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On the negative discrete spectrum of the operator \(-\Delta_ N-\alpha V\) for a class of unbounded domains in \(\mathbb{R}^ d\). (English) Zbl 0888.35075
Greiner, Peter C. (ed.) et al., Partial differential equations and their applications. Lectures given at the 1995 annual seminar of the Canadian Mathematical Society, Toronto, Canada, June 12–23, 1995. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 12, 283-296 (1997).
The symbol \(\Delta_N\) stands for the Neumann Laplacian, \(V\) is a nonnegative function, and \(\alpha\) is a large parameter. Further, \(\Omega= \{(y,t)\in \mathbb{R}^d: t>0, |y|<f(t)\}\), \[ N_-(\alpha V,\Omega,{\mathfrak D})\leq c(d)\alpha^{d/2} \int_\Omega V^{d/2}dx,\;N_-(\alpha V,\Omega,{\mathfrak N})\leq 1+ c(\Omega)\alpha^{d/2} \int_\Omega V^{d/2}dx,\;d\geq 3, \] and besides \(N_-(\alpha V,\Omega,{\mathfrak D})\) has the Weyl type asymptotic behavior. The author studies special asymptotic properties of \(N_-(\alpha V,\Omega,{\mathfrak N})\).
For the entire collection see [Zbl 0878.00060].

35P15 Estimates of eigenvalues in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J25 Boundary value problems for second-order elliptic equations