Nonlinear Schrödinger equation for the twisted Laplacian

Abstract

We establish the local well posedness of solution to the nonlinear Schrödinger equation associated to the twisted Laplacian on C n in certain first order Sobolev space. Our approach is based on Strichartz type estimates, and is valid for a general class of nonlinearities including power type. The case n = 1 represents the magnetic Schrödinger equation in the plane with magnetic potential A(z) = iz, z ∈ C.2010 Mathematics Subject Classification. Primary 42B37, Secondary 35G20, 35G25.

Sharp estimates for the Schrödinger equation associated to the twisted Laplacian

“…Ratnakumar [7], P.K. Ratnakumar and V. K. Sohani [8,9] and Z. Zhang and S. Zheng [17]. The obtained results in these references are analogues of the well known regularity properties for the classical Schrödinger equation…”

“…Ratnakumar and V.K. Sohani [8,9] the previous Ratnakumar result was extended to the non-homogeneous (and possibly non-linear) Schödinger equation associated to L. Also, the Schrödinger equation associated to L, but with non-linear polynomials terms can be found in Z. Zhang and S. Zheng [17].…”

Nonlinear Schrödinger equation for the twisted Laplacian in the critical case

“…Uniqueness: Uniqueness in C((T * , T * ),W 1,2 L (C n ))∩L γ loc (T * , T * ),W 1,ρ L (C n ) will follow from estimate (2.7) with m = 0 in Proposition 2.5, see uniqueness in [6]. Blowup alternative: We prove blowup alternative by method of contradiction.…”

“…From Strichartz estimates and estimate (3.10), u ∈ C([t 0 , t 0 + T ],W 1,2 L ) ∩ L q ((t 0 , t 0 + T ),W 1,p L (C n )) for every admissible pair (q, p). In view of Lemma A.1 in [6], u is also a solution to the initial value problem (1.1), (1.2). Similarly solution exists on the interval [t 0 − T ′ , t 0 ] for some T ′ > 0.…”

A Hardy–Sobolev inequality for the twisted Laplacian

“…We refer the reader to the classic paper by Avron et al . [3], which discusses the magnetic Laplacian for a constant magnetic field, to a (relatively) new paper by Yajima [16] on the Schrödinger equation for the magnetic Laplacian and to [11][12][13], which treat the local and global well-posedness of the Schrödinger equation for the twisted Laplacian.…”