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Dispersive estimates for Schrödinger operators in dimensions one and three. (English) Zbl 1086.81077

Summary: We consider \(L^{1}\rightarrow L^{\infty}\) estimates for the time evolution of Hamiltonians \(H = - \Delta + V\) in dimensions \(d=1\) and \(d=3\) with bound \(t^{-\frac d2}\). We require decay of the potentials but no regularity. In \(d =1\) the decay assumption is \(\int (1+| x|)| V(x)| dx <\infty\), whereas in \(d=3\) it is \(| V(x)\}| \leq C(1+| x|)^{-3-}\).

MSC:

81U05 \(2\)-body potential quantum scattering theory
35Q40 PDEs in connection with quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35P15 Estimates of eigenvalues in context of PDEs
47E05 General theory of ordinary differential operators
47F05 General theory of partial differential operators
47N50 Applications of operator theory in the physical sciences
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