## The essential spectrum of Neumann Laplacians on some bounded singular domains.(English)Zbl 0741.35043

The authors deal with the eigenvalue problem of Laplacian (with Neumann B.C.) in a singular domain and give a very interesting result on the structure of the spectrum and the geometry of the domain. If a bounded domain $$\Omega$$ has a smooth boundary, $$-\Delta_ N^ \Omega$$ has only point spectrums, which do not accumulate. However if $$\partial\Omega$$ is singular, eigenvalues sometimes accumulate on some point, which becomes an essential spectrum. The domain the authors are dealing with has infinite slits (comb domain) and so the boundary $$\partial\Omega$$ is very singular. Hence there arise essential spectrums. They construct a domain with such structure of spectrum. Moreover they give a method to control such essential spectrum by constructing the domain appropriately. They prove the following striking result. Theorem. For any closed set $$S\subset[0,\infty)$$, there exists a bounded domain $$\Omega$$ such that $$\sigma_{ess}(-\Delta_ N^ \Omega)=S$$, $$\sigma_{ac}(-\Delta_ N^ \Omega)=\emptyset$$.
Reviewer: S.Jimbo (Tsushima)

### MSC:

 35P05 General topics in linear spectral theory for PDEs 35R30 Inverse problems for PDEs

### Keywords:

inverse spectral problem; singular domain
Full Text:

### References:

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