A criteria for correct solvability in L_p(R) of a general Sturm-Liouville equation

Traces and inverse nodal problems for a class of Sturm-Liouville operators with retarded argument

In this study, we investigate the traces and solutions of inverse nodal problems of discontinuous Sturm-Liouville operators with retarded argument and with a finite number of transmission conditions.

Expansion Theorem for Sturm-Liouville problems transmission conditions

The purpose of this paper is to extend some spectral properties of regular Sturm-Liouville problems to the special type discontinuous boundary-value problem, which consists of a Sturm-Liouville equation together with eigenparameter-dependent boundary conditions and two supplementary transmission conditions. We construct the resolvent operator and Green's function and prove theorems about expansions in terms of eigenfunctions in modified Hilbert space $L_{2}[a, b]$. \vskip0.3cm\noindent Keywords: Boundary-value problems, transmission conditions, Resolvent operator, expansion theorem.

On a Sturm-Liouville Type Problem with Retarded Argument

In this work a Sturm-Liouville type problem with retarded argument which contains spectral parameter in the boundary conditions and with transmission conditions at the point of discontinuity are investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions.

On some a priori majorant of eigenvalues of Sturm--Liouville problems

Let $M_γ$ be precise a priori majorant of first eigenvalues of Sturm--Liouville problems $-y"+qy=λy,\quad y(0)=y(1)=0$, where $q\leqslant 0$ and $\int_0^1 |q|^γ\,dx=1$, $γ\in (0,1/2)$. It is shown that the inequality $M_γ<π^2$ is true.

A survey of the regular weighted Sturm-Liouville problem - The non-definite case

This is a survey of the non-definite Sturm-Liouville problem from its inception in the early 1900's until 1986.

On the main equation of inverse sturm-liouville operator with discontinuous coefficient

In this work, a boundary value problem for Sturm-Liouville operator with discontinuous coefficient is examined. The main equation is obtained which has an important role in solution of inverse problem for boundary value problem and uniqueness of its solution is proved. Uniqueness theorem for the solution of the inverse problem is given.

Inverse Scattering Problem for Sturm--Liouville Operators

On the space $L^{2}(\mathbb{R})$ the Sturm-Liouville operator $L$ with certain behavior of the potential at infinity is considered. It is proved that $L$ is uniquely determined by its scattering data. The recovery of $L$ is reduced to the solving of a certain linear integral equation.

A note on Sturm-Liouville problems whose spectrum is the set of prime numbers

We show that there is no classical regular Sturm-Liouville problem on a finite interval whose spectrum consists of infinitely many distinct primes numbers. In particular, this answers in the negative a question raised by Zettl in his book on Sturm-Liouville theory. We also show that there {\it may} exist such a problem if the parameter dependence is nonlinear.

Sampling Theory for Sturm-Liouville Problem with Boundary and Transmission Conditions Dependent on Eigenparameter

In this paper, we investigate the sampling analysis associated with discontinuous Sturm-Liouville problem which has transmission conditions at two points of discontinuity also contains an eigenparameter in a boundary condition and two transmission conditions. We establish briefly spectral properties of the problem and then we prove the sampling theorem associated with the problem.

On non-definite Sturm-Liouville problems with two turning points

This is an inaugural study of the Dirichlet problem associated with a regular non-definite Sturm-Liouville equation in the case of two turning points. We give a priori lower bounds on the Richardson numbers associated with this problem thereby complementing pioneering results by Atkinson and Jabon (1984) in the one turning point case

p-Laplacian Fractional Sturm-Liouville Problem for Diffusion Operator via Impulsive Condition

In this study, the existence results of solution is given for fractional p-Laplacian Stum-Liouville problem for diffusion operator of order with impulsive conditions. The derivatives are described in Riemann-Liouville and Caputo sense. The Riemann-Liouville integral operator is used to acquire the integral representation of solution. The existence of solution is demonstrate via Schaefer fixed point theorem.

Correct Solvability, Embedding Theorems and Separability for the Sturm-Liouville Equation

We consider the equation - y"(x)+q(x)y(x)=f(x), x\in R and the weighted function space S_p^{(2)}(R,q)=\{y\in AC_{\loc}^{(1)}(R):\|y"-qy\|_p+\|q^{1/p}y\|_p<\infty\}; p\in[1,\infty), f\in L_p(R)$ and $0\le q\in L_1^{\loc}(R)$. We show that there exists an embedding S_p^{(2)}(R,q)\hookrightarrow L_p(R) if and only if equation above is correctly solvable in $L_p(R).

A Generalization of Classical Symmetric Orthogonal Functions Using a Symmetric Generalization of Sturm-Liouville Problems

In this paper, usual Sturm-Liouville problems are extended for symmetric functions so that the corresponding solutions preserve the orthogonality property. Two basic examples, which are special cases of a generalized Sturm-Liouville problem, are then introduced. First example generalizes the associated Legendre functions having extensive applications in physics and the second example introduces a generic differential equation with various sub-cases that have orthogonal solutions. For instance, this generic equation possesses a symmetric differential equation containing a basic solution of symm…

The number of Dirac-weighted eigenvalues of Sturm-Liouville equations with integrable potentials and an application to inverse problems

In this paper, we further Meirong Zhang, et al.'s work by computing the number of weighted eigenvalues for Sturm-Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm-Liouville equation with a general integrable potential, if its weight is a positive linear combination of $n$ Dirac Delta functions, then it has at most $n$ (may be less than $n$, or even be $0$) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of …

The Sturm--Liouville problem with singular potential and the extrema of the first eigenvalue

On the basis of the theory of Sturm--Liouville problem with distribution coefficients we get the infima and suprema of the first eigenvalue of the problem $-y" + (q-λ) y=0, y'(0) -k_0^2 y(0) = y'(1) + k_1^2 y(1) = 0$, where $q$ belongs to the set of constant-sign summable functions on $[0,1]$ such that $\int_0^1 q dx=\pm 1$.

Fractional Sturm-Liouville eigenvalue problems, I

We introduce and present the general solution of three two-term fractional differential equations of mixed Caputo/Riemann Liouville type. We then solve a Dirichlet type Sturm-Liouville eigenvalue problem for a fractional differential equation derived from a special composition of a Caputo and a Riemann-Liouville operator on a finite interval where the boundary conditions are induced by evaluating Riemann-Liouville integrals at those end-points. For each $1/2<α<1$ it is shown that there is a finite number of real eigenvalues, an infinite number of non-real eigenvalues, that the number of such r…

Sturm-Liouville operators on time scales

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.

Comparison Criteria for Discrete Fractional Sturm-Liouville Equations

In this study, we give the Sturm comparison theorems for discrete fractional Sturm-Liouville (DFSL) equations within Riemann-Liouville and Grünwald-Letnikov sense. The emergence of Sturm-Liouville equations began as one dimensional Schrödinger equation in quantum mechanics and one of the most important results is Sturm comparison theorems [27]. These theorems give information about the properties of zeros of two equations having different potentials.

Ramanujan's master theorem for Sturm-Liouville operator

In this paper we prove an analogue of Ramanujan's master theorem in the setting of Sturm Liouville operator.

Mochizuki-Trooshin type theorem for Sturm-Liouville problem on time scales

In this paper, we consider an interior inverse Sturm-Liouville problem on time scale T and give a Mochizuki-Trooshin type theorem.

A new kind of uniqueness theorems for inverse Sturm-Liouville problems

We prove Marchenko-type uniqueness theorems for inverse Sturm-Liouville problems. Moreover, we prove a generalization of Ambarzumyans theorem.

A Sturm Liouville theorem for quadratic operator pencils

We establish a Sturm{Liouville theorem for quadratic operator pencils counting their unstable real roots, with applications to stability of waves. Such pencils arise, for example, in reduction of eigenvalue systems to higher-order scalar problems.

Riesz bases of root vectors of indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions. II

We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions affinely dependent on the eigenparameter. We give sufficient conditions under which the root vectors of this Sturm-Liouville problem can be selected to form a Riesz basis of a corresponding weighted Hilbert space.

Asymptotic properties of eigenvalues and eigenfunctions of a Sturm-Liouville problem with discontinuous weight function

In this paper, by using the similar methods of [O. Sh. Mukhtarov and M. Kadakal, Some spectral properties of one Sturm-Liouville type problem with discontinuous weight, Siberian Mathematical Journal, 46 (2005) 681-694] we extend some spectral properties of regular Sturm-Liouville problems to those which consist of a Sturm-Liouville equation with discontinuous weight at two interior points together with spectral parameter-dependent boundary conditions. We give an operator-theoretic formulation for the considered problem and obtain asymptotic formulas for the eigenvalues and eigenfunctions.

Sturm Liouville Equations in the frame of fractional operators with Mittag-Leffler kernels and their discrete versions

Very recently, some authors have studied new types of fractional derivatives whose kernels are nonsingular. In this article, we study Sturm-Liouville Equations ($SLEs$) in the frame of fractional operators with Mittag-Leffler kernels. We formulate some Fractional Sturm-Liouville Problems ($FSLPs$) with the diffferential part containing the left and right sided derivatives. We investigate the self-adjointness, eigenvalue and eigenfunction properties of the corresponding Fractional Sturm-Liouville Operators ($FSLOs$) by using fractional integration by parts formulas. The nabla discrete version o…

Half-inverse spectral problems for Sturm--Liouville operators with singular potentials

Half-inverse spectral problem for a Sturm--Liouville operator consists in reconstruction of this operator by its spectrum and half of the potential. We give the necessary and sufficient conditions for solvability of the half-inverse spectral problem for the class of Sturm--Liouville operators with singular potentials from the space $W^{-1}_2(0,1)$ and provide the reconstruction algorithm.

On the "movement" of the zeros of eigenfunctions of the Sturm-Liouville problem

We study the dependence of the zeros of eigenfunctions of Sturm-Liouville problem on the parameters that define the boundary conditions. As a corollary, we obtain Sturm oscillation theorem, which states that the $n$-th eigenfunction has $n$ zeros.

A Sub-Density Theorem of Sturm-Liouville Eigenvalue Problem with Finitely Many Singularities

We study the distribution of the Sturm-Liouville eigenvalues of a potential with finitely many singularities. There is an asymptotically periodical structure on this class of eigenvalues as described by the entire function theory. We describe the singularities of its potential function explicitly in its eigenvalue asymptotics.

Riesz Bases of Root Vectors of Indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions, I

We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions, one being affinely dependent on the eigenparameter. We give sufficient conditions under which a basis of each root subspace for this Sturm-Liouville problem can be selected so that the union of all these bases constitutes a Riesz basis of a corresponding weighted Hilbert space.

Modified Expansion Theorem for Sturm-Liouville problem with transmission conditions

This paper is devoted to the derivation of expansion a associated with a discontinuous Sturm-Liouville problems defined on $[-π, 0)\cup(0,π]$. We derive an eigenfunction expansion theorem for the Green's function of the problem as well as a theorem of uniform convergence of a certain class of functions.

Inverse Sturm-Liouville problems with fixed boundary conditions

Necessary and sufficient conditions for two sequences $\{μ_n\}_{n=0}^\infty$ and $\{ a_n\}_{n=0}^\infty$ to be the spectral data for a certain Sturm-Liouville problem are well known. We add two more conditions so that the same two sequences become necessary and sufficient for being the spectral data for a Sturm-Liouville problem with fixed boundary conditions.

Inverse problems for a conformable fractional Sturm-Liouville operator

In this paper, a Sturm-Liouville boundary value problem equiped with conformable fractional derivates is considered. We give some uniqueness theorems for the solutions of inverse problems according to the Weyl function, two given spectra and classical spectral data. We also study on half-inverse problem and prove a Hochstadt and Lieberman-type theorem.

Inverse Sturm-Liouville problems with summable potential

We describe the necessary and sufficient conditions for two sequences {μ_n}^\infty_n=0 and {a_n}^\infty_n=0 to be correspondingly the set of eigenvalues and the set of norming constants of a Sturm-Liouville problem with real summable potential q and in advance fixed separated boundary conditions.

Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions

Inverse problems of recovering the coefficients of Sturm-Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: (1) from the sequences of eigenvalues and norming constants; (2) from two spectra. Necessary and sufficient conditions for the solvability of these inverse problems are obtained.

On the norming constants of the Sturm-Liouville problem

We derive new asymptotic formulae for the norming constants of Sturm-Liouville problem with summable potentials, which generalize and make more precise previously known formulae. Moreover, our formulae take into account the smooth dependence of norming constants on boundary conditions. We also find some new properties of the remainder terms of asymptotics.

An Inverse Problem for Sturm-Liouville Operators on the Half-line Having Bessel-type Singularity in an Interior Point

We study the inverse problem of recovering Sturm-Liouville operators on the half-line with a Bessel-type singularity inside the interval from the given Weyl function. The corresponding uniqueness theorem is proved, a constructive procedure for the solution of the inverse problem is provided, also necessary and sufficient conditions for the solvability of the inverse problem are obtained.

Riesz bases generated by the spectra of Sturm-Liouville problems

Let $\{λ_n^2\}_{n = 0}^\infty$ be the spectra of a Sturm-Liouville problem on $[0,π]$. We investigate the question: Do the systems $\{\cos(λ_n x)\}_{n= 0}^\infty$ or $\{\sin(λ_n x)\}_{n = 0}^\infty$ form Riesz bases in ${L^2}[0,π]$? The answer is almost always positive.

正则 Sturm-Liouville 算子的局部化与景观函数

Localization and the landscape function for regular Sturm-Liouville operators

本文研究了正则 Sturm-Liouville 算子本征函数的局部化问题，导出了局部化系数的渐近与非渐近上下界，并利用第一本征函数刻画了景观函数，最后通过数值实验验证了理论结果。

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 $W^{\al-1}_{2}(0,1)$, $\al\in[0,1]$, 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.

Sturm-Liouville hypergroups without the compactness axiom

We establish a positive product formula for the solutions of the Sturm-Liouville equation $\ell(u) = λu$, where $\ell$ belongs to a general class which includes singular and degenerate Sturm-Liouville operators. Our technique relies on a positivity theorem for possibly degenerate hyperbolic Cauchy problems and on a regularization method which makes use of the properties of the diffusion semigroup generated by the Sturm-Liouville operator. We show that the product formula gives rise to a convolution algebra structure on the space of finite measures, and we discuss whether this structure satisfi…

Completeness of eigenfunctions of Sturm-Liouville problems with discontinuities at three points

In this work, we study discontinuous Sturm-Liouville type problems with eigenparameter dependent boundary condition and transmission conditions at three interior points. A self-adjoint linear operator A is defined in a suitable Hilbert space H such that the eigenvalues of such a problem coincide with those of A. We show that the eigenvalues of the problem are analytically simple, and the eigenfunctions of A are complete in H.

关于下半有界（奇异）Sturm-Liouville算子的半双线性形式

On Sesquilinear Forms for Lower Semibounded (Singular) Sturm-Liouville Operators

本文针对下半有界奇异Sturm-Liouville算子，利用广义边界值及主/非主解，详细刻画了其自伴延拓对应的半双线性形式，覆盖所有端点类型情况。

Inverse nodal problems for Sturm-Liouville equation with nonlocal boundary conditions

In this paper, a Sturm--Liouville problem with some nonlocal boundary conditions of the Bitsadze-Samarskii type is studied. We show that the coefficients of the problem can be uniquely determined by a dense set of nodal points. Moreover, we give an algorithm for the reconstruction of the potential function and some other coefficients in the boundary conditions.

Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x

We consider a singular Sturm-Liouville expression with the indefinite weight sgn x. To this expression there is naturally a self-adjoint operator in some Krein space associated. We characterize the local definitizability of this operator in a neighbourhood of $\infty$. Moreover, in this situation, the point $\infty$ is a regular critical point. We construct an operator $A=(\sgn x)(-d^2/dx^2+q)$ with non-real spectrum accumulating to a real point. The obtained results are applied to several classes of Sturm-Liouville operators.

New Estimations for Sturm-Liouville Problems in Difference Equations

In this paper, Sturm-Liouville problem for difference equations is considered with potential function q(n). The representations of solutions are obtained by variation of parameters method. These solutions are proved, using summation by parts. Also, estimation of asymptotic expansion of the solutions are established.

Spectral partitions for Sturm-Liouville problems

We look for best partitions of the unit interval that minimize certain functionals defined in terms of the eigenvalues of Sturm-Liouville problems. Via Γ-convergence theory, we study the asymptotic distribution of the minimizers as the number of intervals of the partition tends to infinity. Then we discuss several examples that fit in our framework, such as the sum of (positive and negative) powers of the eigenvalues and an approximation of the trace of the heat Sturm-Liouville operator.

Variational methods for fractional $q$-Sturm--Liouville Problems

In this paper, we formulate a regular $q$-fractional Sturm--Liouville problem (qFSLP) which includes the left-sided Riemann--Liouville and the right-sided Caputo $q$-fractional derivatives of the same order $α$, $α\in (0,1)$. We introduce the essential $q$-fractional variational analysis needed in proving the existence of a countable set of real eigenvalues and associated orthogonal eigenfunctions for the regular qFSLP when $α>1/2$ associated with the boundary condition $y(0)=y(a)=0$. A criteria for the first eigenvalue is proved. Examples are included. These results are a generalization of th…

A Hilbert space on left-definite Sturm-Liouville difference equation

We investigate the discrete Sturm-Liouville equations. In the present situation, the right hand side of the equation does not give rise to a positive definite quadratic form and we use instead the left hand side to define such a form. We prove in this paper that this form determines a Hilbert space (such problems are called left-definite).

Multi-interval dissipative Sturm-Liouville boundary-value problems with distributional coefficients

The paper investigates spectral properties of multi-interval Sturm-Liouville operators with distributional coefficients. Constructive descriptions of all self-adjoint and maximal dissipative/accumulative extensions in terms of boundary conditions are given. Sufficient conditions for the resolvents of these operators to be operators of the trace class and for the systems of root functions to be complete are found. Results of paper are new for one-interval boundary value problems as well.

Analyticity and uniform stability of the inverse singular Sturm--Liouville spectral problem

We prove that the potential of a Sturm--Liouville operator depends analytically and Lipschitz continuously on the spectral data (two spectra or one spectrum and the corresponding norming constants). We treat the class of operators with real-valued distributional potentials in the Sobolev class W^{s-1}_2(0,1), s\in[0,1].

关于Meixner-Pollaczek多项式与Sturm-Liouville问题

On the Meixner-Pollaczek polynomials and the Sturm-Liouville problems

本文系统研究了Meixner-Pollaczek多项式，运用中心差分算子分析Sturm-Liouville问题，给出递推关系的两个线性无关解及相关差分方程，并讨论了第二类函数的建立。

基于 Sturm-Liouville 理论的圆盘上卷积的离散谱

Discrete Spectra of Convolutions on Disks using Sturm-Liouville Theory

本文基于 Sturm-Liouville 理论系统研究了圆盘上函数卷积的离散谱方法，并探讨了零值边界条件和导数边界条件下的理论表现。

Inverse problem of the spectral analysis for the Sturm-Liouville operator with non-separated boundary conditions and spectral parameter in the boundary condition

This work deals with an inverse problem for the Sturm-Liouville operator with non-separated boundary conditions, one of which linearly depends on a spectral parameter. Uniqueness theorem is proved, solution algorithm is constructed and sufficient conditions for solvability of inverse problem are obtained. As spectral data, we only use the spectrum of one boundary value problem and some sequence of signs.

A direct method for solving inverse Sturm-Liouville problems

We consider two main inverse Sturm-Liouville problems: the problem of recovery of the potential and the boundary conditions from two spectra or from a spectral density function. A simple method for practical solution of such problems is developed, based on the transmutation operator approach, new Neumann series of Bessel functions representations for solutions and the Gelfand-Levitan equation. The method allows one to reduce the inverse Sturm-Liouville problem directly to a system of linear algebraic equations, such that the potential is recovered from the first element of the solution vector.…