Schrödinger equations of higher order

Abstract

We are interested in finding the sharp regularity with respect to the time variable of the coefficients of a Schrödinger type operator in order to have a well-posed Cauchy Problem in H ∞ . We consider both the cases of the first derivative that breaks down at a point t0 and of Log-Lipschitz coefficients.

Well-posedness of the Cauchy problem for p-evolution equations

“…Literature about well-posedness in Sobolev spaces of the Cauchy problem for hyperbolic operators is really wide; coming up to p 2, many results of well-posedness in Sobolev spaces are available under the assumption that the coefficients a j of (1.1) are real (see, for instance, [1][2][3][4]7,9]). On the contrary, when the coefficients a j (t, x) for 1 j p − 1 are not real, we only know results for p = 2, 3; all these results show that, in order to have a well-posed Cauchy problem in Sobolev spaces, a suitable decay in x for the imaginary part of the coefficients is needed.…”

Loss of regularity for p-evolution type models

“…Such a factorization procedure was proposed in [1,2] to develop a C 1 approach for p-evolution type models. The hierarchy conditions (3.1) simplify to σ 1(2p−|α|) σ 1(2p−|α 1 |) for 1 |α 1 | |α|.…”

“…There exists a C 1 approach for Schrödinger equations of higher-order [1,3] with non-regular coefficients. The notion C 1 approach is related to the strategy that the non-Lipschitz behavior is described, among other things, by some singular behavior of the first derivative with respect to t at (let us say) t = 0 (we call such conditions local conditions).…”

“…We follow an idea from [1], where a special factorization procedure allows to decompose Schrödinger operators of higherorder into Schrödinger operators of first-order possessing pseudodifferential structure modulo a remainder which can be considered in a suitable pseudodifferential calculus. But this factorization procedure gives restrictions to the x-dependence of coefficients in (1.1).…”

Weighted energy estimates for p-evolution equations in SG classes

“…For general p ≥ 2, many results are known when the coefficients a j (t, x) are real valued, see for instance [1][2][3]11,13,15]. When the coefficients a j (t, x) are complex valued for some 1 ≤ j ≤ p−1, then we know from [8,20] that some decay conditions for |x| → ∞ must be required on the imaginary part of the coefficients in order to obtain H ∞ well posedness.…”