On the Schrödinger equation and the eigenvalue problem. (English) Zbl 0554.35029

There are two results proven in this paper. The first is a lower bound for the k-th eigenvalue of the Dirichlet boundary problem, i.e. the Laplace operator \(\Delta\) on \(L^ 2(D)\) with Dirichlet boundary conditions on the bounded domain D in \({\mathbb{R}}^ n\). More particular the authors show that for all \(k\in N\) \(\lambda_ k\geq (nC_ n/(n+2))k^{n/2}V(D)^{-2/N}\) where V(D) is the volume of D and \(C_ n\) is the so called Weyl constant (or classical constant) which is related to the volume of the unit ball in \({\mathbb{R}}^ n\). This is an accordance to a conjecture of Pólya which says that \(\lambda_ n\geq C_ nk^{n/2}V(D)^{-2/n}.\) The proof is a ”pure analysis proof” which rests on properties of the eigenfunctions and some estimates on their moments.
The second result is an upper bound for the number of eigenvalues N(\(\alpha)\) below a given bound -\(\alpha\) (\(\alpha\geq 0)\) for the Schrödinger operator \(-\Delta +q\) in \(L^ 2({\mathbb{R}}^ n)\), where q is a multiplication operator with \(\int (q+\alpha)_-^{n/2}dx<\infty,\) i.e., if \(n\geq 3\) then \[ n(\alpha)\leq (n(n-2)/4e)^{-n/2}(\omega_{n- 1})^{-1}\int_{{\mathbb{R}}^ n}(q(x)+\alpha)_-^{n/2}dx. \] \(\omega\) \({}_{n-1}\) being the volume of the unit sphere in \({\mathbb{R}}^ n\). This kind of estimate has a long history going back to Cwickel, Lieb, Rosenbljum and Simon [see M. Reed and B. Simon, Methods of modern mathematical physics IV: Analysis of operators (1978; Zbl 0401.47001)]. Note, for \(\alpha =0\) n(\(\alpha)\) is sometimes referred to as the number of bound states.
Reviewer: H.Cycon


35J10 Schrödinger operator, Schrödinger equation
35P15 Estimates of eigenvalues in context of PDEs
47A10 Spectrum, resolvent


Zbl 0401.47001
Full Text: DOI


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