Non-selfadjoint matrix Sturm-Liouville operators with spectral singularities

In this paper we investigate discrete spectrum of the non-selfadjoint matrix Sturm-Liouville operator L generated in L²(R₊,S) by the differential expression

     

Sturm-Liouville operators in the half axis with local perturbations

We give conditions which imply equivalence of the Lebesgue measure with respect to a measure μ generated as an average of spectral measures corresponding to Sturm-Liouville operators in the half axis. We apply this to prove that some spectral properties of these operators hold for large sets of boundary conditions if and only if they hold for large sets of positive local perturbations.     

Sturm-Liouville operators and their spectral functions

Assume that the differential operator −DpD+q in L²(0,∞) has 0 as a regular point and that the limit-point case prevails at ∞. If p≡1 and q satisfies some smoothness conditions, it was proved by Gelfand and Levitan that the spectral functions σ(t) for the Sturm-Liouville operator corresponding to the boundary conditions (pu′)(0)=τu(0), , satisfy the integrability condition . The boundary condition u(0)=0 is exceptional, since the corresponding spectral function does not satisfy such an integrability condition. In fact, this situation gives an example of a differential operator for which one can construct an analog of the Friedrichs extension, even though the underlying minimal operator is not semibounded. In the present paper it is shown with simple arguments and under mild conditions on the coefficients p and q, including the case p≡1, that there exists an analog of the Friedrichs extension for nonsemibounded second order differential operators of the form −DpD+q by establishing the above mentioned integrability conditions for the underlying spectral functions.     

Relative oscillation theory for Sturm-Liouville operators extended

We extend relative oscillation theory to the case of Sturm-Liouville operators Hu=r⁻¹(−^′(pu^′)+qu) with different p's. We show that the weighted number of zeros of Wronskians of certain solutions equals the value of Krein's spectral shift function inside essential spectral gaps.     

Regular singular Sturm-Liouville operators and their zeta-determinants

We consider Sturm-Liouville operators on the line segment [0,1] with general regular singular potentials and separated boundary conditions. We establish existence and a formula for the associated zeta-determinant in terms of the Wronski-determinant of a fundamental system of solutions adapted to the boundary conditions. This generalizes the earlier work of the first author, treating general regular singular potentials but only the Dirichlet boundary conditions at the singular end, and the recent results by Kirsten-Loya-Park for general separated boundary conditions but only special regular singular potentials.     

三组谱的Sturm-Liouville反问题

考虑与三组谱关联的逆Sturm-Liouville问题,证明了若对于给定的两组数列,在一定条件下,可划分为三组数列,使其分别为区间[0,a]上三个Sturm-Liouville问题的部分特征值,则通过三组谱的部分特征值能唯一确定区间[0,a]上的势函数q(x).     

边界条件中带有谱参数的不连续Sturm-Liouville算子的特征值问题

本文研究了具有转移条件且边界条件含特征参数的Sturm-Liouville算子L的特征值问题.首先,使用微分算子谱分析经典的方法,得到λ是该边值问题的特征值的充要条件,证明了该边值问题最多有可数个实的特征值、没有有限值的聚点.其次,通过渐近估计证得,所研究的Sturm-Liouville算子L有可数个离散的特征值且下方有界.     

Dependence of eigenvalues of regular Sturm-Liouville operators on the boundary condition

In this paper, we study the continuous dependence of eigenvalue of Sturm-Liouville differential operators on the boundary condition by using of implicit function theorem. The work not only provides a new and elementary proof of the above results, but also explicitly presents the expressions for derivatives of the n-th eigenvalue with respect to given parameters. Furthermore, we obtain the new results of the position and number of the generated double eigenvalues under the real coupled boundary condition.     

Multiplicities and relative position of eigenvalues of a quadratic pencil of Sturm-Liouville operators

For boundary value problems generated by a second-order differential equation with regular nonseparated boundary conditions, criteria for the eigenvalues to be multiple are given and the relative position of the eigenvalues is studied. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 369–381, March, 2000.     

Inverse nodal problems for the Sturm-Liouville differential operators on a star-type graph

We study inverse nodal problems for the second order differential operators on a star-type graph satisfying the standard matching conditions at the interior vertex. We prove uniqueness theorems and obtain a constructive solution to the inverse problems of this class. Original Russian Text Copyright ? 2009 Yurko V. A. The author was supported by the Russian Foundation for Basic Research (Grant 07-01-00003) and the National Science Council of Taiwan (Grant 07-01-92000-NSC-a). __________ Saratov. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 2, pp. 469–475, March–April, 2009.     

Sturm-Liouville算子方程边值问题的最小极值解

利用 Banach空间中度量广义逆理论 ,证明了 LP(a,b)空间中 Sturm-Liouville算子方程边值问题最小极值解的存在性 ,并借助 Banach空间几何方法给出了最小极值解存在的等价条件     

L~1-uniqueness of Sturm-Liouville operators

In this paper, we give two sufficient and necessary conditions for the L1-uniqueness of one-dimensional Sturm-Liouville operator Lf = af″ + bf′- V f, f∈D = C 0 ∞ (I), where I is an open interval of R and V≥0.     

含p-Laplace算子的Sturm-Liouville边值问题正解的性质

研究了含p-Laplace算子的Sturm-Liouville边值问题正解的性质.利用p-Laplace算子的性质,使用L’Hpital(洛必达)法则和闭区间上连续函数的最值性定理,研究了含p-Laplace算子的Sturm-Liouville边值问题,得到了其正解存在的两个必要条件.最后给出了主要结论的应用.结论丰富了边值问题研究领域的内容,为利用计算机使用迭代技术求这类边值问题的近似解提供了新的渠道,推广了一些文献的结论.     

Eigenvalue asymptotics for Sturm-Liouville operators with singular potentials

We derive eigenvalue asymptotics for Sturm-Liouville operators with singular complex-valued potentials from the space , α∈[0,1], and Dirichlet or Neumann-Dirichlet boundary conditions. We also give application of the obtained results to the inverse spectral problem of recovering the potential from these two spectra.     

Reconstruction of Sturm-Liouville differential operators with singularities inside the interval

The inverse spectral problem for Sturm-Liouville differential operators on a finite interval is studied for an arbitrary and finite number of regular singular points inside the interval. A uniqueness theorem is proved; necessary and sufficient conditions and a procedure for the solution of the inverse problem are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 143–156, July, 1998.This research was supported by the Ministry of Education (KTsFE) under grant No. 96-1.7-4 and by the Russian Foundation for Basic Research under grant No. 97-01-00566.     

Inverse indefinite Sturm-Liouville problems with three spectra

We prove that the potential q(x) of an indefinite Sturm-Liouville problem on the closed interval [a,b] with the indefinite weight function w(x) can be determined uniquely by three spectra, which are generated by the indefinite problem defined on [a,b] and two right-definite problems defined on [a,0] and [0,b], where point 0 lies in (a,b) and is the turning point of the weight function w(x).     

The similarity problem for J-nonnegative Sturm-Liouville operators

Sufficient conditions for the similarity of the operator with an indefinite weight r(x)=(sgnx)|r(x)| are obtained. These conditions are formulated in terms of Titchmarsh-Weyl m-coefficients. Sufficient conditions for the regularity of the critical points 0 and ∞ of J-nonnegative Sturm-Liouville operators are also obtained. This result is exploited to prove the regularity of 0 for various classes of Sturm-Liouville operators. This implies the similarity of the considered operators to self-adjoint ones. In particular, in the case r(x)=sgnx and , we prove that A is similar to a self-adjoint operator if and only if A is J-nonnegative. The latter condition on q is sharp, i.e., we construct such that A is J-nonnegative with the singular critical point 0. Hence A is not similar to a self-adjoint operator. For periodic and infinite-zone potentials, we show that J-positivity is sufficient for the similarity of A to a self-adjoint operator. In the case q≡0, we prove the regularity of the critical point 0 for a wide class of weights r. This yields new results for “forward-backward” diffusion equations.     

一类带不定权函数的奇型Sturm-Liouville算子的局部可定性(英文)

研究一类带不定权函数的奇型Sturm-Liouville算子,给出相应自伴算子在无穷点邻域的局部可定性.     

常型Sturm-Liouville问题的左定边值条件

本文刻画了常型Sturm-Liouville问题的左定边值条件．通过Sturm-Liouville微分算式的系数、区间端点以及边值条件给出了其左定性的充要条件．应用自伴边值条件分类,确切地给出了所有可能的左定边值条件．