Scattering theory approach to the identification of the Helmholtz equation: A nearfield solution

1981 ◽  
Vol 69 (2) ◽  
pp. 483-488 ◽  
Author(s):  
Arthur B. Weglein ◽  
Manuel T. Silvia
Keyword(s):  
Helmholtz Equation ◽  
Scattering Theory ◽  
Theory Approach

scholarly journals A version of Runge's theorem for the Helmholtz equation with applications to scattering theory

1989 ◽  
Vol 32 (1) ◽  
pp. 107-119 ◽  
Author(s):  
R. L. Ochs
Keyword(s):  
Wave Function ◽  
Helmholtz Equation ◽  
Connected Domain ◽  
Scattering Theory ◽  
Polar Coordinates ◽  
Differentiable Functions ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

Let D be a bounded, simply connected domain in the plane R2 that is starlike with respect to the origin and has C2, α boundary, ∂D, described by the equation in polar coordinateswhere C2, α denotes the space of twice Hölder continuously differentiable functions of index α. In this paper, it is shown that any solution of the Helmholtz equationin D can be approximated in the space by an entire Herglotz wave functionwith kernel g ∈ L2[0,2π] having support in an interval [0, η] with η chosen arbitrarily in 0 > η < 2π.

scholarly journals SCATTERING THEORY APPROACH TO RANDOM SCHRÖDINGER OPERATORS IN ONE DIMENSION

1999 ◽  
Vol 11 (02) ◽  
pp. 187-242 ◽  
Author(s):  
V. KOSTRYKIN ◽  
R. SCHRADER
Keyword(s):  
Lyapunov Exponent ◽  
Scattering Theory ◽  
Schrödinger Operators ◽  
Spectral Shift ◽  
One Dimension ◽  
Random Schrödinger Operators ◽  
Shift Function ◽  
Schrodinger Operators ◽  
Theory Approach ◽  
Spectral Shift Function

Methods from scattering theory are introduced to analyze random Schrödinger operators in one dimension by applying a volume cutoff to the potential. The key ingredient is the Lifshitz–Krein spectral shift function, which is related to the scattering phase by the theorem of Birman and Krein. The spectral shift density is defined as the "thermodynamic limit" of the spectral shift function per unit length of the interaction region. This density is shown to be equal to the difference of the densities of states for the free and the interacting Hamiltonians. Based on this construction, we give a new proof of the Thouless formula. We provide a prescription how to obtain the Lyapunov exponent from the scattering matrix, which suggest a way how to extend this notion to the higher dimensional case. This prescription also allows a characterization of those energies which have vanishing Lyapunov exponent.

scholarly journals Scattering theory approach to electrodynamic Casimir forces

2009 ◽  
Vol 80 (8) ◽  
Author(s):  
Sahand Jamal Rahi ◽  
Thorsten Emig ◽  
Noah Graham ◽  
Robert L. Jaffe ◽  
Mehran Kardar
Keyword(s):  
Scattering Theory ◽  
Theory Approach ◽  
Casimir Forces

scholarly journals AN APPROXIMATION PROPERTY OF IMPORTANCE IN INVERSE SCATTERING THEORY

2001 ◽  
Vol 44 (3) ◽  
pp. 449-454 ◽  
Author(s):  
David Colton ◽  
Brian D. Sleeman
Keyword(s):  
Helmholtz Equation ◽  
Scattering Theory ◽  
Sampling Method ◽  
Approximation Property ◽  
Wave Functions ◽  
Field Pattern ◽  
Primary 35R30 ◽  
Linear Sampling Method ◽  
Secondary 35P25 ◽  
Far Field Pattern

AbstractA key step in establishing the validity of the linear sampling method of determining an unknown scattering obstacle $D$ from a knowledge of its far-field pattern is to prove that solutions of the Helmholtz equation in $D$ can be approximated in $H^1(D)$ by Herglotz wave functions.To this end we establish the important property that the set of Herglotz wave functions is dense in the space of solutions of the Helmholtz equation with respect to the Sobolev space $H^1(D)$ norm.AMS 2000 Mathematics subject classification: Primary 35R30. Secondary 35P25

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