Wave front set of solutions to the fractional Schrödinger equation

x ∈ R n .

(1.1)

Here i = √ -1, u : R × R n → C, and the fractional Laplacian (-∆) θ/2 is formally defined by

where F is the Fourier transform. Moreover, the potential V (x) is assumed to be a real-valued function. The fractional Schrödinger equation arises as an extension of quantum mechanics when the Brownian paths in Feynman path integrals are replaced by Lévy flights, yielding a fractional kinetic operator (-∆) θ/2 (see Laskin [16]). Moreover, Longhi [17] proposed an optical realization in which the transverse light evolution in suitably designed optical cavities follows an effective fractional Schrödinger equation with a potential.

In recent years, fractional Schrödinger equations have begun to be studied extensively from various mathematical perspectives, including scattering theory, stationary problems, nonlinear dynamics, and Strichartz-type estimates (see, for instance, [22,4,2,14,5,1]). In this paper, we study the propagation of singularities of solutions in terms of the wave front set. The wave front set is introduced by Hörmander to describe the precise information (position and direction) of the singularity of solutions to PDEs. In particular, Hörmander has proved that the singularities of solutions to hyperbolic PDEs propagate along null bi-characteristic curves (e.g. Hörmander [8,9,7]). Hyperbolic PDEs possess the so-called finite propagation property. Consequently, the wave front set of their solutions can be characterized independently of the growth rate of lower-order terms, such as potential terms and the spatial decay rate of initial data. When θ = 1, the principal symbol of the equation in (1.1) coincides with that of the wave equation. When θ = 2, the equation in (1.1) is the Schrödinger equation. The propagation of singularities for solutions to the Schrödinger equation has been extensively investigated in the literature (e.g. Lascar [15], Sakurai [20], Hassell and Wunsch [6] and Nakamura [19,18]). Unlike hyperbolic equations, the Schrödinger evolution enjoys a dispersive smoothing effect, in the sense that the spatial decay of the initial data can yield improved regularity of the solution. As a result, the propagation of singularities behaves differently from the hyperbolic case. For the Schrödinger setting, in Yajima [21], it has been shown that the regularity of the solution (or the fundamental solution) can change drastically depending on whether the potential is sub-quadratic or super-quadratic at infinity (see also Kato, Nakahashi and Tadano [13]).

The growth assumption on the potential imposed below is designed to bridge the hyperbolic and dispersive regimes. Very recently, Zhu [23] established a characterization theorem for the wave front set of solutions to the fractional Schrödinger equation in the case without a potential term. The proof follows Nakamura’s approach, and Zhu further applied the theorem to the analysis of singularities for gravity-capillary water waves.

Furthermore, in the case 1 < θ < 2, there exists a constant ν < θ θ-1 such that for any multi-index α ∈ Z n ≥0 , there exists a constant C α > 0 satisfying

) with χ(x 0 ) ̸ = 0 and a conic neighborhood Γ of ξ 0 such that for any N ∈ N, there exists a constant C N > 0 satisfying

Here, Γ is called a conic neighborhood of ξ 0 if Γ is an open neighborhood of ξ 0 in R n \ {0} and αξ ∈ Γ holds for any ξ ∈ Γ and α > 0.

Our main result describes the singularities of solutions to the fractional Schrödinger equation in terms of the framework introduced by Kato, Kobayashi and Ito [12]. To this end, we employ the wave packet transform originally introduced by Córdoba and Fefferman [3].

The function φ appearing in the wave packet transform is referred to as a wave packet or a window function. Moreover, for scaled wave packets, we often use the following notation for a parameter b satisfying 0 < b < 1 and λ ≥ 1:

The equivalence of the following definitions of the wave front set is established by Kato, Kobayashi, and Ito, and it will be used in the proof of the main theorem.

Proposition 1.1 (Kato, Kobayashi and Ito [12]).

Then, the following conditions are equivalent:

(ii) There exists a conic neighborhood V of (x 0 , ξ 0 ) such that for any N ∈ N, any a ≥ 1, and any φ ∈ S(R n ) \ {0}, there exists a constant C N,a,φ > 0 satisfying

The following theorem is our main result, which provides a characterization of the wave front set W F (u(t)) for the fractional Schrödinger equation via the wave packet transform. Theorem 1.2. Suppose Assumption 1.1 holds. Let u 0 ∈ L 2 (R n ) and let u(t, x) be the solution to (1.1) in the class C(R; L 2 (R n )). Let b be a parameter satisfying 0 < b < (2 -θ)/2. Then, (x 0 , ξ 0 ) / ∈ W F (u(t)) if and only if there exists a conic neighborhood V = K × Γ of (x