An Inverse Eigenvalue Problem for an Arbitrary, Multiply Connected, Bounded Domain in $R^3 $ with Impedance Boundary Conditions

1992 ◽  
Vol 52 (3) ◽  
pp. 725-729 ◽  
Author(s):  
E. M. E. Zayed
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Bounded Domain ◽  
Inverse Eigenvalue Problem ◽  
Impedance Boundary ◽  
Multiply Connected ◽  
Impedance Boundary Conditions ◽  
Inverse Eigenvalue

scholarly journals The wave equation approach to an inverse eigenvalue problem for an arbitrary multiply connected drum inℝ2with Robin boundary conditions

2001 ◽  
Vol 25 (11) ◽  
pp. 717-726 ◽  
Author(s):  
E. M. E. Zayed ◽  
I. H. Abdel-Halim
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Inverse Eigenvalue Problem ◽  
Robin Boundary Conditions ◽  
Two Dimensional ◽  
Equation Approach ◽  
Multiply Connected ◽  
Inverse Eigenvalue ◽  
Negative Laplacian ◽  
Robin Boundary

The spectral functionμˆ(t)=∑j=1∞exp(−itμj1/2), where{μj}j=1∞are the eigenvalues of the two-dimensional negative Laplacian, is studied for small|t|for a variety of domains, where−∞<t<∞andi=−1. The dependencies ofμˆ(t)on the connectivity of a domain and the Robin boundary conditions are analyzed. Particular attention is given to an arbitrary multiply-connected drum inℝ2together with Robin boundary conditions on its boundaries.

scholarly journals AN INVERSE PROBLEM FOR A GENERAL DOUBLY-CONNECTED BOUNDED DOMAIN: AN EXTENSION TO HIGHER DIMENSIONS

1997 ◽  
Vol 28 (4) ◽  
pp. 277-295
Author(s):  
E. M. E. ZAYED
Keyword(s):  
Inverse Problem ◽  
Boundary Conditions ◽  
Spectral Function ◽  
Bounded Domain ◽  
Higher Dimensions ◽  
Piecewise Smooth ◽  
Bounding Surface ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions ◽  
Negative Laplacian

The spectral function $\Theta(t)=\sum_{\nu=1}^\infty \exp(-t\lambda_\nu)$, where $\{\lambda_\nu\}_{\nu=1}^\infty$ are the eigenvalues of the negative Laplacian $-\nabla^2=-\sum_{i=1}^3(\frac{\partial}{\partial x_i})^2$ in the $(x^1, x^2, x^3)$-space, is studied for an arbitrary doubly connected bounded domain $\Omega$ in $R^3$ together with its smooth inner bounding surface $\tilde S_1$ and its smooth outer bounding surface $\tilde S_2$, where piecewise smooth impedance boundary conditions on the parts $S_1^*$, $S_2^*$ of $\tilde S_1$ and $S_3^*$, $S_4^*$ of $\tilde S_2$ are considered, such that $\tilde S_1=S_1^*\cup S_2^*$ and $\tilde S_2=S_3^*\cup S_4^*$.

scholarly journals Toeplitz nonnegative realization of spectra via companion matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 230-245
Author(s):  
Macarena Collao ◽  
Mario Salas ◽  
Ricardo L. Soto
Keyword(s):  
Eigenvalue Problem ◽  
Half Plane ◽  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
Inverse Eigenvalue ◽  
Companion Matrices ◽  
Solution Matrix

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

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