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The Structure of the Spectrum of Fourth‐Order Differential Operators with Random Coefficients
Author(s) -
Teichert H.
Publication year - 1990
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.3211480122
Subject(s) - mathematics , ergodic theory , eigenfunction , spectrum (functional analysis) , differential operator , order (exchange) , operator (biology) , combinatorics , mathematical analysis , pure mathematics , eigenvalues and eigenvectors , physics , quantum mechanics , chemistry , finance , repressor , transcription factor , economics , biochemistry , gene
In this paper it is proved that under certain conditions on the coefficients the random operator H = \documentclass{article}\pagestyle{empty}\begin{document}$H = \frac{1}{{r\left( {x_t } \right)}}\left[ {\frac{{d^2 }}{{dt^2 }}\left( {\frac{1}{{p\left( {x_t } \right)}}\frac{{d^2 }}{{dt^2 }}} \right) + q\left( {x_t } \right)} \right], \in R^1$\end{document} being a stationary, ergodic Markov process with compact State space K , has almost surely pure point spectrum and exponentially decreasing eigenfunctions. The method used here can be extended to operators corresponding to certain matrix Sturm‐Liouville problems.