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Semiclassical limit of the spectral decomposition of a Schrödinger operator in one dimension. (English) Zbl 0785.34058

Let \(L_ h=-h^ 2 {d^ 2 \over dx^ 2} +V(x)\), with \(V(x)\) continuous and \(1 \geq h>0\). Let \(A:D \to H=L^ 2(\mathbb{R})\) be a selfadjoint extension of \(L_ h\) operating on \(C^ 2_ 0 (\mathbb{R})\). Finally, let \(A=\int \lambda dE(\lambda)\) be the spectral decomposition of \(A\). The following theorem concerning the semiclassical limit of the projections \(E(\Delta)\) is proven:
Theorem. Assume that \(V(x)-\sup \Delta \geq c^ 2\), \(c>0\), in an inverval \(I\) and that \(J\) is a subinterval of \(I\) whose distance to the boundary of \(I\) is \(d>0\), then \[ \| E(\Delta)f \| \leq 8 \sqrt 5 e^{-{cd \over 2h}} \| f \| \] if either \(| J | \leq d/2\) and \(0<h \leq 1\) or \(\exp(-cd/h) \leq 1/2\) and in both cases \(f\) is a square integrable function with support in \(J\). Under the same conditions, \[ \| \chi_ J E(\Delta)f \| \leq 8 \sqrt 5 e^{-{cd \over 2h}} \| f \| \] where \(f \in H\) and \(\chi_ J\) is the characteristic function of \(J\). Some consequences concerning the relation between the spectral properties of the global operator and the one localized to the well are discussed.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47E05 General theory of ordinary differential operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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