A note on the wave packet transforms

Wave front set for solutions to Schrodinger equations with time-dependent variable coefficients

“…In the previous studies of the wave front set, G. B. Folland used the basic wave packet as a window. T. Ōkaji [9] obtained the same conclusion with a general window under some conditions. After that, K. Kato-K. Kobayashi-S. Ito [5] removed some restriction of window completely.…”

Wave front set of solutions to Schrödinger equations with perturbed harmonic oscillators

“…G. B. Folland [6] has shown that the conclusion follows if the wave packet ϕ is an even and nonzero function in S(R n ) and b = 1/2. In T. Ōkaji [22], the proof of Proposition 2.2 for b = 1/2 is given if ϕ satisfies x α ϕ(x)dx = 0 for some α ∈ (N ∪ {0}) n . In [14], the condition x α ϕ(x)dx = 0 have been removed.…”

“…In [2], W. Craig, T. Kappeler and W. Strauss have shown for solutions that for a point x 0 = 0 and a conic neighborhood Γ of x 0 , x r u 0 (x) ∈ L 2 (Γ) implies ξ r û(t, ξ) ∈ L 2 (Γ ′ ) for a conic neighborhood Γ ′ of x 0 and for t = 0, though they have considered more general operators. Several mathematicians have studied in this direction ( [4], [5], [20], [22], [23]).…”

Singularity for Solutions of Linearized KdV Equations

“…holds for all x ∈ K, ξ ∈ Γ with b −1 ≤ |ξ| ≤ b, λ ≥ 1 by taking t = t 0 , since ϕ λ (0) = ϕ 0,λ and x(t 0 ; λ) = x. Then Theorem 1.1 of [12] which is a refinement of Theorem 2.2 of [19] and Theorem 3.22 of [9], shows that ( 15) is equivalent to (x 0 , ξ 0 ) / ∈ WF(u(t 0 , •)), i.e., we have (i) in Theorem 1.6. Hence we prove P (MM 0 , ϕ 0 ) by the induction with respect to N for some small constant M 0 .…”

“…By using the method of characteristics, we can obtain an integral equation which has the solutions to Hamilton equation associated to (1). We can characterize the wave front sets of the solutions just by the asymptotic behavior of solution with respect to the parameter λ which is equivalent to |ξ| to the integral equation with the aide of characterization of wave front set via wave packet transform (Theorem 3.22 of [ [9]], Theorem 2.2 of [ [19]] and Theorem 1.1 of [ [12]]). This is the merit of using the wave packet transform in the study of characterization of wave front set.…”