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  Volume 3, Issue 4, Article 63
Lower Bounds for the Infimum of the Spectrum of the Schrödinger Operator in $\mathbb{R}^n$ and the Sobolev Inequalities

    Authors: Ed J.M. Veling,  
    Keywords: Optimal lower bound, Infimum spectrum, Schrödinger operator, Sobolev inequality.  
    Date Received: 15/04/02  
    Date Accepted: 27/05/02  
    Subject Codes:


    Editors: Sever S. Dragomir,  

This article is concerned with the infimum $ e_{1}$ of the spectrum of the Schrödinger operator $ tau=-Delta+q$ in $ mathbf{R}^{N}$, $ Ngeq1$. It is assumed that $ q_{_{-}}=max(0,-q)in L^{p}(mathbf{R}^{N})$, where $ pgeq1$ if $ N=1$, $ p>N/2$ if $ Ngeq2.$ The infimum $ e_{1}$ is estimated in terms of the $ L^{p}$-norm of $ q_{_{-}}$ and the infimum $ lambda_{N,theta}$ of a functional $ Lambda_{N,theta}(nu)=Vertnabla vVert_{2}^{theta}Vert vVert _{2}^{1-theta}Vert vVert_{r}^{-1},$ with $ nu$ element of the Sobolev space $  H^{1}(mathbf{R}^{N})$, where $ theta=N/(2p)$ and $ r=2N/allowbreak (N-2theta)$. The result is optimal. The constant $ lambda_{N,theta}$ is known explicitly for $ N=1$; for $ Ngeq2$, it is estimated by the optimal constant $ C_{N,s}$ in the Sobolev inequality, where $ s=2theta=N/p$. A combination of these results gives an explicit lower bound for the infimum $ e_{1}$ of the spectrum. The results improve and generalize those of Thirring [A Course in Mathematical Physics III. Quantum Mechanics of Atoms and Molecules, Springer, New York 1981] and Rosen [Phys. Rev. Lett., 49 (1982), 1885-1887] who considered the special case $ N=3.$ The infimum $ lambda_{N,theta}$ of the functional $ Lambda_{N,theta}$ is calculated numerically (for $ N=2,3,4,5,$ and $ 10$) and compared with the lower bounds as found in this article. Also, the results are compared with these by Nasibov [Soviet. Math. Dokl., 40 (1990), 110-115].

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