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 , where is a positive real number, is the identity matrix, is a continuously differentiable function from to the Hermitian matrices, and and are bounded. It is shown that, if is large with respect to , there are small positive numbers such that, for every solution to the equation is square integrable, but is not. It is also shown that, if is large with respect to , there is a real number close to zero such that is in the essential spectrum of any selfadjoint operator in associated with . These results generalize the results of Hartman and Putnam, who proved these statements for the scalar case .

**[1]**N. Dunford and J. T. Schwartz,*Linear operators*. II:*Spectral theory. Self-adjoint operators in Hilbert space*, Wiley, New York, 1963. MR**32**#6181.**[2]**Seymour Goldberg,*Unbounded linear operators: Theory and applications*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0200692****[3]**Philip Hartman and Calvin R. Putnam,*The gaps in the essential spectra of wave equations*, Amer. J. Math.**72**(1950), 849–862. MR**0038533****[4]**Robert M. Kauffman,*On the growth of solutions in the oscillatory case*, Proc. Amer. Math. Soc.**51**(1975), 49–54. MR**0374559**, 10.1090/S0002-9939-1975-0374559-2

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1975-0374560-9

Keywords:
Rate of exponential growth,
essential spectrum,
self-adjoint operator,
Fredholm operator

Article copyright:
© Copyright 1975
American Mathematical Society