A SMOOTHING PROPERTY OF SCHRÖDINGER EQUATIONS ALONG THE SPHERE

MITSURU SUGIMOTO

We shall consider the following Cauchy problem of the Schrödinger equation

\{ \begin{matrix}(i∂_{t} + △_{x})u(t, x) & = 0 \\ u(0, x) & = φ(x), \end{matrix} (1)

where u = u(t, x) ∈ S′(R_{t} × R_{x}^{n}) and φ(x) ∈ S(R^{n}). Since the solution operator e^{it△} conserves the L^{2}-norm, the solution u(t, x) can be defined for φ(x) ∈ L^{2}(R^{n}), as well, and belongs to L^{2}(R_{x}^{n}) for any fixed t. But, by integrating the solution in t, the extra gain of regularity in x can be observed. For example, in the case n = 1, we have the inequality

sup_{x∈R} ∥|D_{x}|^{1/2}u(·, x)∥_{L^{2}(R_{t})} ≤ ∥φ∥_{L^{2}(R)}

by an appropriate use of Plancherel’s theorem. In the higher dimensional case n ≥ 2, Sjölin [4] shows that ⟨D_{x}⟩^{1/2}u ∈ L_{loc}^{2}(R_{t} × R_{x}^{n}) for φ(x) ∈ L^{2}(R^{n}).

On the other hand, Kato and Yajima [3] showed their global version, that is, the inequality

∥|x|^{α−1}|D_{x}|^{α}u∥_{L^{2}(R_{t}×R_{x}^{n})} ≤ C∥φ∥_{L^{2}(R^{n})}, (2)

where 0 ≤ α < 1/2 in the case n ≥ 3 and 0 < α < 1/2 in the case n = 2. This result has been extended to the case 1 − n/2 < α < 1/2 in the case n ≥ 2 by Sugimoto [5].

As for the problem whether we can obtain the inequality (2) with α = 1/2, Hoshiro [2] proved, in the case n ≥ 3 and 0 ≤ α < 1/2,

∥|x|^{α−1}Ω^{1/2−α}|D_{x}|^{α}u∥_{L^{2}(R_{t}×R_{x}^{n})} ≤ C∥φ∥_{L^{2}(R^{n})}. (3)

Here Ω^{σ} denotes the homogeneous extension of the elliptic self-adjoint operator (1 − △_{S^{n−1}})^{σ/2} on the sphere S^{n−1}, and △_{S^{n−1}} denotes the Laplace-Beltrami operator on S^{n−1}. For example, for k ∈ 2N, we have

Ω^{k} = (1 − ∑_{i<j} (x_{j} \frac{∂}{∂x_{i}} − x_{j} \frac{∂}{∂x_{i}} )^{2})^{k/2}.

If we remark that Ω^{σ} behaves like |x|^{σ}|D_{x}|^{σ} in the sense of the orders of differential and decay, the estimate (3) means that the gain of regularity is exactly the order 1/2 just like the local smoothing property proved by Sjölin [4].

The objective of this article is to investigate why the sphere S^{n−1} has such a special meaning. For the purpose, we shall replace the Laplacian −△_{x} in (1) by more general elliptic operators L_{p} = p(D)^{2}, where p(ξ) is the square root of a positive definite quadratic form. The case p(ξ) = |ξ| corresponds to the case L_{p} = −△_{x}. We shall give an answer to the question by finding the surface which has the same property in this generalized case.

Our main theorem below says that the surface

Σ′_{p} = {∇p(ξ); p(ξ) = 1}

plays the same role as the sphere S^{n−1} does in the special case p(ξ) = |ξ|. We can replace Ω^{1/2−α} in the estimate (3) by Ω_{p′}^{1/2−α}, which is defined by using the Laplace-Beltrami operator on Σ′_{p} instead of △_{S^{n−1}}. Furthermore, it says that the estimate (3) can be extended to the case n ≥ 2 and 1 − n/2 < α < 1/2.

Theorem 1. Suppose n ≥ 2 and 1 − n/2 < α < 1/2. Then the solution u(t, x) ∈ S′(R_{t} × R_{x}^{n}) to the problem

\{ \begin{matrix}(i∂_{t} − L_{p})u(t, x) & = 0 \\ u(0, x) & = φ(x) ∈ S(R_{x}^{n}) \end{matrix}

satisfies

∥|x|^{α−1}Ω_{p′}^{1/2−α}|p(D)|^{α}u(t, x)∥_{L^{2}(R_{t}×R_{x}^{n})} ≤ C∥φ∥_{L^{2}(R_{x}^{n})}.

The surface Σ′_{p} is the set of the positions of particles at the time t = 1, which started from the origin at t = 0 and traveled along the classical orbits. This theorem suggests that the classical orbit propagates some information on unknown regularity properties.

We remark that Theorem 1 cannot be obtained from a simple modification of the proof of the estimate (3). Since the method in [2] depends on the special function theory, it can be applied to only the special case L_{p} = −△_{x}. The change of variable does not directly imply our theorem either.

The essential part of the proof is to show a refinement of the limiting absorption principle, the fact that the resolvent (L_{p}−μ)^{−1} has a meaning even if μ is a spectrum. This principle was first shown by Agmon [1], which says that (L_{p} −μ)^{−1} has the weak limit as a bounded operator on a weighted L^{2}-space of the order greater than 1/2, when μ tends to the spectrum. More precisely, for each d ∈ R, we have

sup_{ε>0} ∥(1 + |x|)^{−k}(L_{p} − (d + iε)^{2})^{−1}v∥_{L^{2}(R^{n})} ≤ C_{d}∥(1 + |x|)^{l}v∥_{L^{2}(R^{n})}

if k > 1/2 and l > 1/2. To prove Theorem 1, we need the following estimate which says that we can attain the critical order 1/2 for the weight if we consider an effect along the surface Σ_{p′}:

Theorem 2. Let d ∈ R and Ψ ∈ C_{0}^{∞}(R^{n}). Suppose k, l > 1/2. Then we have

sup_{ε>0} ∥(1 + |x|)^{−k}Ω_{p′}^{k−1/2}(L_{p} − (d + iε)^{2})^{−1}Ψ(D)v∥_{L^{2}(R^{n})}
≤ C_{d,Ψ} ∥(1 + |x|)^{l}(Ω_{p′}^{1/2−l})^{∗}v∥_{L^{2}(R^{n})}.

References

[1] S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa 2 (1975), 151–218.

[2] T. Hoshiro, On weighted L^{2} estimates of solutions to wave equations, J. Anal. Math. 72 (1997), 127–140.

[3] T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1 (1989), 481–496.

[4] P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J. 55 (1987), 699–715.

[5] M. Sugimoto, Global smoothing properties of generalized Schrödinger equations, J. Anal. Math. 76 (1998), 191–204.

Department of Mathematics

Graduate School of Science

Osaka University

Toyonaka, Osaka 560-0043, Japan

  E-mail address: sugimoto@math.wani.osaka-u.ac.jp