On weak and strong solution operators for evolution equations coming from quadratic operators

Abstract

We identify, through a change of variables, solution operators for evolution equations with generators given by certain simple first-order differential operators acting on Fock spaces. This analysis applies, through unitary equivalence, to a broad class of supersymmetric quadratic multiplicationdifferentiation operators acting on L 2 (R n ) which includes the elliptic and weakly elliptic quadratic operators. We demonstrate a variety of sharp results on boundedness, decay, and return to equilibrium for these solution operators, connecting the short-time behavior with the range of the symbol and the long-time behavior with the eigenvalues of their generators. This is particularly striking when it allows for the definition of solution operators which are compact and regularizing for large times for certain operators whose spectrum is the entire complex plane.

Perturbations of Gibbs semigroups and the non-selfadjoint harmonic oscillator

“…This is the extension of Mehler's formula obtained in [1] for H ϑ . The semigroup property e −H ϑ (τ +σ) = e −H ϑ τ e −H ϑ σ is valid for all τ, σ ∈ S ϑ .…”

“…According to the framework of [1,26], when seen as a family of bounded operators in τ , the holomorphic semigroup e −H θ τ has a bounded extension (in the uniform operator norm) to the maximal semi-modulus…”

“…Assume that ϑ = 0. Let V satisfy (1). Then there exist constants K > 0 and n 0 ∈ N such that α n ≥ Kn and β ± n ≥ Kn…”

Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians

“…To show that the graph  ( −2 ) is a positive Lagrangian in * ℂ × * ℂ with respect to the symplectic form 1 we must show that…”

“…These classes cater for the presence of potentials with certain properties and symbols q of greater generality than quadratic forms, but they do not admit the quadratic form q to have a nonzero real part. We also note the recent preprint containing interesting ideas on Schrödinger equations of more general type than we study. These ideas are different from those explored in this paper.…”