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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the existence of eigenvalues of differential operators dependent on a parameter


Authors: Sh. Strelitz and S. Abramovich
Journal: Trans. Amer. Math. Soc. 258 (1980), 407-429
MSC: Primary 34B10; Secondary 30E25, 34A20
MathSciNet review: 558181
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Abstract: In this paper we obtain results about the existence of eigenvalues for a system which depends polynomially on $ \lambda $,

\begin{displaymath}\begin{array}{*{20}{c}} {{{u'}_k}(x)\, = \,\sum\limits_{j\, =... ...\, = \,1}^N {a_{kj}^i{u_j}({x_i})\, = \,0,} } } \\ \end{array} \end{displaymath}

, $ k\, = \,1,...,\,N$. In order to get these results we prove that this system can be reduced to a standard system of the form

\begin{displaymath}\begin{array}{*{20}{c}} {{{y'}_k}(x)\, = \,\sum\limits_{j = 1... ...)\, = \,{a_k}(\lambda ),} & {{y_n}(1)\, = \,0,} \\ \end{array} \end{displaymath}

$ k\, = \,1,...,\,n$.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0558181-7
PII: S 0002-9947(1980)0558181-7
Keywords: Eigenvalues, algebraic functions, order of entire functions, asymptotic expansion
Article copyright: © Copyright 1980 American Mathematical Society