In this paper we consider linear, time dependent Schrodinger equations of the form i partial derivative(t)psi = K-0 psi + V(t)psi, where K-0 is a positive selfadjoint operator with discrete spectrum and whose spectral gaps are asymptotically constant. We give a strategy to construct bounded perturbations V(t) such that the Hamiltonian K-0 + V(t) generates unbounded orbits. We apply our abstract construction to three cases: (i) the Har- monic oscillator on N, (ii) the half-wave equation on and (iii) the Dirac-Schrodinger equation on Zoll manifolds. In each case, V(t) is a smooth and periodic in time pseudodifferential operator and the Schrodinger equation has solutions fulfilling the optimal lower bound estimate parallel to psi(t)parallel to r greater than or similar to vertical bar t vertical bar as vertical bar t vertical bar >> 1.

Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations / Maspero, Alberto. - In: MATHEMATICAL RESEARCH LETTERS. - ISSN 1073-2780. - 26:4(2019), pp. 1197-1215. [10.4310/MRL.2019.v26.n4.a11]

Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations

Maspero, Alberto
2019

Abstract

In this paper we consider linear, time dependent Schrodinger equations of the form i partial derivative(t)psi = K-0 psi + V(t)psi, where K-0 is a positive selfadjoint operator with discrete spectrum and whose spectral gaps are asymptotically constant. We give a strategy to construct bounded perturbations V(t) such that the Hamiltonian K-0 + V(t) generates unbounded orbits. We apply our abstract construction to three cases: (i) the Har- monic oscillator on N, (ii) the half-wave equation on and (iii) the Dirac-Schrodinger equation on Zoll manifolds. In each case, V(t) is a smooth and periodic in time pseudodifferential operator and the Schrodinger equation has solutions fulfilling the optimal lower bound estimate parallel to psi(t)parallel to r greater than or similar to vertical bar t vertical bar as vertical bar t vertical bar >> 1.
26
4
1197
1215
https://arxiv.org/abs/1801.06813v2
Maspero, Alberto
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/106099
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