Singular Endpoints Research Articles

Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators

Let H be a Hilbert space and let A be a simple symmetric operator in H with equal deficiency indices d : = n ± ( A ) < ∞ . We show that if, for all λ in an open interval I ⊂ R , the dimension of defect subspaces N λ ( A ) ( = Ker ( A ⁎ − λ ) ) coincides with d, then every self-adjoint extension A ˜ ⊃ A has no continuous spectrum in I and the point spectrum of A ˜ is nowhere dense in I. Application of this statement to differential operators makes it possible to generalize the known results by Weidmann to the case of an ordinary differential expression with both singular endpoints and arbitrary equal deficiency indices of the minimal operator.

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.

Self-adjoint extensions for linear Hamiltonian systems with two singular endpoints

This paper is concerned with self-adjoint extensions for a linear Hamiltonian system with two singular endpoints. The domain of the closure of the corresponding minimal Hamiltonian operator H 0 is described by properties of its elements at the endpoints of the discussed interval, decompositions of the domains of the corresponding left and right maximal Hamiltonian operators are provided, and expressions of the defect indices of H 0 in terms of those of the left and right minimal operators are given. Based on them, characterizations of all the self-adjoint extensions for a Hamiltonian system are obtained in terms of square integrable solutions. As a consequence, the characterizations of all the self-adjoint extensions are given for systems in several special cases.

Sturm-Liouville Operators with Singularities and Generalized Nevanlinna Functions

The Titchmarsh–Weyl function, which was introduced in Fulton (Math Nachr 281(10):1418–1475, 2008) for the Sturm-Liouville equation with a hydrogen-like potential on (0, ∞), is shown to belong to a generalized Nevanlinna class $${\bf N_\kappa}$$ . As a consequence, also in the case of two singular endpoints for the Fourier transformation defined by means of Frobenius solutions there exists a scalar spectral function. This spectral function is given explicitly for potentials of the form $${\dfrac{q_0}{x^2}+\dfrac{q_1}{x},\,-\dfrac 14\le q_0 < \infty}$$ .

Titchmarsh–Weyl m ‐functions for second‐order Sturm–Liouville problems with two singular endpoints

AbstractIn this paper we consider some cases of Sturm–Liouville problems with two singular endpoints at x = 0 and x = ∞ which have a simple spectrum, and show that the simplicity of the spectrum can be built into the definition of a Titchmarsh–Weyl m ‐function from which the eigenfunction expansion can be constructed. The use of initial conditions at a point interior to the interval (0,∞) is avoided in favor of Frobenius solutions near the regular singular point x = 0. In contrast to the classical theory associated with a regular left endpoint, the growth behaviour of the associated spectral functions can be on the order of λβ for any β ∈ (0,∞). Application of the theory to the Bessel equation on (0,∞) and to the radial part of the separated hydrogen atom on (0,∞) is given. In the case of the hydrogen atom a single Titchmarsh–Weyl m ‐function is obtained which completely describes both the discrete negative spectrum and the continuous positive spectrum. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Asymptotics of the Spectral Density for Radial Dirac Operators with Divergent Potentials

We study the asymptotics of the spectral density of one-dimensional Dirac systems on the half-line with an angular momentum term and a potential tending to infinity at infinity. The problem has two singular end-points; however, as the spectrum is simple, the derivative of the spectral matrix has only one non-zero eigenvalue which we take to be the spectral density. Our main result shows that, assuming sufficient regularity of the potential, there are no points of spectral concentration for large values of the spectral parameter outside a neighbourhood of a discrete set of exceptional points.

Large amplitude oscillations for a class of symmetric polynomial differential systems in R³

In this paper we study a class of symmetric polynomial differential systems in R³, which has a set of parallel invariant straight lines, forming degenerate heteroclinic cycles, which have their two singular endpoints at infinity. The global study near infinity is performed using the Poincaré compactification. We prove that for all n &lt;FONT FACE=Symbol&gt;Î&lt;/FONT&gt; N there is epsilonn &gt; 0 such that for 0 &lt; epsilon &lt; epsilonn the system has at least n large amplitude periodic orbits bifurcating from the heteroclinic loop formed by the two invariant straight lines closest to the x-axis, one contained in the half-space y &gt; 0 and the other in y &lt; 0.

The Fourth-order Bessel–type Differential Equation

The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear differential equations, with a regular singularity at the origin and an irregular singularity at the point of infinity of the complex plane. There is a Bessel-type differential equation for each even-order integer; the equation of order two is the classical Bessel differential equation. These even-order Bessel-type equations are not formal powers of the classical Bessel equation. When the independent variable of these equations is restricted to the positive real axis of the plane they can be written in the Lagrange symmetric (formally self-adjoint) form of the Glazman–Naimark type, with real coefficients. Embedded in this form of the equation is a spectral parameter; this combination leads to the generation of self-adjoint operators in a weighted Hilbert function space. In the second-order case one of these associated operators has an eigenfunction expansion that leads to the Hankel integral transform. This article is devoted to a study of the spectral theory of the Bessel-type differential equation of order four; considered on the positive real axis this equation has singularities at both end-points. In the associated Hilbert function space these singular end-points are classified, the minimal and maximal operators are defined and all associated self-adjoint operators are determined, including the Friedrichs self-adjoint operator. The spectral properties of these self-adjoint operators are given in explicit form. From the properties of the domain of the maximal operator, in the associated Hilbert function space, it is possible to obtain a virial theorem for the fourth-order Bessel-type differential equation. There are two solutions of this fourth-order equation that can be expressed in terms of classical Bessel functions of order zero and order one. However it appears that additional, independent solutions essentially involve new special functions not yet defined. The spectral properties of the self-adjoint operators suggest that there is an eigenfunction expansion similar to the Hankel transform, but details await a further study of the solutions of the differential equation.

ON THE ESSENTIAL SPECTRA OF GENERAL DIFFERENTIAL OPERATORS

In this paper, it is shown in the cases of one and two singular end-points and when all solutions of the equation $M[u]-\lambda uw=0$, and its adjoint $M^+[v] -\lambda wv = 0$ are in $L_w^2 (a, b)$ (the limit circle case) with $f\in L^2_w(a,b)$ for $M[u]-\lambda wu=wf$ that all well-posed extensions of the minimal operator $T_0(M)$ generated by a general ordinary quasi-differential expression $M$ of $n$-th order with complex coefficients have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly slovable operators have all the standard essential spectra to be empty. These results extend those of formally symmetric expression $M$ studied in [1] and [12], and also extend those proved in [8] in the case of one singular end-point of the interval [a,b).

Subordinacy Analysis and Absolutely Continuous Spectra for Sturm-Liouville Equations with Two Singular Endpoints

AbstractThe Gilbert-Pearson characterization of the spectrum is established for a generalized Sturm-Liouville equation with two singular endpoints. It is also shown that strong absolute continuity for the one singular endpoint problem guarantees absolute continuity for the two singular endpoint problem. As a consequence, we obtain the result that strong nonsubordinacy, at one singular endpoint, of a particular solution guarantees the nonexistence of subordinate solutions at both singular endpoints.

Singular non-selfadjoint differential operators

A characterisation is obtained of all the regularly solvable operators and their adjoints generated by general ordinary quasidifferential expressions inThe domains of these operators are described in terms of boundary conditions involving thesolutions ofM[u] =λwuand the adjoint equationat both singular end-pointsaandb.These results are an extension of those proved in [3], by Evans and Ibrahim, to the case of two singular end-points of the interval (a, b), and a generalisation of those in [10] and [13] concerning selfadjoint andJ-selfadjoint differential operators, whereJdenotes complex conjugation.

Boundary value problems for the Laplace tidal wave equation

This paper discusses the eigenvalue problem associated with the Laplace tidal wave equation (LTWE) given, for μ ϵ (—1,1), by 1 − μ 2 μ 2 − τ 2 y ′ ( μ ) ′ + 1 μ 2 − τ 2 s τ μ 2 + τ 2 μ 2 − τ 2 + s 2 1 + μ 2 y ( μ ) = λ y ( μ ) , ( LTWE ) where s and τ are parameters, with s an integer and 0 &lt; τ &lt; 1, and λ determines the eigenvalues. This ordinary differential equation is derived from a linear system of partial differential equations, which system serves as a mathematical model for the wave motion of a thin layer of fluid on a massive, rotating gravitational sphere. The problems raised by this differential equation are significant, for both the analytic and numerical studies of Sturm-Liouville equations, in respect of the interior singularities, at the points ± τ , and of the changes in sign of the leading coefficient (1 - μ 2 )/( μ 2 - τ 2 ) over the interval (-1, 1). Direct sum space methods, quasi-derivatives and transformation theory are used to determine three physically significant, well-posed boundary value problems from the Sturm-Liouville eigenvalue problem (LTWE), which has singular end-points ± 1 and, additionally, interior singularities at ± τ . Self-adjoint differential operators in appropriate Hilbert function spaces are constructed to represent each of the three well-posed boundary value problems derived from LTWE and it is shown that these three operators are unitarily equivalent. The qualitative nature of the common spectrum is discussed and finite energy properties of functions in the domains of the associated differential operators are studied. This work continues the studies of LTWE made by earlier workers, in particular Hough, Lamb, Longuet-Higgins and Lindzen.

On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints

SynopsisThe theory of subordinacy is extended to all one-dimensional Schrödinger operatorsfor which the corresponding differential expressionL= –d2/(dr2) +V(r) is in the limit point case at both ends of an interval (a,b), withV(r) locally integrable. This enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions ofLu = xu, x εℝ, asr→aandr→b. The result provides a rigorous but straightforward method of direct spectral analysis which has very general application, and somefurther properties of the spectrum are deduced from the underlying theory.

Boundary conditions for differential operators with intermediate deficiency index

The problem of assigning boundary conditions at singular endpoints is considered for differential operators which are neither limit-point or limit-circle at the singular endpoint. A solution is obtained by using Niessen subspaces. Additional asymptotic information yields a concrete description of the boundary conditions including a characterization of the Friedrich's extension.

Absolutely Continuous Spectra of Second Order Differential Operators with Short and Long Range Potentials

For ordinary second order differential operators with one or two singular endpoints, the problem is considered of determining when a continuous spectrum is absolutely continuous or is of class $C^{(1)} $. Operators are considered which have a smooth part plus perturbation terms. The perturbation terms considered are short and long range type potentials and also potentials of a highly oscillatory character. The absolutely continuous spectrum found is either a ray $[\lambda _0 ,\infty )$ or a whole line $( - \infty ,\infty )$. For a certain class of equations, the spectrum is also found to be bounded below. The theory developed is applied to the energy operator of the hydrogen atom.

Wirtinger’s Inequality

General forms of Wirtinger-type inequalities are proved in both one and n dimensions. Since singular endpoints and unbounded intervals are allowed, a large class of new one-dimensional results are generated as well as previously known results. In the (usual) case that the admissible functions are identically zero on the boundary $\partial G$ of a bounded domain G in $E^n $, the sharp form of Wirtinger’s inequality in G is proved without any regularity hypotheses on $\partial G$. If the admissible functions are not so restricted, the companion inequality is proved for domains with ${\bf C}^2 $ boundaries.

On the Spectrum of a Boundary Value Problem with Two Singular Endpoints

On the Spectrum of a Boundary Value Problem with Two Singular Endpoints