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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Gaps in the essential spectrum for second order systems
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by Robert M. Kauffman PDF
Proc. Amer. Math. Soc. 51 (1975), 55-61 Request permission

Abstract:

Consider the equation $({D^2} + A + \alpha E)f = 0$, where $\alpha$ is a positive real number, $E$ is the $n \times n$ identity matrix, $A$ is a continuously differentiable function from $[0,\infty )$ to the $n \times n$ Hermitian matrices, and $A$ and $A’$ are bounded. It is shown that, if $\alpha$ is large with respect to $||A|{|_\infty }$, there are small positive numbers $\lambda$ such that, for every solution $f$ to the equation $({D^2} + A + \alpha E)f = 0,{e^{ - \lambda t}}f(t)$ is square integrable, but ${e^{\lambda t}}f(t)$ is not. It is also shown that, if $\alpha$ is large with respect to $||A|{|_\infty }$, there is a real number $\lambda$ close to zero such that $\lambda$ is in the essential spectrum of any selfadjoint operator in ${L_2}$ associated with ${D^2} + A + \alpha E$. These results generalize the results of Hartman and Putnam, who proved these statements for the scalar case $n = 1$.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 51 (1975), 55-61
  • MSC: Primary 34C10; Secondary 34B05
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0374560-9
  • MathSciNet review: 0374560