Abstract

Counting Eigenvalues in Spectral Gaps A. Hinz We report on a joint project with M. Brown, M. Eastham, D. McCormack (Cardiff), H. Kalf, T. Kriecherbauer, and K. Schmidt (Munich), announced at the 1996 Gregynog Workshop. They include an analytic proof of the existence of {\em Welsh eigenvalues} (cf. Brown, Eastham, Hinz, Kriecherbauer, McCormack, and Schmidt, J. Math. Anal. Appl. 225(1998), 347-357; more on that in K. Schmidt's talk) and a method for counting eigenvalues in compact subintervals of gaps in the essential spectrum of perturbed one-dimensional Mathieu operators. Together these eigenvalues form intervals of dense point spectrum of the corresponding higher-dimensional Schr\"{o}dinger operators, as found out during the 1989 Gregynog Workshop (cf. Hempel, Herbst, Hinz, and Kalf, J. London Math. Soc. (2) 43(1991), 295-304). The resulting code is employed to allow a closer look at existing asymptotic formulas (i.e. with respect to large coupling constants of the perturbation).

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