## 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

### MSC:

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

Zbl 0401.47001
Full Text:

### References:

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