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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Gaps in the essential spectrum for second order systems


Author: Robert M. Kauffman
Journal: Proc. Amer. Math. Soc. 51 (1975), 55-61
MSC: Primary 34C10; Secondary 34B05
MathSciNet review: 0374560
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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 $ \vert\vert A\vert{\vert _\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 $ \vert\vert A\vert{\vert _\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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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0374560-9
PII: S 0002-9939(1975)0374560-9
Keywords: Rate of exponential growth, essential spectrum, self-adjoint operator, Fredholm operator
Article copyright: © Copyright 1975 American Mathematical Society