B. M. Brown, M. S. P. Eastham
We are concerned with the Sturm-Liouville problem on the half line. We show that when the potential $q$ is subject only to power decay at infinity the $L^2$ solution may be continued into a sector of the so-called un-physical sheet. This gives rise to resonance free regions and the numerical estimation of resonances outside these regions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We consider Sturm-Liouville operators with singular potentials from the class on star-type graph with cycle, which consist the edges with commensurable lengths. Asymptotic representation for eigenvalues for such operators is obtained. Recovering of the characteristic function the Sturm-Liouville operators with the singular potentials is considered.
We solve the inverse spectral problem of recovering the singular potentials $q\in W^{-1}_{2}(0,1)$ of Sturm-Liouville operators by two spectra. The reconstruction algorithm is presented and necessary and sufficient conditions on two sequences to be spectral data for Sturm-Liouville operators under consideration are given.
We study spectral properties of energy-dependent Sturm-Liouville equations, introduce the notion of norming constants and establish their interrelation with the spectra. One of the main tools is the linearization of the problem in a suitable Pontryagin space.
In this article we obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the nonself-adjoint operator generated by a system of Sturm-Liouville equations with summable coefficients and the quasiperiodic boundary conditions. Then using these asymptotic formulas, we find the conditions on the potential for which the root funcions of this operator form a Riesz basis.
In this paper, we present a new approachment for Sturm-Liouville problem having special potentials. We acquire the representations of solutions and asymptotic formulas for solutions with regard to initial conditions. Also, a few applications are given to show the requirement of Sturm-Liouville difference equations having potential function in view of suitability to the spectral theory. The approximate numerical outcomes for the eigenfunctions are compared with each other.
An inverse spectral problem for the Sturm-Liouville operator with a singular potential from the class $W_2^{-1}$ is solved by the method of spectral mappings. We prove the uniqueness theorem, develop a constructive algorithm for solution, and obtain necessary and sufficient conditions of solvability for the inverse problem in the self-adjoint and the non-self-adjoint cases
In this paper we study a Sturm--Liouville operator $Ly=-y"+q(x)y$ in the space $L_2[0,π]$ with Direchlet boundary conditions. Here the potential $q$ is a first order distribution: $q\in W_2^{-1}[0,π]$. Such operators were defined in our previous papers. Here we clerify two leading terms in asymptotic formulae for eigenfunctions of such operators and for functions of biorthogonal system.
The inverse spectral problem is solved for the class of Sturm-Liouville operators with singular real-valued potentials from the space $W^{-1}_2(0,1)$. The potential is recovered via the eigenvalues and the corresponding norming constants. The reconstruction algorithm is presented and its stability proved. Also, the set of all possible spectral data is explicitly described and the isospectral sets are characterized.
The matrix Sturm-Liouville operator on a finite interval with singular potential of class $W_2^{-1}$ and the general self-adjoint boundary conditions is studied. This operator generalizes the Sturm-Liouville operators on geometrical graphs. We investigate the inverse problem that consists in recovering the considered operator from the spectral data (eigenvalues and weight matrices). The inverse problem is reduced to a linear equation in a suitable Banach space, and a constructive algorithm for the inverse problem solution is developed. Moreover, we obtain the spectral data characterization for…
We consider the inverse Sturm-Liouville problem with one discontinuous point on three-star graph, we deduced the distribution of the eigenvalues, and proved that one spectrum could uniquely determine the unknown potential and jump information when we know the whole potentials on two edges and half potential on another edge.
For a particular family of long-range potentials $V$, we prove that the eigenvalues of the indefinite Sturm--Liouville operator $A = \mathrm{sign}(x)(-Δ+ V(x))$ accumulate to zero asymptotically along specific curves in the complex plane. Additionally, we relate the asymptotics of complex eigenvalues to the two-term asymptotics of the eigenvalues of associated self-adjoint operators.
We introduce a novel approach for dealing with eigenvalue problems of Sturm-Liouville operators generated by the differential expression \begin{equation*} Ly=\frac{1}{r}\left( -(p\left[ y^{\prime }+sy\right] )^{\prime }+sp\left[ y^{\prime }+sy\right] +qy\right) \end{equation*} which is based on norm resolvent convergence of classical Sturm-Liouville operators. This enables us to describe the continuous dependence of the $n$-th eigenvalue on the space of self-adjoint boundary conditions and the coefficients of the differential equation after giving the inequalities among the eigenvalues. Moreov…
The limit distribution of the discrete spectrum of the Sturm-Liouville problem with complex-valued polynomial potential on an interval, on a half-axis, and on the entire axis is studied. It is shown that at large parameter values, the eigenvalues are concentrated along the so-called limit spectral graph; the curves forming this graph are classified. Asymptotics of eigenvalues along curves of various types in the graph are calculated.
The works of V. A. Vinokurov have shown that eigenvalues and normalized eigenfunctions of Sturm-Liouville problems are analytic in potentials, considered as mappings from the Lebesgue space to the space of real numbers and the Banach space of continuous functions respectively. Moreover, the first-order Frechet derivatives are known and paly an important role in many problems. In this paper, we will find the second-order Frechet derivatives of eigenvalues in potentials, which are also proved to be negative definite quadratic forms for some cases.
Boundary value problems for Sturm-Liouville operators with potentials from the class $W_2^{-1}$ on a star-shaped graph are considered. We assume that the potentials are known on all the edges of the graph except two, and show that the potentials on the remaining edges can be constructed by fractional parts of two spectra. A uniqueness theorem is proved, and an algorithm for the constructive solution of the partial inverse problem is provided. The main ingredient of the proofs is the Riesz-basis property of specially constructed systems of functions.
The Sturm-Liouville operator with singular potentials on the lasso graph is considered. We suppose that the potential is known a priori on the boundary edge, and recover the potential on the loop from a part of the spectrum and some additional data. We prove the uniqueness theorem and provide a constructive algorithm for the solution of this partial inverse problem.
We deal with the Sturm--Liouville operator $Ly=l(y)=-\dfrac{d^2y}{dx^2}+q(x)y,$ with Dirichlet--Neumann boundary conditions $ y(0)=y'(π)=0 $ in the space $L_2[0,π]$. We assume that the potential $q$ is complex-valued and has the form $q(x)=u'(x)$, where $u\in L_2[0,π]$. Here the derivative is treated in the distributional sense. Our aim is to obtain the detailed asymptotic formulae for eigenvalues and eigen- and associated functions of the operator $L$.
We consider Sturm-Liouville operators on a half line $[a,\infty), a>0$, with potentials that are growing at most quadratically at infinity. Such operators arise naturally in the analysis of hyperbolic manifolds, or more generally manifolds with cusps. 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. Despite being the natural objects in the context of hyperbolic geometry, spectral geometry of such operators has only recently been studied in the context of analytic…
We consider Sturm-Liouville operators on geometrical graphs without cycles (trees) with singular potentials from the class $W_2^{-1}$. We suppose that the potentials are known on a part of the graph, and study the so-called partial inverse problem, which consists in recovering the potentials on the remaining part of the graph from some parts of several spectra. The main results of the paper are the uniqueness theorem and a constructive procedure for the solution of the partial inverse problem. Our method is based on the completeness and the Riesz-basis property of special systems of vector fun…
Explicit sharp bounds for all nodes of Sturm-Liouville operators with potentials in $L^1$ balls
本文通过研究势函数q∈L¹[0,1]的非线性泛函，推导出Sturm-Liouville算子所有节点的显式紧界表达式。
In this paper, inverse spectral problems for Sturm-Liouville operators on a tree (a graph without cycles) are studied. We show that if the potential on an edge is known a priori, then b - 1 spectral sets uniquely determine the potential functions on a tree with b external edges. Constructive solutions, based on the method of spectral mappings, are provided for the considered inverse problems.
Ambarzumyan-type theorem for vectorial Sturm-Liouville operator with impulses
本文研究了带 Neumann 条件的向量脉冲 Sturm-Liouville 问题，证明了其 Ambarzumyan 定理，即若问题特征值与零势一致则势为零。
In this work, we consider the Sturm-Liouville operator on a finite interval $[0,1]$ with discontinuous conditions at $1/2$. We prove that if the potential is known a priori on a subinterval $[b,1]$ with $b\ge1/2$, then parts of two spectra can uniquely determine the potential and all parameters in discontinuous conditions and boundary conditions. For the case $b<1/2$, parts of either one or two spectra can uniquely determine the potential and a part of parameters.
In this article we obtain asymptotic formulas for the eigenvalues and eigenfunctions of the non-self-adjoint operator generated in space of vector-functions by the Sturm-Liouville equation with m by m matrix potential and the boundary conditions whose scalar case (m=1) are strongly regular. Using these asymptotic formulas, we find a condition on the potential for which the root functions of this operator form a Riesz basis.
In this study, inverse nodal problem is solved for p-Laplacian Schrödinger equation with energy-dependent potential with the Drichlet conditions. Asymptotic estimates of eigenvalues, nodal points and nodal lengths are given by using Prüfer substitution. Especially, an explicit formula for potential function is given by using nodal lengths. Results are more general than classical p- Laplacian Sturm Liouville problem. For the proofs, it is used the methods given in the references <cite>lav3</cite>, <cite>Wang</cite>.
We consider spectral problems for the Sturm-Liouville operator with arbitrary complex-valued potential q(x) and degenerate boundary conditions. We solve corresponding inverse problem, and also study the completeness property and the basis property of the root function system.
Stability of the inverse Sturm-Liouville problem on a quantum tree
本文研究度量树上分布势Sturm-Liouville算子的反谱问题，基于谱映射思想提出稳定的势重构算法，证明了该反问题的一致稳定性与局部稳定性。
In this study, the theorem on necessary and sufficient conditions for the solvability of inverse problem for Sturm-Liouville operator with discontinuous coefficient is proved and the algorithm of reconstruction of potential from spectral data (eigenvalues and normalizing numbers) is given.
The spectrum of the singular indefinite Sturm-Liouville operator $$A=\text{\rm sgn}(\cdot)\bigl(-\tfrac{d^2}{dx^2}+q\bigr)$$ with a real potential $q\in L^1(\mathbb R)$ covers the whole real line and, in addition, non-real eigenvalues may appear if the potential $q$ assumes negative values. A quantitative analysis of the non-real eigenvalues is a challenging problem, and so far only partial results in this direction were obtained. In this paper the bound $$|λ|\leq |q|_{L^1}^2$$ on the absolute values of the non-real eigenvalues $λ$ of $A$ is obtained. Furthermore, separate bounds on the imagin…
It is well known that a potential $q$ of the Sturm-Liouville operator $Ly= -y" +q(x)y$ on the finite interval $[0, π]$ can be uniquely recovered by the spectrum $\{λ_k\}_1^\infty$ and norming constants $\{α_k\}_1^\infty$ of this operator with Dirichlet boundary conditions. Given potential $q$ belonging to Sobolev space $W^θ_2[0, π]$ with $θ> -1$ we associate its $2N$-approximation $q_N$ constructed by the final sets $\{λ_k\}_1^N$ and $\{α_k\}_1^N$. The main result claims that for $-1\leqslantτ<θ$ the estimate $\|q -q_N\|_τ\leqslant CN^{θ-τ}$ holds, where $\|\cdot\|_τ$ is the norm in $W^τ_2$ an…
Ambarzumyan-type theorem for the Sturm-Liouville operator on the lasso graph
本文在由线段和环组成的任意长度套索图上研究 Sturm-Liouville 算子，证明了若其特征值与零势能情形一致，则势能必为零，从而建立 Ambarzumyan 型定理。
The indefinite Sturm-Liouville operator $A = (\sgn x)(-d^2/dx^2+q(x))$ is studied. It is proved that similarity of $A$ to a selfadjoint operator is equivalent to integral estimates of Cauchy integrals. Also similarity conditions in terms of Weyl functions are given. For operators with a finite-zone potential, the components $\Aess$ and $\Adisc$ of $A$ corresponding to essential and discrete spectrums, respectively, are considered. A criterion of similarity of $\Aess$ to a selfadjoint operator is given in terms of Weyl functions for the Sturm-Liouville operator $-d^2/dx^2+q(x)$ with a finite-zo…
The Sturm-Liouville operator on a star-shaped graph is considered. We assume that the potential is known a priori on all the edges except one, and study the partial inverse problem, which consists in recovering the potential on the remaining edge from the part of the spectrum. A constructive method is developed for the solution of this problem, based on the Riesz-basicity of some sequence of vector functions. The local solvability of the inverse problem and the stability of its solution are proved.
The main focus of this paper is the following matrix Cauchy problem for the Dirac system on the interval $[0,1]:$ \[ D'(x)+\left[\begin{array}{cc} 0 & σ_1(x)\\ σ_2(x) & 0 \end{array} \right] D(x)=iμ\left[\begin{array}{cc} 1 & 0\\ 0 &-1 \end{array} \right]D(x),\quad D(0)=\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right], \] where $μ\in\mathbb{C}$ is a spectral parameter, and $σ_j\in L_2[0,1]$, $j=1,2$. We propose a new approach for the study of asymptotic behaviour of its solutions as $μ\to \infty$ and $|{\rm Im}\,μ|\le d$. As an application, we obtain new, sharp asymptotic formulas for …
The perturbation of the Sturm-Liouville operator on a finite interval with Dirichlet boundary conditions by a convolution operator is considered. Local stability and global unique solvability of the inverse problem of recovering the convolution kernel from the spectrum, provided that the potential is given a priori, is known. In the present work, we establish uniform full stability of this inverse problem involving a uniform estimate of deviations of the convolution kernel via deviations of the spectrum and the potential within balls of any fixed radii.
We consider the Sturm-Liouville operator Lu=u''-q(x)u with regular but not strongly regular boundary conditions. Under some supplementary assumptions we prove that the set of potentials q(x) that ensure an asymptotically multiple spectrum is everywhere dense in the space of summable functions.
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.
We obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators with general regular boundary conditions. Using these formulas, we find sufficient conditions on the potential q such that the root functions of these operators do not form a Riesz basis.
We obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators with some regular boundary conditions. Using these formulas, we find sufficient conditions on the potential q such that the root functions of these operators do not form a Riesz basis.
Uniform stability of recovering the Sturm-Liouville operators with frozen argument
本文研究从谱及附加数据反演具有冻结变量的Sturm-Liouville算子，证明势函数对输入数据具有Lipschitz连续依赖的一致稳定性。
Inverse Sturm-Liouville problem with singular potential and spectral parameter in the boundary conditions
本文首次研究边界条件含奇异势及谱参数多项式依赖的逆 Sturm-Liouville 问题，证明局部可解性与稳定性，给出子谱唯一性充要条件并建立重构方法。
Uniform stability of the inverse Sturm-Liouville problem on a star-shaped graph
本文研究了星形图上逆斯图尔姆-刘维尔问题的均匀稳定性，通过谱数据恢复势函数并证明其稳定性。
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.
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.
The inverse spectral problem is studied for the Sturm-Liouville operator with a complex-valued potential and arbitrary entire functions in one of the boundary conditions. We obtain necessary and sufficient conditions for uniqueness, and develop a constructive algorithm for the inverse problem solution. The main results are applied to the Hochstadt-Lieberman half-inverse problem. As an auxiliary proposition, we prove local solvability and stability for the inverse Sturm-Liouville problem by the Cauchy data in the non-self-adjoint case.
We give a complete description of the set of spectral data (eigenvalues and specially introduced norming constants) for Sturm--Liouville operators on the interval $[0,1]$ with matrix-valued potentials in the Sobolev space $W_2^{-1}$ and suggest an algorithm reconstructing the potential from the spectral data that is based on Krein's accelerant method.
The matrix Sturm-Liouville operator on a finite interval with the boundary conditions in the general self-adjoint form and with the singular potential from the class $W_2^{-1}$ is studied. This operator generalizes Sturm-Liouville operators on geometrical graphs. We investigate structural and asymptotical properties of the spectral data (eigenvalues and weight matrices) of this operator. Furthermore, we prove the uniqueness of recovering the operator from its spectral data, by using the method of spectral mappings.
The matrix Sturm-Liouville operator with an integrable potential on the half-line is considered. We study the inverse spectral problem, which consists in recovering of this operator by the Weyl matrix. The main result of the paper is the necessary and sufficient conditions on the Weyl matrix of the non-self-adjoint matrix Sturm-Liouville operator. We also investigate the self-adjoint case and obtain the characterization of the spectral data as a corollary of our general result.
It is constructively proved that for class $A_{r,γ}=\{q\in L_{1,loc}(0,1): q\leq 0, \int_0^1 rq^γ\,dx\leqslant 1\}$, where $r\in C[0,1]$ is uniformly positive weight and $γ>1$, there exists a unique potential $\hat q\in A_{r,γ}$ such that minimal eigenvalue $λ_0(\hat q)$ of boundary problem $$-y"+\hat qy=λy, y(0)=y(1)=0 $$ is equal to $M_{r,γ}=\sup_{q\in A_{r,γ}}λ_0(q)$. For case $γ=1$ we obtain that there exists a unique potential $\hat q\inΓ_{r,γ}$ with analogous property. Here $Γ_{r,γ}$ is a closure of $A_{r,γ}$ in the space $W_{2,loc}^{-1}(0,1)$ of generalized functions.
Partial Inverse Spectral Problems for Sturm-Liouville Operators with Frozen Arguments on a Star-Shaped Graph
本文研究星形图上具有冻结参数的 Sturm-Liouville 算子的部分逆谱问题，提出了一种基于向量函数系统 Riesz 基性质的构造性算法，利用部分谱数据重构单条边上的未知势函数。
On the Fundamental Eigenvalues gap of Sturm-Liouville Operators
本文利用直接优化方法，在单阱势和单垒权函数及凸性约束下，研究区间上Sturm-Liouville算子基本特征值间隙的极小化子。
In this paper, we study spectral problems for the Sturm-Liouville operator with arbitrary complexvalued potential q(x) and two-point boundary conditions. All types of mentioned boundary conditions are considered. We ivestigate in detail the completeness property and the basis property of the root function system.
Solving the inverse Sturm-Liouville problem with singular potential and with polynomials in the boundary conditions
本文首次针对边界条件含谱参数多项式且具复值奇异势的逆 Sturm-Liouville 问题提出构造性解，通过将其转化为 Banach 空间线性方程推导出全新系数重构公式，并证明了从 Weyl 函数恢复的唯一性。
On Molchanov's criterion for compactness of the resolvent for a non self-adjoint Sturm-Liouville operator
本文研究了非自伴 Sturm-Liouville 算子预解式紧性的 Molchanov 判据，指出其必要性但非充分性，并证明了在取值于特定扇形区域的势函数下该判据仍成立。
In this study, the inverse problem of the scattering theory on the half line for a piecewise continuous Sturm-Liouville equation with boundary condition depending quadratic on the spectral parameter is considered. The scattering data of the problem is defined, some properties of the scattering data are investigated. The main equation is derived and uniqueness algorithm to the potential with given scattering data is studied.
In this paper we study a Sturm--Liouville operator $Ly=-y''+q(x)y$ in the space $L_2[0,π]$ with Direchlet boundary conditions. Here the potential $q$ is a fitst order distribution $q\in W_2^{-1}[0,π]$. Such operators were defined in our previous papers. Here we study an asymptotic behaviour of eigenfunctions with uniform estimates of rests. We obtain this estimates also for potentials from Sobolev spaces $q\in W_2^{θ-1}$, where $θ\in[0,1/2)$.
We study the multi-interval boundary-value Sturm-Liouville problems with distributional potentials. For the corresponding symmetric operators boundary triplets are found and the constructive descriptions of all self-adjoint, maximal dissipative and maximal accumulative extensions and generalized resolvents in terms of homogeneous boundary conditions are given. It is shown that all real maximal dissipative and maximal accumulative extensions are self-adjoint and all such extensions are described.
Stability of the inverse Sturm-Liouville problem on a graph with a cycle
本文研究带环图上Sturm-Liouville算子的反谱问题，通过多组谱与环上拟周期问题相关符号序列恢复势函数，证明了该问题的局部稳定性并探究了一致稳定性。
In this paper we propose four different methods to determine Sturm-Liouville operator on an interval $(a,b)$ in case, when a potential $q(x)$ is a distribution from the Sobolev space with negative index of smoothness, i.e. (q\in W_2^{-θ}), where (θ\le 1). The main and second terms of asymptotic series for eigenvalues and eigenfunctions of these operators are obtained and the remaining terms are estimated depending on the class of smoothness of potential /(q/). Particular families of potentials, not belonging to the space (W_2^{-1}) are studied as well.
Transmutation operators for the Sturm--Liouville equation in impedance form with an integrable potential
本文研究阻抗形式Sturm--Liouville算子的换位算子，证明特定换位算子可表示为积分形式，核函数满足Goursat问题，并用逐次逼近法和Picard定理确立其存在性。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。