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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-1980-0558181-7
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 $\lambda$, \[ \begin {array}{*{20}{c}} {{{u’}_k}(x) = \sum \limits _{j = 1}^n {{b_{kj}}(x, \lambda ){u_j}(x),} } & {\sum \limits _{i = 0}^p {\sum \limits _{j = 1}^N {a_{kj}^i{u_j}({x_i}) = 0,} } } \\ \end {array} \] , $k = 1,..., N$. In order to get these results we prove that this system can be reduced to a standard system of the form \[ \begin {array}{*{20}{c}} {{{y’}_k}(x) = \sum \limits _{j = 1}^n {{a_{kj}}(x, \lambda ) {y_j}(x)} ,} & {{y_k}(0) = {a_k}(\lambda ),} & {{y_n}(1) = 0,} \\ \end {array} \] $k = 1,..., n$.


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Keywords: Eigenvalues, algebraic functions, order of entire functions, asymptotic expansion
Article copyright: © Copyright 1980 American Mathematical Society