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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
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