We study the partial Gelfand–Shilov regularizing effect and the exponential decay for the solutions to evolution equations associated with a class of accretive non-selfadjoint quadratic operators, which fail to be globally hypoelliptic on the whole phase space. By taking advantage of the associated Gevrey regularizing effects, we study the null-controllability of parabolic equations posed on the whole Euclidean space associated with this class of possibly non-globally hypoelliptic quadratic operators. We prove that these parabolic equations are null-controllable in any positive time from thick control subsets. This thickness property is known to be a necessary and sufficient condition for the null-controllability of the heat equation posed on the whole Euclidean space. Our result shows that this geometric condition turns out to be a sufficient one for the null-controllability of a large class of quadratic differential operators.

Galerkin discretizations of integral equations in $\mathbb{R}^{d}$ requirethe evaluation of integrals $I = \int_{S^{(1)}}\int_{S^{(2)}}g(x,y){\rm d}y{\rm d}x$ where S (1),S (2) are d-simplices and g has a singularityat x = y. We assume that g is Gevrey smooth for x $\ne$ y andsatisfies bounds for the derivatives which allow algebraic singularitiesat x = y. This holds for kernel functions commonly occurring in integralequations. We construct a family of quadrature rules $\mathcal{Q}_{N}$ usingN function evaluations of g which achieves exponential convergence|I – $\mathcal{Q}_{N}$ | ≤C exp(–r N γ ) with constants r, γ > 0.

In this paper, we studythe linear Schrödinger equation over the d-dimensional torus,with small values of the perturbing potential.We consider numerical approximations of the associated solutions obtainedby a symplectic splitting method (to discretize the time variable) in combination with the Fast Fourier Transform algorithm (to discretize the space variable).In this fully discrete setting, we prove that the regularity of the initialdatum is preserved over long times, i.e. times that are exponentially longwith the time discretization parameter. We here refer to Gevrey regularity, and our estimatesturn out to be uniform in the space discretization parameter.This paper extends [G. Dujardin and E. Faou, Numer. Math.97 (2004) 493–535], where a similar result has been obtained inthe semi-discrete situation, i.e. when the mere time variable is discretized and spaceis kept a continuous variable.