scholarly journals Towards a Method for the Accurate Solution of the Schrödinger Wave Equation in Many Variables. II. A Simple Numerical Illustration

1959 ◽  
Vol 12 (4) ◽  
pp. 441 ◽  
Author(s):  
IM Bassett
Keyword(s):  
Wave Equation ◽  
Eigenvalue Problem ◽  
Accurate Solution ◽  
Numerical Illustration

The method described in Part I is here applied to an eigenvalue problem in two Variables

scholarly journals Towards a Method for the Accurate Solution of the Schrödinger Wave Equation in Many Variables. I. Formulation of the Method for an Eigenvalue Problem without Symmetry Conditions

1959 ◽  
Vol 12 (4) ◽  
pp. 430
Author(s):  
IM Bassett
Keyword(s):  
Wave Equation ◽  
Eigenvalue Problem ◽  
Digital Computer ◽  
Accurate Solution ◽  
New Method ◽  
Electronic Digital Computer

The aim of this paper and those following is to formulate and explore a new method, suitable for use with an electronic digital computer, for the solution of eigenvalue. eigenfunction problems in many variables, with the aim of applying the method to the Schrodinger wave equation.

A Separation Principle for Gyroscopic Conservative Systems

1997 ◽  
Vol 119 (1) ◽  
pp. 110-119 ◽  
Author(s):  
L. Meirovitch
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Separation Theorem ◽  
Separation Principle ◽  
Numerical Illustration ◽  
Closed Form Solutions ◽  
Conservative Systems ◽  
Algebraic Eigenvalue Problem

Closed-form solutions to differential eigenvalue problems associated with natural conservative systems, albeit self-adjoint, can be obtained in only a limited number of cases. Approximate solutions generally require spatial discretization, which amounts to approximating the differential eigenvalue problem by an algebraic eigenvalue problem. If the discretization process is carried out by the Rayleigh-Ritz method in conjunction with the variational approach, then the approximate eigenvalues can be characterized by means of the Courant and Fischer maximin theorem and the separation theorem. The latter theorem can be used to demonstrate the convergence of the approximate eigenvalues thus derived to the actual eigenvalues. This paper develops a maximin theorem and a separation theorem for discretized gyroscopic conservative systems, and provides a numerical illustration.

scholarly journals Spectral enclosures for the damped elastic wave equation

2021 ◽  
Vol 4 (6) ◽  
pp. 1-10
Author(s):  
Biagio Cassano ◽  
◽  
Lucrezia Cossetti ◽  
Luca Fanelli ◽  
◽  
...  
Keyword(s):  
Wave Equation ◽  
Eigenvalue Problem ◽  
Elastic Wave ◽  
Spectral Properties ◽  
Point Spectrum ◽  
Damping Coefficient ◽  
Elastic Wave Equation ◽  
The One

<abstract><p>In this paper we investigate spectral properties of the damped elastic wave equation. Deducing a correspondence between the eigenvalue problem of this model and the one of Lamé operators with non self-adjoint perturbations, we provide quantitative bounds on the location of the point spectrum in terms of suitable norms of the damping coefficient.</p></abstract>

Isospectral sets for transmission eigenvalue problem

2020 ◽  
Vol 28 (1) ◽  
pp. 63-69 ◽  
Author(s):  
Chuan-Fu Yang ◽  
Sergey A. Buterin
Keyword(s):  
Wave Equation ◽  
Eigenvalue Problem ◽  
Wave Speed ◽  
Spherically Symmetric ◽  
Variable Speed ◽  
Transmission Eigenvalues ◽  
Constructive Methods ◽  
Transmission Eigenvalue ◽  
Symmetric Variable ◽  
Transmission Eigenvalue Problem

AbstractWe consider the boundary value problem {R(a,q)}: {-y^{\prime\prime}(x)+q(x)y(x)=\lambda y(x)} with {y(0)=0} and {y(1)\cos(a\sqrt{\lambda})=y^{\prime}(1)\frac{\sin(a\sqrt{\lambda})}{\sqrt{% \lambda}}}. Motivated by the previous work [T. Aktosun and V. G. Papanicolaou, Reconstruction of the wave speed from transmission eigenvalues for the spherically symmetric variable-speed wave equation, Inverse Problems 29 2013, 6, Article ID 065007], it is natural to consider the following interesting question: how does one characterize isospectral sets corresponding to problem {R(1,q)}? In this paper applying constructive methods we answer the above question.

Topographic waves in open domains. Part 2. Bay modes and resonances

1989 ◽  
Vol 200 ◽  
pp. 77-93 ◽  
Author(s):  
Thomas F. Stocker ◽  
E. R. Johnson
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Eigenvalue Problem ◽  
Cutoff Frequency ◽  
Flat Channel ◽  
Continuous Part ◽  
Depth Profiles ◽  
Countably Infinite ◽  
Infinite Set ◽  
Algebraic Eigenvalue Problem

The topographic wave equation is solved in a domain consisting of a channel with a terminating bay zone. For exponential depth profiles the problem reduces to an algebraic eigenvalue problem. In a flat channel adjacent to a shelf–like bay zone the solutions form a countably infinite set of orthogonal bay modes: the spectrum of eigenfrequencies is purely discrete. A channel with transverse topography allows wave propagation towards and away from the bay: the spectrum has a continuous part below the cutoff frequency of free channel waves. Above this cutoff frequency a finite number (possibly zero) of bay-trapped solutions occur. Bounds for this number are given. At particular frequencies below the cutoff the system is in resonance with the incident wave. These resonances are shown to be associated with bay modes.

Boundary value problems for the Laplace tidal wave equation

1990 ◽  
Vol 428 (1874) ◽  
pp. 157-180 ◽  
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Eigenvalue Problem ◽  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Tidal Wave ◽  
Transformation Theory ◽  
Sturm Liouville ◽  
Well Posed

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 < τ < 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.

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