Frequency-dependent time decay of Schrödinger flows

  • Luca Fanelli

    Università di Roma La Sapienza, Italy
  • Veronica Felli

    Università di Milano Bicocca, Italy
  • Marco Antonio Fontelos

    ICMAT-CSIC, Madrid, Spain
  • Ana Primo

    Universidad Autónoma de Madrid, Spain
Frequency-dependent time decay of Schrödinger flows cover
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Abstract

We show that the presence of negative eigenvalues in the spectrum of the angular component of an electromagnetic Schrödinger Hamiltonian generically produces a lack of the classical time-decay for the associated Schrödinger flow . This is in contrast with the fact that dispersive estimates (Strichartz) still hold, in general, also in this case. We also observe an improvement of the decay for higher positive modes, showing that the time decay of the solution is due to the first nonzero term in the expansion of the initial datum as a series of eigenfunctions of a quantum harmonic oscillator with a singular potential.\linebreak A completely analogous phenomenon is shown for the heat semigroup, as expected.

Cite this article

Luca Fanelli, Veronica Felli, Marco Antonio Fontelos, Ana Primo, Frequency-dependent time decay of Schrödinger flows. J. Spectr. Theory 8 (2018), no. 2, pp. 509–521

DOI 10.4171/JST/204