In this paper we prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schrödinger equations on the one dimensional torus. We show that for any initial condition even in x, regular enough and of size ε sufficiently small, the lifespan of the solution is of order ε −N for any N ∈ N if some non resonance conditions are fulfilled. After a paralinearization of the equation we perform several para-differential changes of variables which diagonalize the system up to a very regularizing term. Once achieved the diagonalization, we construct modified energies for the solution by means of Birkhoff normal forms techniques. * This research was supported by PRIN 2015 "Variational methods, with applications to problems in mathematical physics and geometry".The following is a subclass of the previous class made of those operators which are autonomous, i.e. they depend on the variable t only through the function U .Definition 2.5 (Autonomous smoothing operator). We define, according to the notation of Definition 2.2, the class of autonomous non-homogeneous smoothing operator R −ρ K,0,N [r, aut] as the subspace of R −ρ K,0,N [r] made of those maps (U, V ) → R(U )V satisfying estimates (2.8) with K ′ = 0, the time dependence being only through U = U (t). In the same way, we denote by ΣR −ρ K,0,p [r, N, aut] the space of maps (U, V ) → R(U, V ) of the form (2.9) with K ′ = 0 and where the last term belongs to R −ρ K,0,N [r, aut].0,N [r, aut]. This inclusion follows by the multi-linearity of R in each argument, and by estimate (2.6). For further details we refer to the remark after Definition 2.2.3 in [7]. Spaces of MapsIn the following, sometimes, we shall treat operators without having to keep track of the number of lost derivatives in a very precise way. We introduce some further classes.. The following is a subclass of the class defined in 2.16 made of those symbols which depend on the variable t only through the function U .Definition 2.18 (Autonomous non-homogeneous Symbols). We denote by Γ m K,0,p [r, aut] the subspace of Γ m K,0,p [r] made of the non-homogeneous symbols (U, x, ξ) → a(U ; x, ξ) that satisfy estimate (2.18) with K ′ = 0, the time dependence being only through U = U (t).Remark 2.19. A symbol a(U; ·) of Γ m p defines, by restriction to the diagonal, the symbol a(U, . . . , U ; ·) for Γ m K,0,p [r, aut] for any r > 0. For further details we refer the reader to the first remark after Definition 2.1.3 in [7].

We prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schrödinger equations on the circle. After a paralinearization of the equation, we perform several paradifferential changes of coordinates in order to transform the system into a paradifferential one with symbols which, at the positive order, are constant and purely imaginary. This allows to obtain a priori energy estimates on the Sobolev norms of the solutions. * This research was supported by PRIN 2015 "Variational methods, with applications to problems in mathematical physics and geometry".

No abstract

The study of dispersive properties of Schrödinger operators with point interactions is a fundamental tool for understanding the behavior of many body quantum systems interacting with very short range potential, whose dynamics can be approximated by non linear Schrödinger equations with singular interactions. In this work we proved that, in the case of one point interaction in R 3 , the perturbed Laplacian satisfies the same L p − L q estimates of the free Laplacian in the smaller regime q ∈ [2, 3). These estimates are implied by a recent result concerning the L p boundedness of the wave operators for the perturbed Laplacian. Our approach, however, is more direct and relatively simple, and could potentially be useful to prove optimal weighted estimates also in the regime q ≥ 3.

In this paper we consider an abstract class of quasi-linear para-differential equations on the circle. For each equation in the class we prove the existence of a change of coordinates which conjugates the equation to a diagonal and constant coefficient para-differential one. In the case of Hamiltonian equations we also put them in Poincaré-Birkhoff normal form. By means of the Bony para-linearization formula we prove that it is possible to apply this transformation to quasi-linear Hamiltonian perturbations of the Schrödinger and beam equations. In this way we obtain the first long time existence result without requiring any symmetry on the initial data in the case of quasi-linear PDEs with super-linear dispersion law. We also prove the local in time existence of solutions for quasi-linear (not necessarily Hamiltonian) perturbations of the Benjamin-Ono equation.