Spectral and pseudospectral functions of various dimensions for\n symmetric systems

2016/11/09 by Vadim Mogilevskii, Mogilevskii, Vadim
Mathematics · Engineering · #Spectral Theory in Mathematical Physics #Holomorphic and Operator Theory #Material Science and Thermodynamics

paper · pdf · doi:10.48550/arxiv.1611.03174

Abstract

The main object of the paper is a symmetric system J y'-B(t)y= l D(t) y\ndefined on an interval cI=[a,b) with the regular endpoint a. Let\n f( cd, l) be a matrix solution of this system of an arbitrary dimension and\nlet (Vf)(s)=\∫\_ cI f^*(t,s) D(t)f(t) ,dt be the Fourier transform\nof the function f( cd)\∈ L_ D²( cI). We define a pseudospectral function of\nthe system as a matrix-valued distribution function s( cd) of the dimension\nn_ s such that V is a partial isometry from L_ D²( cI) to\nL²( s; bC^{n_ s}) with the minimally possible kernel. Moreover, we find the\nminimally possible value of n_ s and parameterize all spectral and\npseudospectral functions of every possible dimensions n_ s by means of a\nNevanlinna boundary parameter. The obtained results develop the results by Arov\nand Dym; A.~Sakhnovich, L.~Sakhnovich and Roitberg; Langer and Textorius.\n

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