Nonseparated Boundary Conditions Research Articles

Dissipative Sturm-Liouville Operators with Nonseparated Boundary Conditions

A space of boundary values is constructed for the minimal symmetric singular Sturm-Liouville operator in the Hilbert space \(L_{w}^{2}(a,b)(-\infty \le a < b \le \infty )\) with defect index (2,2) (in Weyl’s limit-circle cases at singular points a and b). A description of all maximal dissipative, maximal accretive, selfadjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at a and b. We investigate maximal dissipative operators with, generally speaking, nonseparated boundary conditions. We construct a selfadjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of dilation. We also construct a functional model of a dissipative operator and define its characteristic function. We prove a theorem on completeness of the system of eigenfunctions and associated functions of the dissipative operators.

Spectral analysis of dissipative Dirac operators with general boundary conditions

A space of boundary values is constructed for minimal symmetric Dirac operator in L A 2((−∞,∞); C 2) with defect index (2,2) (in Weyl's limit-circle cases at ±∞). A description of all maximal dissipative (accretive), selfadjoint, and other extensions of such a symmetric operator is given in terms of boundary conditions at ±∞. We investigate maximal dissipative operators with, generally speaking, nonseparated (nondecomposed) boundary conditions. In particular, if we consider separated boundary conditions, at ±∞ the nonselfadjoint (dissipative) boundary conditions are prescribed simultaneously. We construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and define its characteristic function. We prove the theorem on completeness of the system of eigenvectors and associated vectors of the dissipative Dirac operators.

Parallel Numerical Solution of ABD and BABD linear systems arising from BVPs

We consider linear systems with coefficient matrices having the ABD or the Bordered ABD (BABD) structures. These systems arise in the discretization of BVPs for ordinary and partial differential equations with separated and non-separated boundary conditions, respectively. We describe the cyclic reduction algorithm for the solution of BABD linear systems which allowed us to write the codes BABDCR and GBABDCR (the latter code is suitable for matrices with a more generic BABD structure). A comparison of the GBABDCR code with respect to the well-known sequential code COLROW on ABD linear systems is then analysed. We report some tests on an OpenMP Fortran 90 parallel version of the GBABDCR code and finally we discuss about the use of GBABDCR inside the BVP code BVPSOLVER.

The Inverse Spectral Problem for Differential Operators with Nonseparated Boundary Conditions

The Inverse Spectral Problem for Differential Operators with Nonseparated Boundary Conditions

Generation of Analytic Semi-Groups by Second-Order Differential Operators with Nonseparated Boundary Conditions

Generation of Analytic Semi-Groups by Second-Order Differential Operators with Nonseparated Boundary Conditions

Sturm–Liouville problems with non-separated eigenvalue dependent boundary conditions

Sturm–Liouville differential equations are studied under non-separated boundary conditions whose coefficients are first degree polynomials in the eigenparameter. Situations are examined where there are at most finitely many non-real eigenvalues and also where there are only finitely many real ones.

Multiplicities and relative position of eigenvalues of a quadratic pencil of Sturm-Liouville operators

For boundary value problems generated by a second-order differential equation with regular nonseparated boundary conditions, criteria for the eigenvalues to be multiple are given and the relative position of the eigenvalues is studied.

Lax pairs and a new spectral method for linear and integrable nonlinear PDEs

A new transform method for solving initial-boundary value problems for linear and integrable nonlinear PDEs in two independent variables has been recently introduced in [1]. For linear PDEs this method involves: (a) formulating the given PDE as the compatibility condition of two linear equations which, by analogy with the nonlinear theory, we call a Lax pair; (b) formulating a classical mathematical problem, the so-called Riemann-Hilbert problem, by performing a simultaneous spectral analysis of both equations defining the Lax pair; (c) deriving certain global relations satisfied by the boundary values of the solution of the given PDE. Here this method is used to solve certain problems for the heat equation, the linearized Korteweg-deVries equation and the Laplace equation. Some of these problems illustrate that the new method can be effectively used for problems with complicated boundary conditions such as changing type as well as nonseparable boundary conditions. It is shown that for simple boundary conditions the global relations (c) can be analyzed using only algebraic manipulations, while for complicated boundary conditions, one needs to solve an additional Riemann-Hilbert problem. The relationship of this problem with the classical Wiener-Hopf technique is pointed out. The extension of the above results to integrable nonlinear equations is also discussed. In particular, the Korteweg-deVries equation in the quarter plane is linearized.

Second-order odes with non-separated boundary conditions and generation of analytic semigroups

it is well-known that the differential operator associated to the system is the infinitesimal generator of an analytic semigroup of operators in a variety of spaces. The aim of this work is to extend this result to the case of non-separated boundary conditions. Such boundary conditions were studied by Birkhoff

On the Determinant of One-Dimensional Elliptic Boundary Value Problems

We discuss the $\zeta-$regularized determinant of elliptic boundary value problems on a line segment. Our framework is applicable for separated and non-separated boundary conditions.

Boundary-valued shape-preserving interpolating splines

This article describes a general-purpose method for computing interpolating polynomial splines with arbitrary constraints on their shape and satisfying separable or nonseparable boundary conditions. Examples of applications of the related Fortran code are periodic shape-preserving spline intepolation and the construction of visually pleasing closed curves.

Scattering theory for finitely many sphere interactions supported by concentric spheres

We study stationary scattering theory for finitely many sphere interactions formally given by the Hamiltonian H=−Δ+∑j=1Nαjδ(|x|−Rj) and its generalizations to the case of interactions of the second type and interactions with nonseparated boundary conditions. In a previous publication [J. Math. Phys. 29, 660–664 (1988)], it was shown that the self-adjoint Hamiltonian H{αl},{R} corresponding to H may be defined as a limit in norm resolvent convergence of a family Hε of local scaled short-range Hamiltonians. In this paper we also study scattering theory corresponding to Hε and show that the scattering quantities associated with Hε converge to those of H{αl},{R} as ε→0.

Algorithm 770: BVSPIS—a package for computing boundary-valued shape-preserving interpolating splines

This article describes a software package for computing interpolating polynomial splines with arbitrary constraints on their shape and satisfying separable or nonseparable boundary conditions.

The Discretized Mie-Formalism for plane wave scattering on dielectric objects with non-separable geometries

The knowledge of the scattering behaviour of different irregularly shaped particles is of particular interest in various fields of physics, astronomy and chemistry. In this paper, we present a uniform description of electromagnetic wave scattering on dielectric axisymmetric objects and infinitely extended cylinders with non-circular cross section. The analysis is based on the Discretized Mie-Formalism which provides an appropriate method to solve the corresponding Helmholtz equation with non-separable boundary conditions. Numerical results for spheres, spheroids, Chebyshev particles and cylinders with circular and elliptical cross sections are given.

Convergence and instability in PCG methods for bordered systems

Bordered almost block diagonal systems arise from discretizing a linearized first-order system of n ordinary differential equations in a two-point boundary value problem with nonseparated boundary conditions. The discretization may use spline collocation, finite differences, or multiple shooting. After internal condensation, if necessary, the bordered almost block diagonal system reduces to a standard finite difference structure, which can be solved using a preconditioned conjugate gradient method based on a simple matrix splitting technique. This preconditioned conjugate gradient method is “guaranteed” to converge in at most 2 n + 1 iterations. We exhibit a significant collection of two-point boundary value problems for which this preconditioned conjugate gradient method is unstable, and hence, convergence is not achieved.

On differential operators with nonseparated boundary conditions

We study the inverse spectral problem for Sturm-Liouville operators with nonseparated boundary conditions on a finite interval. The inverse problem for such operators was investigated in [1-5] and other papers. In particular, the characterization of the spectrum for the periodic case was obtained in [1, 2]. Later, Plaksina [3, 4] obtained similar results for other types of boundary conditions. Another approach to this problem is applied here. We present the solution of the inverse problem for Sturm-Liouville operators with nonseparated boundary conditions, give a characterization of spectrum for such operators, and investigate stability. The main results are stated in Theorems 1 and 3. 1. Consider the self-adjoint boundary value problem L = L(q(x) , a, d, b):

Codes for almost block diagonal systems

We present a new set of codes for solving almost block diagonal systems of linear equations and for performing multiplicative operations with matrices represented using the same data structures. These data structures arise when solving ordinary differential equation boundary value problems with non-separated boundary conditions by finite differences, and when using spline collocation methods. Our codes are written in a modular form using the BLAS and are intended to take advantage of vector architecture and, to a limited extent, parallelism.

Finitely many sphere interactions in quantum mechanics: Nonseparated boundary conditions

Let Ḣ be the closure of the restriction of the three-dimensional Laplacian −Δ on the domain C∞0(R3\Σ), where Σ=∪Nj=1∂K(0,Rj) and ∼(K(0,Rj)) is a closed ball of radius Rj centered at the origin in R3. It is well known that Ḣ is a closed symmetric operator with deficiency indices (∞,∞). In this paper all self-adjoint (s.a.) extensions of Ḣ are constructed; these extensions contain as particular cases the quantum Hamiltonian describing concentric δ- and δ′-sphere interactions. It is also shown that the s.a. extensions of Ḣ may be obtained as norm-resolvent limits of momentum cutoff and scaled separable potentials.

INVERSE PROBLEMS OF SPECTRAL ANALYSIS FOR STURM-LIOUVILLE OPERATORS WITH NONSEPARATED BOUNDARY CONDITIONS

This is a continuation of Mat. Sb. 131(173) (1986), no. 1, 3-26=Math. USSR Sb. 59 (1988), no. 1, 1-23. The author proves sufficient conditions that must be satisfied by the spectral data of any two similar Sturm-Liouville boundary value problems with nonseparated boundary conditions. Characteristic properties are obtained for conformal mappings of domains connected with such problems onto the upper half-plane.Bibliography: 4 titles.

Second Order Differential Systems with Nonseparated Boundary Conditions

Second Order Differential Systems with Nonseparated Boundary Conditions