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Universal lower bounds on eigenvalue splittings for one dimensional Schrödinger operators. (English) Zbl 0579.34014

Consider a one dimensional Schrödinger operator \(H=-d^ 2/dx^ 2+V\) where V is a continuous function. Let us denote by \(E_ n\), \(n=0,1,2,..\). the eigenvalues of H below the essential spectrum of H. It is well known that the eigenvalues \(E_ n\) are nondegenerate. We give estimates on how close \(E_ n\) and \(E_{n-1}\) can be, that is we provide lower bounds on \(E_ n-E_{n-1}\). Those bounds depend only on qualitative properties of V. Suppose that \(V(x)>E_ n+\alpha^ 2\) for \(x\not\in [a,b]\) and set \(\lambda =\sup \{| V(x)-E|;x\in [a,b],E\in [E_{n-1},E_ n]\},\) then \(E_ n-E_{n-1}\geq Ce^{- \lambda (b-a)}\) with a computable constant C (depending on \(\alpha\) and \(\lambda)\). Such an estimate is ”best possible” in the sense that the exponential factor is saturated precisely in tunneling examples. The method can be extended to Sturm-Liouville operators.
The method of proof relies heavily on ODE-techniques, especially on the method of Prüfer angles. We have, in addition, a similar result on the difference \(E_ 1-E_ 0\) in the multidimensional case by completely different methods.

MSC:

34L99 Ordinary differential operators
47E05 General theory of ordinary differential operators
81Q99 General mathematical topics and methods in quantum theory
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