The Weyl Symbol of Schrödinger Semigroups

Abstract : In this paper, we study the Weyl symbol of the Schrödinger semigroup e −tH , H = −∆ + V , t > 0, on L 2 (R n), with nonnegative potentials V in L 1 loc. Some general estimates like the L ∞ norm concerning the symbol u are derived. In the case of large dimension, typically for nearest neighbor or mean field interaction potentials, we prove estimates with parameters independent of the dimension for the derivatives ∂ α x ∂ β ξ u. In particular, this implies that the symbol of the Schrödinger semigroups belongs to the class of symbols introduced in [2] in a high-dimensional setting. In addition, a commutator estimate concerning the semigroup is proved.
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Submitted on : Wednesday, September 26, 2018 - 11:13:13 AM
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Laurent Amour, Lisette Jager, Jean Nourrigat. The Weyl Symbol of Schrödinger Semigroups. Annales Henri Poincaré, Springer Verlag, 2015, 16 (6), pp.1479 - 1488. ⟨10.1007/s00023-014-0344-2⟩. ⟨hal-01881696⟩



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