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Eigenvalue asymptotics of the Neumann Laplacian of regions and manifolds with cusps. (English) Zbl 0783.35040
Summary: We study the eigenvalue asymptotics of a Neumann Laplacian $$-\Delta^ \Omega_ N$$ in unbounded regions $$\Omega$$ of $$\mathbb{R}^ 2$$ with cusps at infinity (a typical example is $$\Omega=\{(x,y) \in \mathbb{R}^ 2:x>1,\;| y |<e^{-x^ 2}\})$$ and prove that $N_ E(-\Delta^ \Omega_ N)\sim N_ E(H_ V)+{\textstyle{E\over 2}}\text{ Vol}(\Omega),$ where $$H_ V$$ is the canonical one-dimensional Schrödinger operator associated to the problem. We establish a similar formula for manifolds with cusps and derive the eigenvalue asymptotics of a Dirichlet Laplacian $$-\Delta^ \Omega_ D$$ for a class of cusp-type regions of infinite volume.

##### MSC:
 35P20 Asymptotic distributions of eigenvalues in context of PDEs 35J15 Second-order elliptic equations 58J50 Spectral problems; spectral geometry; scattering theory on manifolds
##### Keywords:
Neumann Laplacian; unbounded regions; Dirichlet Laplacian
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##### References:
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