Self-adjoint Boundary Conditions Research Articles

Minimal spectral functions of an ordinary differential operator

AbstractLetl[y]be a formally self-adjoint differential expression of an even order on the interval [0,b〉(b ≤ ∞) and letL0be the corresponding minimal operator. By using the concept of a decomposing boundary triplet, we consider the boundary problem formed by the equationl[y] − λy = f,f ∈ L2[0, b〉, and the Nevanlinna λ-dependent boundary conditions with constant values at the regular endpoint 0. For such a problem we introduce the concept of them-function, which in the case of self-adjoint separated boundary conditions coincides with the classical characteristic (Titchmarsh–Weyl) function. Our method allows one to describe all minimal spectral functions of the boundary problem, i.e. all spectral functions of the minimally possible dimension. We also improve (in the case of intermediate deficiency indicesn±(L0)and non-separated boundary conditions) the known estimate of the spectral multiplicity of the (exit space) self-adjoint extensionà ⊃ L0. Results are obtained for expressionsl[y]with operator-valued coefficients and arbitrary (equal or unequal) deficiency indicesn±(L0).

An abstract approach to weak convergence of spectral shift functions and applications to multi-dimensional Schrödinger operators

We study the manner in which a sequence of spectral shift functions \xi(\cdot;H_j,H_{0,j}) associated with abstract pairs of self-adjoint operators (H_j, H_{0,j}) in Hilbert spaces \mathcal H_j , j\in\mathbb N , converge to a limiting spectral shift function \xi(\cdot;H,H_0) associated with a pair (H,H_0) in the limiting Hilbert space \mathcal H as j\to\infty (mimicking the infinite volume limit in concrete applications to multi-dimensional Schrödinger operators). Our techniques rely on a Fredholm determinant approach combined with certain measure theoretic facts. In particular, we show that prior vague convergence results for spectral shift functions in the literature actually extend to the notion of weak convergence. More precisely, in the concrete case of multi-dimensional Schrödinger operators on a sequence of domains \Omega_j exhausting \mathbb R^n as j\to\infty , we extend the convergence of associated spectral shift functions from vague to weak convergence and also from Dirichlet boundary conditions to more general self-adjoint boundary conditions on \partial\Omega_j .

Boundary pairs and boundary conditions for general (not necessarily definite) first‐order symmetric systems with arbitrary deficiency indices

AbstractWe investigate in the paper general (not necessarily definite) first‐order symmetric systems of differential equations in the framework of extension theory of symmetric linear relations. For this aim we first introduce the new notion of a boundary pair \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\lbrace \mathcal H_0\oplus \mathcal H_1, \Gamma \rbrace$\end{document} for A*, where A is a symmetric linear relation, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal H_0$\end{document} is a boundary Hilbert space, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal H_1$\end{document} is a subspace in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal H_0$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Gamma :A^*\rightarrow \mathcal H_0\oplus \mathcal H_1$\end{document} is a (possibly multivalued) linear mapping satisfying the abstract Green's identity. Unlike known concept of a boundary pair for A* our definition is suitable for relations A with possibly unequal deficiency indices n±(A).Next, the general symmetric system Jy′(t) − B(t)y(t) = Δ(t)f(t) on an interval \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal I=(a,b),\; -\infty \le a < b\le \infty ,$\end{document} is considered in the paper. We characterize explicitly the corresponding minimal relation \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$T_{\textit {min}}$\end{document}, which enables us to construct a special (so‐called decomposing) boundary pair \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\lbrace \mathcal H_0\oplus \mathcal H_1, \Gamma \rbrace$\end{document} for the maximal relation \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$T_{\textit {max}}$\end{document}. It turns out that the system is definite if and only if the mapping Γ is single‐valued, in which case a decomposing boundary pair turns into the decomposing boundary triplet \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Pi =\lbrace \mathcal H,\Gamma _0,\Gamma _1\rbrace$\end{document} for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$T_{\textit {max}}$\end{document}. By using such a triplet we describe in terms of boundary conditions proper extensions of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$T_{\textit {min}}$\end{document} in the case of the regular endpoint a and arbitrary (possibly unequal) deficiency indices \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$n_\pm (T_{\textit {min}})$\end{document}. We also show that self‐adjoint decomposing boundary conditions exist only for Hamiltonian systems; moreover, we describe all such conditions in the compact form. These results are generalizations of the known results by Rofe‐Beketov on regular differential operators. Finally, we characterize all maximal dissipative and accumulative separated boundary conditions for arbitrary (not necessarily Hamiltonian) definite symmetric systems

Fourth order canonical forms of singular self-adjoint boundary conditions

Canonical forms of self-adjoint boundary conditions are well known in the second order (Sturm–Liouville) case for both regular and singular problems. These are critical for the theoretical investigation of the eigenvalues as well as their numerical computation. Recently canonical forms have been found for fourth order regular problems. These are much more complicated than the second order ones. Here we find canonical forms for fourth order singular problems with one or both endpoints of the domain interval singular and with arbitrary deficiency index d. In the regular fourth order case d=4, in the singular case d can assume any value between 0 and 4 and, moreover, depends on the nature of the singularities at the two endpoints. These different values of d and their dependence on the endpoints introduces serious additional complications.

A note on relative oscillation theory for symplectic difference systems with general boundary conditions

We develop an analog of classical oscillation theory for discrete symplectic eigenvalue problems with general self-adjoint boundary conditions which, rather than measuring of the spectrum of one single problem, measures the difference between the spectra of two different problems. We prove formulas connecting the numbers of eigenvalues in a given interval for two symplectic eigenvalue problems with different self-adjoint boundary conditions. We derive as corollaries generalized interlacing properties of eigenvalues.

Matrix representations of Sturm–Liouville problems with transmission conditions

We identify a class of Sturm–Liouville equations with transmission conditions such that any Sturm–Liouville problem consisting of such an equation with transmission condition and an arbitrary separated or real coupled self-adjoint boundary condition has a representation as an equivalent finite dimensional matrix eigenvalue problem. Conversely, given any matrix eigenvalue problem of certain type and an arbitrary separated or real coupled self-adjoint boundary condition and transmission condition, we construct a class of Sturm–Liouville problems with this specified boundary condition and transmission condition, each of which is equivalent to the given matrix eigenvalue problem.

On the Product of Self-Adjoint Sturm-Liouville Differential Operators in Direct Sum Spaces

In this paper, the second-order symmetric Sturm-Liouville differential expressions $\tau _1,\tau _2,\ldots,\tau _n$ with real coefficients on any finite number of intervals are studied in the setting of the direct sum of the $L_w^2$-spaces of functions defined on each of the separate intervals. It is shown that the characterization of singular self-adjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximal domain of the product operators, it is an exact parallel of that in the regular case. This characterization is an extension of those obtained in [6]}, [7], [8], [9], [12], [14] and [15].

Canonical forms of self-adjoint boundary conditions for differential operators of order four

Canonical forms of regular self-adjoint boundary conditions for differential operators are well known in the second order i.e. Sturm–Liouville case. In this paper we find canonical forms for fourth order self-adjoint boundary conditions.

Small-energy analysis for the self-adjoint matrix Schrödinger operator on the half line

The matrix Schrödinger equation with a self-adjoint matrix potential is considered on the half line with the most general self-adjoint boundary condition at the origin. When the matrix potential is integrable and has a first moment, it is shown that the corresponding scattering matrix is continuous at zero energy. An explicit formula is provided for the scattering matrix at zero energy. The small-energy asymptotics are established also for the related Jost matrix, its inverse, and various other quantities relevant to the corresponding direct and inverse scattering problems.

Self-adjoint extensions and spectral analysis in the generalized Kratzer problem

We present a mathematically rigorous quantum-mechanical treatment of a one-dimensional non-relativistic motion of a particle in the potential field , For g2>0 and g1<0, the potential is known as the Kratzer potentialVK(x) and is usually used to describe molecular energy and structure, interactions between different molecules and interactions between non-bonded atoms. We construct all self-adjoint Schrödinger operators with the potential V(x) and represent rigorous solutions of the corresponding spectral problems. Solving the first part of the problem, we use a method of specifying self-adjoint extensions by (asymptotic) self-adjoint boundary conditions. Solving spectral problems, we follow Krein's method of guiding functionals. This work is a continuation of our previous works devoted to the Coulomb, Calogero and Aharonov–Bohm potentials.

FAST SOLITONS ON STAR GRAPHS

We define the Schrödinger equation with focusing, cubic nonlinearity on one-vertex graphs. We prove global well-posedness in the energy domain and conservation laws for some self-adjoint boundary conditions at the vertex, i.e. Kirchhoff boundary condition and the so-called δ and δ′ boundary conditions. Moreover, in the same setting, we study the collision of a fast solitary wave with the vertex and we show that it splits in reflected and transmitted components. The outgoing waves preserve a soliton character over a time which depends on the logarithm of the velocity of the ingoing solitary wave. Over the same timescale, the reflection and transmission coefficients of the outgoing waves coincide with the corresponding coefficients of the linear problem. In the analysis of the problem, we follow ideas borrowed from the seminal paper [17] about scattering of fast solitons by a delta interaction on the line, by Holmer, Marzuola and Zworski. The present paper represents an extension of their work to the case of graphs and, as a byproduct, it shows how to extend the analysis of soliton scattering by other point interactions on the line, interpreted as a degenerate graph.

The classification of self-adjoint boundary conditions of differential operators with two singular endpoints

For general even order linear ordinary differential equations with real coefficients and endpoints which are regular or singular and for arbitrary deficiency index d, the self-adjoint domains are determined by d linearly independent boundary conditions. These conditions are of three types: separated, coupled, and mixed. We give a construction for all conditions of each type and determine the number of conditions of each type possible for a given self-adjoint domain. Our construction gives a direct alternative to the recent construction of Everitt and Markus which uses the theory of symplectic spaces. We believe our construction will prove useful in the spectral analysis of these operators and in obtaining canonical forms of self-adjoint boundary conditions. Such forms are known only in the second order, i.e. Sturm–Liouville, case. Even for regular problems of order four no such forms are available. In the case when all d conditions are separated this construction yields explicit non-real conditions for all orders greater than two. It is well known that no such conditions exist in the second order case.

Inverse spectral problem for a string equation with partial information

We consider an inverse spectral problem for a string equation, subject to the self-adjoint boundary conditions and with partial information given on the density function p(x) which is assumed to be a positive piecewise continuously differentiable function defined in the interval [0, 1]. We investigate the situation where (i) the density p is known only on the subinterval [a, 1] for some a ∊ (0, 1) and (ii) only part of one spectrum or two spectra of the problem is given. We will, under suitable circumstances, prove that (i) and (ii) are sufficient to uniquely determine the values of the density function p on the entire interval [0, 1].

Bounds for the spectrum of a matrix differential operator with a damping term

We consider a second-order matrix ordinary regular differential nonselfadjoint operator with a damping term and selfadjoint boundary conditions. An estimate for the resolvent and bounds for the spectrum are established. An operator is said to be stable if its spectrum lies in the right half-plane. By the obtained bounds, stability and instability conditions are established.

Sesquilinear forms corresponding to a non-semibounded Sturm–Liouville operator

Let −DpD be a differential operator on the compact interval [−b, b] whose leading coefficient is positive on (0, b] and negative on [−b, 0), with fixed, separated, self-adjoint boundary conditions at b and −b and an additional interface condition at 0. The self-adjoint extensions of the corresponding minimal differential operator are non-semibounded and are related to non-semibounded sesquilinear forms by a generalization of Kato's representation theorems. The theory of non-semibounded sesquilinear forms is applied to this concrete situation. In particular, the generalized Friedrichs extension is obtained as the operator associated with the unique regular closure of the minimal sesquilinear form. Moreover, among all closed forms associated with the self-adjoint extensions, the regular closed forms are identified. As a consequence, eigenfunction expansion theorems are obtained for the differential operators as well as for certain indefinite Kreĭn–Feller operators with a single concentrated mass.

Fullerenes, zero-modes and self-adjoint extensions

We consider the low-energy electronic properties of graphene cones in the presence of a global Fries–Kekulé Peierls distortion. Such cones occur in fullerenes as the geometric response to the disclination associated with pentagon rings. It is well known that the long-range effect of the disclination deficit-angle can be modelled in the continuum Dirac-equation approximation by a spin connection and a non-Abelian gauge field. We show here that to understand the bound states localized in the vicinity of a pair of pentagons one must, in addition to the long-range topological effects of the curvature and gauge flux, consider the effect of the short-range lattice disruption near the defect. In particular, the radial Dirac equation for the lowest angular-momentum channel sees the defect as a singular endpoint at the origin, and the resulting operator possesses deficiency indices (2, 2). The radial equation therefore admits a four-parameter set of self-adjoint boundary conditions. The values of the four parameters depend on how the pentagons are distributed and determine whether or not there are zero modes or other bound states.

Explicit isospectral flows for the AKNS operator on the unit interval

We consider the AKNS operator on the unit interval. The boundary conditions are self-adjoint separated boundary conditions. For fixed boundary conditions, we make the formulae for flows induced by general tangent vector fields on isospectral sets explicit.

The M-matrix inverse problem for the Sturm—Liouville equation on graphs

We consider an inverse spectral problem for Sturm–Liouville boundary-value problems on a graph with formally self-adjoint boundary conditions at the nodes, where the given information is the M-matrix. Based on the authors' previous results, using Green's function, we prove that the poles of the M-matrix are at the eigenvalues of the associated boundary-value problem and are simple, located on the real axis, and that the residue at a pole is a negative semi-definite matrix with rank equal to the multiplicity of the eigenvalue. We define the so-called norming constants and relate them to the spectral measure and the M-matrix. This enables us to recover, from the M-matrix, the boundary conditions and the potential, up to a unitary equivalence for co-normal boundary conditions.

On the boundary conditions in self-adjoint multi-interval Sturm–Liouville problems

In this paper, we completely solve a problem whose two-interval case has been studied recently by Wang et al. in [A. Wang, J. Sun, A. Zettl, Two-interval Sturm–Liouville operators in modified Hilbert spaces, J. Math. Anal. Appl. 328 (2007) 390–399]. The problem is to give an explicit description of all boundary conditions defining the self-adjoint differential operators associated with several Sturm–Liouville equations together, i.e., the so-called multi-interval Sturm–Liouville problems. The explicit description obtained in this paper only uses hermitian matrices. As an application, we then show that a generic self-adjoint boundary condition can not come from any direct sum of self-adjoint operators associated with the individual Sturm–Liouville equations, and it depends non-trivially on the multiples used in the definition of the so-called maximum operator.

Matrix Representations of Sturm–Liouville Problems with Finite Spectrum

We identify a class of Sturm–Liouville equations such that any Sturm–Liouville (SL) problem consisting of such an equation and an arbitrary separated or coupled real self-adjoint boundary condition has a representation as an equivalent finite dimensional matrix eigenvalue problem. Conversely, given any matrix eigenvalue problem of certain type and an arbitrary separated or coupled real self-adjoint (SL) boundary condition, we construct a class of Sturm–Liouville problems with the specified boundary condition, each of which is equivalent to the given matrix eigenvalue problem.