Discrete Eigenvalue Problem Research Articles

A parasite-free non-orthogonal frequency-domain finite difference method for the electromagnetic analysis of anisotropic waveguides

A methodology is presented for the development of a discrete vector eigenvalue problem for the dispersive analysis of inhomogeneous, anisotropic waveguiding structures. The methodology is based on the discretization of the the frequency-dependent Maxwell's equations on a non-orthogonal grid using covariant and contravariant representation for the fields and a simple point-matching procedure. The occurrence of spurious modes is avoided by the direct enforcement of Gauss' law in the development of the matrix eigenvalue problem. Results from the application of the proposed method to the dispersive analysis of planar, anisotropic microwave and optical waveguides compare favorably with published data obtained using alternative finite element formulations.

Spherically symmetric random walks. II. Dimensionally dependent critical behavior.

A recently developed model of random walks on a $D$-dimensional hyperspherical lattice, where $D$ is {\sl not} restricted to integer values, is extended to include the possibility of creating and annihilating random walkers. Steady-state distributions of random walkers are obtained for all dimensions $D>0$ by solving a discrete eigenvalue problem. These distributions exhibit dimensionally dependent critical behavior as a function of the birth rate. This remarkably simple model exhibits a second-order phase transition with a nontrivial critical exponent for all dimensions $D>0$.

Discrete Bargmann and Neumann systems and finite-dimensional integrable systems

Abstract The nonlinearization approach of eigenvalue problems is equally well applied to the discrete KdV hierarchy. Two kinds of constraints between the potentials and eigenfunctions are suggested, from which the discrete Schrodinger eigenvalue problem, the spatial part of the Lax pairs of the discrete KdV hierarchy, is nonlinearized to be a discrete Bargmann system and a discrete Neumann system, while the nonlinearization of the time part of the Lax pairs leads to two hierarchies of new finite-dimensional completely integrable systems in the Liouville sense. The solutions of the discrete KdV equation are reduced to solving the compatible system of difference equations and ordinary differential equations.

Perturbation analysis of vibration modes with close frequencies

AbstractThis paper focuses on how parameter changes in a vibration system expressed by a discrete eigenvalue problem affect vibration modes in the special case of closely spaced eigenvalues, i.e. ‘eigenvalue clusters’. For this class of problems, a perturbation analysis is developed that can be used to compute changes in eigenvalues and eigenvectors in response to small parameter changes. The analysis is applied to a 6‐degree‐of‐freedom spring‐mass system containing a single pair of closely spaced eigenvalues.

Finite-dimensional discrete systems and integrable systems through nonlinearization of the discrete eigenvalue problem

A method is developed that extends the nonlinearization procedure of eigenvalue problems to the discrete case, from which the finite-dimensional discrete systems and finite-dimensional completely integrable systems associated with the Toda hierarchy are obtained. A new model for solving discrete soliton equations is proposed, which enables one to get the solutions of the Toda lattice equations by solving the compatible systems of discrete equations and nonlinear ordinary differential equations.

A Two-Dimensional Bisection Method for Solving Two-Parameter Eigenvalue Problems

It has long been known that separation of variables can be applied to the Helmholtz equation in 11 three-dimensional coordinate systems. As a result, a multiparameter eigenvalue problem is formed. In this paper, the well-known bisection method for ordinary eigenvalue problems is generalized to a special class of discrete two-parameter eigenvalue problems. The range of the real roots of the problem is discussed and some numerical results are given.

Involutive solutions for toda and langmuir lattices through nonlinearization of lax pairs

Under the constraint determined by a relation (a n ,b n )T={f(ϕ)} n between the reflectionless potentials and the eigenfunctions of the general discrete Schrodinger eigenvalue problem, the Lax pair of the Toda lattice is nonlinearized to be a finite-dimensional difference system and a nonlinear evolution equation, while the solution varietyN of the former is an invariant set of S-flows determined by the latter, and the constants of the motion for the algebraic system are presented.f maps the solution of the algebraic system into the solution of a certain stationary Toda equation. Similar results concerning the Langmuir lattice are given, and a relation between the two difference systems, which are the spatial parts of the nonlinearized Lax pairs of the Toda lattice and Langmuir lattice, is discussed.

Stable solutions of discrete eigenvalue problems

In this note, a method is proposed for finding explicit closed form solutions of cou- pled boundary value difference problems. It avoids increasing the dimension of the problem. Conditions under which the proposed method is numerically stable are given in terms of the data.

Improved calculations of modal design sensitivities

An iterative method is presented for the improved calculation of modal design sensitivities. The procedure is simple and no complicated formulation is involved. The nonlinear perturbation equation is solved using successive approximation techniques. The first-order perturbation equation provides the starting point for the iterations. For the mode shape sensitivity of a structural system with multiple eigenvalues, more accurate results are obtained by applying the nonlinear equation rather than using the second-order perturbation terms. An example problem shows that a single nonlinear iteration can give significant improvement for both eigenvalue and mode shape sensitivities of a system with multiple eigenvalues. The method can be applied to discrete eigenvalue problems as well as systems with multiple eigenvalues.

A systematic approach to the soliton equations of a discrete eigenvalue problem

The Ablowitz–Ladik (AL) problem is a linear vector difference equation whose isospectral flow equations include several important soliton equations; e.g., the discrete nonlinear Schrödinger equation: iq̇n=qn−1−2qn+qn+1 +‖qn‖2(qn−1+qn+1). There is an established procedure for describing the soliton hierarchy of the more familiar AKNS (Ablowitz, Kaup, Newell and Segur)problem. It is based on the notion of a generator for the hierarchy. In this paper the soliton equations of the AL hierarchy are described and characterized by a generator pair. A new continuous spectral problem is introduced and the AKNS hierarchy is embedded in its hierarchy as a specialization.

Finite elements for the eigenvalue problem of differential operators in unbounded intervals

AbstractThe eigenvalue problem for one‐dimensional differential operators with a possible essential spectrum is discretized with finite elements defined on a bounded interval together with a fundamental system of the differential equation outside of the interval. A non‐pollution property of the discrete spectra is proved and the error in the approximation of isolated eigenvalues and corresponding eigenvectors is estimated. The convergence of some numerical algorithms for the solution of the subsequent discrete nonlinear eigenvalue problem is proved. The method is tested in some numerical examples.

The inverse Sturm-Liouville problem and the Rayleigh-Ritz method

In this paper we present an algorithm for solving the inverse Sturm-Liouville problem with symmetric potential and Dirichlet boundary conditions. The algorithm is based on the Rayleigh-Ritz method for calculating the eigenvalues of a two point boundary value problem, and reduces the inverse problem for the differential equation to a nonstandard discrete inverse eigenvalue problem. It is proved that the solution of the discrete problem converges to the solution of the continuous problem. Finally, we establish the stability of the method and give numerical examples.

Discrete inverse strum-liouville problems

We consider a natural discretization of the inverse Sturm-Liouville problem ?y?+qy=?ry with Dirichlet boundary conditions. We prove thatq andr are uniquely determined at the mesh points by the eigenvalues of the discrete problem, provided that the number of mesh points is even and thatq andr are even functions around the midpoint of the interval. The corresponding fact is false for the continuous problem. In case the number of mesh points is odd we characterize all solutions of the discrete inverse eigenvalue problem.

The Discrete Eigenvalue Problem for Azimuthally Dependent Transport Theory

The discrete eigenvalue problem associated with the one-speed azimuthal Fourier harmonics in plane geometry is discussed. An explicit expression, well-suited to numerical evaluation, is given for the dispersion function, and the reality and maximum number of discrete eigenvalues are demonstrated. From numerical examples, it is found that quite often there are no discrete eigenvalues, particularly for the higher harmonics.

Nonlinear differential−difference equations

A method is presented which enables one to obtain and solve certain classes of nonlinear differential−difference equations. The introduction of a new discrete eigenvalue problem allows the exact solution of the self−dual network equations to be found by inverse scattering. The eigenvalue problem has as its singular limit the continuous eigenvalue equations of Zakharov and Shabat. Some interesting differences arise both in the scattering analysis and in the time dependence from previous work.

On shell-model calculations in the continuum

Shell-model calculations include very often single-particle configurations whose energy is larger than the particle emission threshold in the corresponding channel. Nevertheless the continuum nature of the energy spectrum is not taken into account and only a discrete eigen-value problem is solved. In this paper, the schematic Brown model is extended to include configurations in the energy continuum. It is shown that, if there are potential resonances, an “equivalent” discrete problem can be found, thereby giving an overall justification of the usual procedure. At the same time, the limitations of the “equivalent” discrete problem in reproducing the features of the actual problem are exhibited.