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Two Parameter Asymptotic Spectra in the Uniformly Elliptic Case

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Abstract

In this article we study the abstract two parameter eigenvalue problem

$$\begin{gathered} T_1 u_1 = \left( {\lambda _1 V_{11} + \lambda _2 V_{12} } \right)u_1 , \left\| {u_1 } \right\| = 1 \hfill \\ T_2 u_2 = \left( {\lambda _1 V_{21} + \lambda _2 V_{22} } \right)u_2 , \left\| {u_2 } \right\| = 1 \hfill \\ \end{gathered}$$

where, in the Hilbert spaces Hj, Tj is self-adjoint, bounded below and has compact resolvent, and Vjk are self-adjoint bounded operators, (−1)j+kVjk >> 0, j, k = 1, 2. An eigenvalue λ for this problem is a point in R2 satisfying both equations. Under appropriate conditions, the eigenvalues λn = (λ1 n, λ2 n) are countable and in R2. We aim to describe the set of limit points of λn/∥λn∥, as ∥λn∥ → ∞, in terms of the Vjk.

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Authors and Affiliations

  1. Department of Mathematics & Statistics, University of Calgary, Calgary, Alberta, Canada, T2N 1N4

    P. A. Binding

  2. Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada, S7N 0W0

    P. J. Browne

  3. Department of Mathematics, Shiraz University, Shiraz, Iran, 714354

    K. Seddighi

Authors
  1. P. A. Binding
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  2. P. J. Browne
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  3. K. Seddighi
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Additional information

Research supported in part by grants from the NSERC of Canada and I.W. Killam Foundation.

Research supported in part by grants from the NSERC of Canada.

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Binding, P.A., Browne, P.J. & Seddighi, K. Two Parameter Asymptotic Spectra in the Uniformly Elliptic Case. Results. Math. 31, 1–13 (1997). https://doi.org/10.1007/BF03322147

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