ABSTRACT

Consider the linear 2mth-order Schro¨dinger equation (the LSE-2m), with any integer m ≥ 1, which, for convenience and equivalently, we write as

ut = −i (−Δ)mu in IRN × IR+, u|t=0 = u0, (1)

where Δ is the Laplace operator in IRN , for initial data u0 in some weighted L2-space, to be introduced. Here m = 1 leads to the classic linear Schro¨dinger equation (with t → −t)

iut = −Δu in IRN × IR+, (2)

which very actively entered general PDE theory beginning with quantum mechanics in the 1920s. It is not possible to express the depth of mathematical theory developed for models such as (2), (1), and related semilinear ones. We

and

refer to well-known monographs [378, 63], which cover classes of both linear and nonlinear Schro¨dinger PDEs. Concerning the results that are more closely related to the subject of this

chapter, we note that scattering L2-and Lq,rx,t-theories for (2) have been fully developed in the works by Stein, Tomas, Segal, and Strichartz in the 1970s, with later further involved estimates in more general spaces by Ginibre and Velo, Yajima, Cazenave and Weissler, Montgomery-Smith, Keel, Tao, and many others (see [233] and [395] for references concerning these), as well as optimal Lq,rx,t-estimates for the inhomogeneous Schro¨dinger equation

iut = −Δu+ F (x, t) in IRN × IR, (3)

see also more recent papers [241, 347, 417]. The 2mth-order counterpart (1) has also been under scrutiny for a long

period. We refer to Ablowitz-Segur’s monograph [3], Ivano-Kosevich [216], Turitsyn [389], Karpman [226], and Karpman-Shagalov [227] for physical, symmetry, and other backgrounds of higher-order Schro¨dinger-type semilinear models (see also [418] for extra motivations from nonlinear optics) [320] for first existence and uniqueness results, and more recent papers [22, 23, 89, 189, 194, 195, 296, 318, 319, 419] as an account for further applied and rigorous research, as well as other earlier key references and surveys in this fundamental area of modern PDE theory.