Estimates on modulation spaces for Schrödinger evolution operators with quadratic and sub-quadratic potentials

Abstract

In this paper we give new estimates for the solution to the Schrödinger equation with quadratic and sub-quadratic potentials in the framework of modulation spaces.

Modulation space estimates for Schrödinger type equations with time-dependent potentials

“…This leads to an additional remainder Hu in (3.10) of our paper, compared with (14) in [9]. To handle this remainder, we need to establish a new inequality (2.1) and apply the estimate for oscillatory integrals in Lemma 2.3, which are both trivial in the case of κ = 2 in [9]. Moreover, due to [10], the singularity of the classical Hamiltonian (2.11) is considered in Lemma 2.5 and Lemma 2.6, which generalize the corresponding results in the case of the smooth Hamiltonian with κ = 2 in [9].…”

“…Finally, combining (3.27), (4.6) and (4.11), the general case (p, q) follows from the complex interpolation theorem for the modulation space, i.e., u(t, ·) M Remark. After this paper was submitted, the author was informed kindly by the reviewer of one recent work [9] about the estimates on modulation spaces for Schrödinger equations with smooth potentials. Both the results in the present paper and those in [9] are inspired by [7], [8], and the proofs of main results in this paper and [9] both rely essentially on integration by parts and the inversion formula for the short-time Fourier transform.…”

“…After this paper was submitted, the author was informed kindly by the reviewer of one recent work [9] about the estimates on modulation spaces for Schrödinger equations with smooth potentials. Both the results in the present paper and those in [9] are inspired by [7], [8], and the proofs of main results in this paper and [9] both rely essentially on integration by parts and the inversion formula for the short-time Fourier transform. However, the author would like to point out that the fractional power of negative Laplacian (−∆) κ/2 with κ = 2, which is considered in this article, is much more complicated than the negative Laplacian that is dealt with in [9].…”

“…Both the results in the present paper and those in [9] are inspired by [7], [8], and the proofs of main results in this paper and [9] both rely essentially on integration by parts and the inversion formula for the short-time Fourier transform. However, the author would like to point out that the fractional power of negative Laplacian (−∆) κ/2 with κ = 2, which is considered in this article, is much more complicated than the negative Laplacian that is dealt with in [9]. This leads to an additional remainder Hu in (3.10) of our paper, compared with (14) in [9].…”

“…However, the author would like to point out that the fractional power of negative Laplacian (−∆) κ/2 with κ = 2, which is considered in this article, is much more complicated than the negative Laplacian that is dealt with in [9]. This leads to an additional remainder Hu in (3.10) of our paper, compared with (14) in [9]. To handle this remainder, we need to establish a new inequality (2.1) and apply the estimate for oscillatory integrals in Lemma 2.3, which are both trivial in the case of κ = 2 in [9].…”