Lower bounds on the growth of Sobolev norms in some linear time\n dependent Schr "odinger equations

2018/01/21 by Alberto Maspero, Maspero, Alberto · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1801.06813

openalex publication_date 2018/01/21 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

In this paper we consider linear, time dependent Schr "odinger equations of\nthe form i \∂_t \ψ = K₀ \ψ + V(t) \ψ , where K₀ is a positive\nself-adjoint operator with discrete spectrum and whose spectral gaps are\nasymptotically constant.\n We give a strategy to construct bounded perturbations V(t) such that the\nHamiltonian K₀ + V(t) generates unbounded orbits. We apply our abstract\nconstruction to three cases: (i) the Harmonic oscillator on mathbb R, (ii)\nthe half-wave equation on mathbb T and (iii) the Dirac-Schr "odinger\nequation on the sphere. In each case, V(t) is a smooth and periodic in time\npseudodifferential operator and the Schr "odinger equation has solutions\nfulfilling \‖ \ψ(t) \‖_r gtrsim |t|^r as |t| \≫ 1.\n

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• Long-time behavior of resonant time-dependent perturbations of periodic transport equations on ℝ
• Sobolev instability for perturbations of periodic transport equations

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