Sturm-Liouville Systems Research Articles

Spectrum and resolvent of multi-channel systems with internal energies and common boundary conditions

In the article the spectrum and resolvent of the so-called multichannel systems with nonzero internal energies were investigated. The spectrum and resolvent of multichannel Sturm-Liouville systems with non-zero internal energies mi2 and general boundary conditions were investigated. These systems describe the propagation of partial waves in the theory of quantum physics. The importance of studying the spectral characteristics of these systems is presented in the well-known books of the theory of quantum physics. The finiteness of the number of eigenvalues was proved, the multiplicity of positive eigenvalues was investigated, and as well as the resolvent kernel of the system was found.

Sturm–Liouville systems for the survival probability in first-passage time problems

We derive a Sturm–Liouville system of equations for the exact calculation of the survival probability in first-passage time problems. This system is the one associated with the Wiener–Hopf integral equation obtained from the theory of random walks. The derived approach is an alternative to the existing literature and we tested it against direct calculations from both discrete- and continuous-time random walks in a manageable, but meaningful, example. Within this framework, the Sparre Andersen theorem results to be a boundary condition for the system.

Mathematical theory for topological photonic materials in one dimension

This work presents a rigorous theory for topological photonic materials in one dimension. The main focus is on the existence of interface modes that are induced by topological properties of the bulk structure. For a general 1D photonic structure with time-reversal symmetry, we investigate the existence of an interface mode that is induced by a Dirac point upon perturbation. Specifically, we establish conditions on the perturbation which guarantee the opening of a band gap around the Dirac point and the existence of an interface mode. For a periodic photonic structure with both time-reversal and inversion symmetry, the Zak phase is quantized, taking only two values . We show that the Zak phase is determined by the parity (even or odd) of the Bloch modes at the band edges. For a photonic structure consisting of two semi-infinite systems on the two sides of an interface with distinct topological indices, we show the existence of an interface mode inside the common gap. The stability of the mode under perturbations is also investigated. Finally, we study resonances for finite topological structures. Our results are based on the transfer matrix method and the oscillation theory for Sturm–Liouville operators. The methods and results can be extended to general topological Sturm–Liouville systems in one dimension.

Renormalized Oscillation Theory for Linear Hamiltonian Systems on [0, 1] Via the Maslov Index

Working with a general class of regular linear Hamiltonian systems, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of \({\mathbb {C}}^{2n}\). We verify that our applicability class includes Dirac and Sturm–Liouville systems, as well as a system arising from differential-algebraic equations for which the spectral parameter appears nonlinearly.

Sturm - Liouville systems and nonselfadjoint operators, presented as couplings of dissipative and antidissipative operators with real absolutely continuous spectra

This paper is a continuation of the considerations of the paper [1] and it presents the connection between Sturm-Liouville systems and Livšic operator colligations theory. An usefull representation of solutions of Sturm - Liouville systems is obtained using the resolvent of operators from a large class of nonselfadjoint nondissipative operators, presented as couplings of dissipative and antidissipative operators with real spectra. A connection between Sturm-Liouville systems and the inner state of the corresponding open system of operators from the considered class is presented.

The Maslov Index and Spectral Counts for Linear Hamiltonian Systems on $${\mathbb {R}}$$

Working with a general class of linear Hamiltonian systems specified on $${\mathbb {R}}$$ , we develop a framework for relating the Maslov index to the number of eigenvalues the systems have on intervals of the form $$[\lambda _1, \lambda _2)$$ and $$(-\infty , \lambda _2)$$ . We verify that our framework can be implemented for Sturm–Liouville systems, fourth-order potential systems, and a family of systems nonlinear in the spectral parameter. The analysis is primarily motivated by applications to the analysis of spectral stability for nonlinear waves, and aspects of such analyses are emphasized.

The Maslov and Morse indices for Sturm-Liouville systems on the half-line

We show that for Sturm-Liouville Systems on the half-line \begin{document}$ [0, \infty) $\end{document} , the Morse index can be expressed in terms of the Maslov index and an additional term associated with the boundary conditions at \begin{document}$ x = 0 $\end{document} . Relations are given both for the case in which the target Lagrangian subspace is associated with the space of \begin{document}$ L^2 ((0, \infty), \mathbb{C}^{n}) $\end{document} solutions to the Sturm-Liouville System, and the case in which the target Lagrangian subspace is associated with the space of solutions satisfying the boundary conditions at \begin{document}$ x = 0 $\end{document} . In the former case, a formula of Hormander's is used to show that the target space can be replaced with the Dirichlet space, along with additional explicit terms. We illustrate our theory by applying it to an eigenvalue problem that arises when the nonlinear Schrodinger equation on a star graph is linearized about a half-soliton solution.

Inverse method identification of thermophysical properties based on solotone effect analysis for discontinuous Sturm–Liouville systems

ABSTRACTWe consider one-dimensional Sturm–Liouville systems where conditions for the existence of oscillatory terms within the distribution of the eigenvalues have been established. This so-called solotone effect arises when the underlying partial differential equations contain discontinuities within their coefficients. Models of the Earth contain such discontinuities and research into solotones was originally motivated by vibration problems in geophysics. In this paper, we consider heat conduction in a layered composite and provide what is probably the first example of the presence of a solotone effect within a heat transfer context. Furthermore, we devise a novel method whereby relative values of the specific heat capacity and thermal conductivity of a composite rod may be accurately estimated from oscillations within the eigenvalue spectra. That is, we show how the solotone effect may be used to solve a particular heat transfer inverse problem. The method we present could be used to determine thermophysical properties of bonds or welds, for example when developing new adhesives or solders.

Transfer Function, Stabilizability, and Detectability of Non-autonomous Riesz-spectral Systems

Stability of a state linear system can be identified by controllability, observability, stabilizability, detectability, and transfer function. The approximate controllability and observability of non-autonomous Riesz-spectral systems have been established as well as non-autonomous Sturm-Liouville systems. As a continuation of the establishments, this paper concern on the analysis of the transfer function, stabilizability, and detectability of the non-autonomous Riesz-spectral systems. A strongly continuous quasi semigroup approach is implemented. The results show that the transfer function, stabilizability, and detectability can be established comprehensively in the non-autonomous Riesz-spectral systems. In particular, sufficient and necessary conditions for the stabilizability and detectability can be constructed. These results are parallel with infinite dimensional of autonomous systems.

Exact Null Controllability, Stabilizability, and Detectability of Linear Nonautonomous Control Systems: A Quasisemigroup Approach

In the theory control systems, there are many various qualitative control problems that can be considered. In our previous work, we have analyzed the approximate controllability and observability of the nonautonomous Riesz-spectral systems including the nonautonomous Sturm-Liouville systems. As a continuation of the work, we are concerned with the analysis of stability, stabilizability, detectability, exact null controllability, and complete stabilizability of linear non-autonomous control systems in Banach spaces. The used analysis is a quasisemigroup approach. In this paper, the stability is identified by uniform exponential stability of the associated C0-quasisemigroup. The results show that, in the linear nonautonomous control systems, there are equivalences among internal stability, stabizability, detectability, and input-output stability. Moreover, in the systems, exact null controllability implies complete stabilizability.

Complete stabilizability and stability radius of time-varying Sturm-Liouville systems

There are many physical systems which can be modeled by a time-varying Sturm-Liouville system i.e. a system is inducted by the negative of a time-varying Sturm-Liouville operator. The system is a time-varying Riesz-spectral system. The goals of this paper are to analyze exact controllability, stabilizability, and stability radius of the time-varying Riesz-spectral systems. The steps of analysis of methods of the paper are to define a time-varying Sturm-Liouville system, to construct sufficient and necessary conditions for exact controllability and stabilizability, and to find stability radius. The analysis uses a strongly continuous quasi-semigroup approach. The results show that sufficient and necessary conditions for exact controllability, exact null controllability, and stabilizability can be constructed. Moreover, in the time-varying Sturm-Liouville systems, the exact null controllability implies the complete stabilizability. This is parallel with infinite dimensional of time-invariant systems. Also, the lower bound for the stability radius of the time-varying Sturm-Liouville systems is a reciprocal of the norm of the input-output operator. An example is provided to confirm the results.

Controllability and Observability of Nonautonomous Riesz-Spectral Systems

There are many industrial and biological reaction diffusion systems which involve the time-varying features where certain parameters of the system change during the process. A part of the transport-reaction phenomena is often modelled as an abstract nonautonomous equation generated by a (generalized) Riesz-spectral operator on a Hilbert space. The basic problems related to the equations are existence of solutions of the equations and how to control dynamical behaviour of the equations. In contrast to the autonomous control problems, theory of controllability and observability for the nonautonomous systems is less well established. In this paper, we consider some relevant aspects regarding the controllability and observability for the nonautonomous Riesz-spectral systems including the Sturm-Liouville systems using a C0-quasi-semigroup approach. Three examples are provided. The first is related to sufficient conditions for the existence of solutions and the others are to confirm the approximate controllability and observability of the nonautonomous Riesz-spectral systems and Sturm-Liouville systems, respectively.

Generalized Fourier Series as Green’s Function Expansion for Multi-interval Sturm–Liouville Systems

This study aims to investigate a class of boundary-value transmission problems consisting of Sturm–Liouville equation on finite number disjoint intervals together with eigenparameter-dependent boundary conditions and supplementary transmission conditions at the interior transmittal points. We introduce new Hilbert spaces for self-adjoint realization of the problem, and state the main spectral properties of eigenvalues and eigenfunctions of the considered problem. Then by suggesting own approaches, we have presented a formula for Green’s function and resolvent operator. Finally, we find the resolvent function for corresponding inhomogeneous problem and establish completeness relation of eigenfunctions as generalized Fourier series.

Eigenvalue problem of Sturm–Liouville systems with separated boundary conditions

Let \(\lambda _j\) be the jth eigenvalue of Sturm–Liouville systems with separated boundary conditions, we build up the Hill-type formula, which represent \(\prod \nolimits _{j}(1-\lambda _j^{-1})\) as a determinant of finite matrix. Consequently, we get the Krein-type trace formula based on the Hill-type formula, which express \(\sum \nolimits _{j}{1\over \lambda _j^m}\) as trace of finite matrices. The trace formula can be used to estimate the conjugate point alone a geodesic in Riemannian manifold and to get some infinite sum identities.

Trace Formula for Linear Hamiltonian Systems with its Applications to Elliptic Lagrangian Solutions

In the present paper, we build up trace formulas for both the linear Hamiltonian systems and Sturm-Liouville systems. The formula connects the monodromy matrix of a symmetric periodic orbit with the infinite sum of eigenvalues of the Hessian of the action functional. A natural application is to study the non-degeneracy of linear Hamiltonian systems. Precisely, by the trace formula, we can give an estimation for the upper bound such that the non-degeneracy preserves. Moreover, we could estimate the relative Morse index by the trace formula. Consequently, a series of new stability criteria for the symmetric periodic orbits is given. As a concrete application, the trace formula is used to study the linear stability of elliptic Lagrangian solutions of the classical planar three-body problem. It is well known that the linear stability of elliptic Lagrangian solutions depends on the mass parameter $\bb=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2\in [0,9]$ and the eccentricity $e\in [0,1)$. Based on the trace formula, we estimate the stable region and hyperbolic region of the elliptic Lagranian solutions.

Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics

This paper constructs the small-gain theorem upon a general class of Sturm-Liouville systems. It appears that the feedback connection of two Sturm-Liouville sub-systems is guaranteed of well-posedness, Hurwitz, dissipativity and passivity in L2-spaces provided the loop gain is less than 1. To construct the theorem, spatiotemporal transfer-function and geometrical isomorphism between the space-time domain and the mode-frequency domain are developed, whereof the H∞-norm is extended to be 2D-H∞ norm in mode-frequency domain. On grounds of this small-gain theorem, robust performance of any Sturm-Liouville plant can be formulated as robust stability of a feedback connection, whereupon feedback syntheses can be performed via modal-spectral μ-loopshaping.

Luenberger boundary observer synthesis for Sturm–Liouville systems

A static Luenberger observer of a system with Sturm–Liouville operator is synthesised with the aid of a boundary control formulation. To this aim, approximate observability, detectability and stability of the system is studied and design results are worked out for a typical biochemical case study.

An inverse nodal problem for two-parameter Sturm–Liouville systems

We consider uniqueness and reconstruction of the potentials qj ∊ L1, and of the (separated) boundary condition parameters, from a dense sequence of nodal points for a two-parameter system of Sturm–Liouville equations linked by the eigenvalue pairs (λ, μ). Under the ‘right definiteness’ condition a1(t1)b2(t1) − b1(t1)a2(t2) > 0 for all t1, t2, we show that a single sequence of nodal points for each j = 1, 2, one point for each eigenfunction yjk, k = 1, 2, …, suffices for our purposes.

Completeness Theorems for a Non-Standard Two-Parameter Eigenvalue Problem

We consider two simultaneous Sturm-Liouville systems coupled by two spectral parameters. However, unlike the standard multiparameter problem, we now suppose that the principal part of each of the differential operators is multiplied by a different parameter. In a recent paper, Faierman and Mennicken derived various results concerning the eigenvalues and eigenfunctions, and in particular, they established the oscillation theory for this system. Here we continue this investigation focusing on the completeness of the set of eigenfunctions in a suitable function space.

On sampling theory and basic Sturm–Liouville systems

We investigate the sampling theory associated with basic Sturm–Liouville eigenvalue problems. We derive two sampling theorems for integral transforms whose kernels are basic functions and the integral is of Jackson's type. The kernel in the first theorem is a solution of a basic difference equation and in the second one it is expressed in terms of basic Green's function of the basic Sturm–Liouville systems. Examples involving basic sine and cosine transforms are given.