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A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator. (English) Zbl 0929.34076

The authors refer to papers of E. H. Lieb and W. E. Thirring [Stud. Math. Phys., Essays Honor Valentine Bargmann, 269-303 (1976; Zbl 0342.35044)] published in the 1970s containing families of inequalities which proved to be useful, particularly in proofs of stability of matter. Specifically, they assert that given a Schrödinger operator \(-\Delta+V\) acting on \(L^2(\mathbb{R}^d)\) the sum of the moments of the negative eigenvalues of this operator obeys the inequality: \(\Sigma E_i^\gamma\leq L_{\gamma, d}\int (V_-(x))^{\gamma+d/2}dx\), with \(-E_1<-E_2 <\cdots \leq 0\), and \(V_-(x)\) stands for \(\max(-V (x),0)\). These inequalities have been shown to be true for \(d=1\), \(\gamma>1/2\); \(d=2\), \(\gamma>0\); and \(d\geq 3\), \(\gamma\geq 0\) \((d\) is the dimension of space, \(\gamma\) is the power to which we raise the eigenvalues). In two dimensions there cannot be any bound for the number of negative eigenvalues if \(\gamma=0\). In the case \(d\geq 3\) several bounds have been proved, but a sharp bound is not known.
The well-known form \(-\partial^2_x -c\delta\) and the Hamiltonian corresponding to it have been widely discussed as an example of a solvable form in quantum mechanics. With \(c>0\) the only bound state is \(\psi(x)= \exp\{-c| x|/2\}\), with eigenvalue \(-c^2/ 4\). This fact led to several conjectures in the past.
After such introductory remarks, and related discussion, the authors proceed to prove that if this Dirac potential is optimal, then the sharp constant \(L_{\gamma,d}\) for \(d=1\) and for \(\gamma=1/2\) is \(1/2\). After completing the proof, they extend some of their results to other potentials (aside from Dirac delta) that are measures. In the process of proving their results they discuss a construction of an analog to the Birman-Schwinger kernel for measures.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators

Citations:

Zbl 0342.35044
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