue parameter in the original, non-linearized, problem. The boundary conditions give rise to a 2 × 2 matrix function, the so-called Titchmarsh-Weyl coefficient. The characteristic properties of this function are studied. The formal aspects of the theory of this class of equations turn out to be quite parallel to those for the usual situation when there is no floating singularity. 相似文献
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Discontinuous Sturm-Liouville Problems Containing Eigenparameter in the Boundary Conditions 总被引:3,自引:0,他引:3
M. KADAKAL O. Sh. MUKHTAROV 《数学学报(英文版)》2006,22(5):1519-1528
In this paper, discontinuous Sturm-Liouville problems, which contain eigenvalue parameters both in the equation and in one of the boundary conditions, are investigated. By using an operatortheoretic interpretation we extend some classic results for regular Sturm-Liouville problems and obtain asymptotic approximate formulae for eigenvalues and normalized eigenfunctions. We modify some techniques of [Fulton, C. T., Proc. Roy. Soc. Edin. 77 (A), 293-308 (1977)], [Walter, J., Math. Z., 133, 301-312 (1973)] and [Titchmarsh, E. C., Eigenfunctions Expansion Associated with Second Order Differential Equations I, 2nd edn., Oxford Univ. Pres, London, 1962], then by using these techniques we obtain asymptotic formulae for eigenelement norms and normalized eigenfunctions. 相似文献
7.
We study Sturm–Liouville differential operators on noncompact graphs without cycles (i.e., on trees) with standard matching
conditions in internal vertices. First we establish properties of the spectral characteristics and then we investigate the
inverse problem of recovering the operator from the so-called Weyl vector. For this inverse problem we prove a uniqueness
theorem and propose a procedure for constructing the solution using the method of spectral mappings.
Received: February 13, 2007. 相似文献
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Julio Flores 《Integral Equations and Operator Theory》2006,55(4):487-496
In the setting of K?the function spaces we present some suffcient conditions for a regular integral operator to be strictly
singular. 相似文献
12.
Mathematical Notes - We study Sturm–Liouville differential operators on the time scales consisting of finitely many isolated points and closed intervals. In the author’s previous paper,... 相似文献
13.
本文给出了奇型Sturm—Liouville微分算子限界自伴扩张的充要条件,从而得 到按边值条件分类的所有限界自伴边值条件,并直接回答了奇型Sturm—Liouville问题 的最小特征值不等式中相等的边值条件. 相似文献
14.
Jonathan Eckhardt 《Journal of Difference Equations and Applications》2013,19(11):1875-1887
We establish the connection between Sturm–Liouville equations on time scales and Sturm–Liouville equations with measure-valued coefficients. Based on this connection, we generalize several results for Sturm–Liouville equations on time scales, which have been obtained by various authors in the past. 相似文献
15.
研究了奇型Sturm-Liouville算子的逆问题.对于固定的n∈N,证明了Sturm-Liouville问题(1.3)-(1.5)的第n个特征值λ_n(q,H)关于H是严格单调增加的,及一组不同边界条件下的第n个特征值的谱集合{λ_n(q,H_k)}_(k=1)~(+∞)能够唯一确定(0,πr)上的势函数q(x). 相似文献
16.
We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations and similar operators in detail. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive a non-local self-adjoint operator similar to the Schrödinger operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem. 相似文献
17.
In direct sum spaces with inner product multiples, we study two-interval Sturm–Liouville problems. For singular problems,
we generate self-adjoint realizations for boundary conditions with any real coupling matrix whose determinant is positive.
This contrasts with the usual theory which requires the coupling matrix to have determinant one.
This paper is in final form and no version of it will be submitted for publication elsewhere.
Received: September 28, 2006. 相似文献
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In this paper, we consider the inverse scattering problem for the Sturm–Liouville operator on the half-line [0,∞) with Herglotz function of spectral parameter in the boundary condition. The scattering data of the problem is defined, and its properties are investigated. The main equation is obtained for the solution of the inverse problem and it is shown that the potential is uniquely recovered in terms of the scattering data. 相似文献
20.
Wei Gao GE Wei Ping SUN 《数学学报(英文版)》2005,21(5):1015-1026
By the use of continuation theorems and the duality principle it's proved that for both homogeneous and nonhomogeneous Sturm-Liouville boundary value problems with Laplacian type operators, relative superlinearity implies the existence of infinitely many solutions under a few weakly added conditions. 相似文献