Acoustic Beamforming with Maximum SNR Criterion and Efficient Generalized Eigenvector Tracking

2014 ◽  
pp. 373-382 ◽  
Author(s):  
Toshihisa Tanaka ◽  
Mitsuaki Shiono
Keyword(s):  
Generalized Eigenvector

scholarly journals GENERALIZED EIGENVECTORS FOR RESONANCES IN THE FRIEDRICHS MODEL AND THEIR ASSOCIATED GAMOV VECTORS

2006 ◽  
Vol 18 (01) ◽  
pp. 61-78 ◽  
Author(s):  
HELLMUT BAUMGÄRTEL
Keyword(s):  
Hardy Space ◽  
Half Plane ◽  
Lower Half Plane ◽  
Linear Forms ◽  
Generalized Eigenvalues ◽  
Dirac Type ◽  
Generalized Eigenvectors ◽  
Upper Half Plane ◽  
Friedrichs Model ◽  
Generalized Eigenvector

A Gelfand triplet for the Hamiltonian H of the Friedrichs model on ℝ with multiplicity space [Formula: see text], [Formula: see text], is constructed such that exactly the resonances (poles of the inverse of the Livšic-matrix) are (generalized) eigenvalues of H. The corresponding eigen(anti)linear forms are calculated explicitly. Using the wave matrices for the wave (Möller) operators the corresponding eigen(anti)linear forms on the Schwartz space [Formula: see text] for the unperturbed Hamiltonian H0 are also calculated. It turns out that they are of pure Dirac type and can be characterized by their corresponding Gamov vector λ → k/(ζ0 - λ)-1, ζ0 resonance, [Formula: see text], which is uniquely determined by restriction of [Formula: see text] to [Formula: see text], where [Formula: see text] denotes the Hardy space of the upper half-plane. Simultaneously this restriction yields a truncation of the generalized evolution to the well-known decay semigroup for t ≥ 0 of the Toeplitz type on [Formula: see text]. That is: Exactly those pre-Gamov vectors λ → k/(ζ - λ)-1, ζ from the lower half-plane, [Formula: see text], have an extension to a generalized eigenvector of H if ζ is a resonance and if k is from that subspace of [Formula: see text] which is uniquely determined by its corresponding Dirac type antilinear form.

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