Cutoff Resolvent Estimates and the Semilinear Schrödinger Equation

In this short note we show how cutoff semiclassical resolvent estimates for the Laplacian on a non-compact manifold, with spectral parameter on the real axis, lead to well-posedness results for the semilinear Schrödinger equation. Motivated by the requirements of [Chr3] and [BGT2], and the microlocal inverse estimates of [Chr1, Chr2], we first prove a general theorem for a large class of resolvents. Following the recent work of Nonnenmacher-Zworski [NoZw], we apply the general theorem in the case there is a hyperbolic fractal trapped set.

Let $(M,g)$ be a Riemannian manifold of dimension $n$ without boundary, with (non-negative) Laplace-Beltrami operator $-\Delta$ acting on functions. The Laplace-Beltrami operator is an unbounded, essentially self-adjoint operator on $L^{2}(M)$ with domain $H^{2}(M)$. We assume $(M,g)$ is asymptotically Euclidean in the sense of [NoZw, (3.7)-(3.9)] and that the classical resolvent $(-\Delta-(\lambda^{2}+i\epsilon))^{-1}$ obeys a limiting absorption principle as $\epsilon\to 0+$, $\lambda\neq 0$.

Our first result is that if we have cutoff semiclassical resolvent estimates with a sufficiently small loss, then we have weighted smoothing for the Schrödinger propagator with a loss. Let $\rho_{s}$ be a smooth, non-vanishing weight function satisfying

for some fixed $x_{0}$ and $x$ outside a compact set.

The assumption that $(M,g)$ is asymptotically Euclidean is that there exists $R_{0}>0$ sufficiently large that, on each infinite branch of $M\setminus B(0,R_{0})$, the semiclassical Laplacian $-h^{2}\Delta$ takes the form

with $a_{\alpha}(x,h)$ independent of $h$ for $|\alpha|=2$,

In order to quote the results of [NoZw] we also need the following analyticity assumption: $\exists\theta_{0}\in[0,\pi)$ such that the $a_{\alpha}(x,h)$ are extend holomorphically to

As in [NoZw], the analyticity assumption immediately implies

Recall the free Laplacian $(-\Delta_{0}-\lambda^{2})^{-1}$ on ${\mathbb{R}}^{n}$ has a holomorphic continuation from $\,\mathrm{Im}\,\lambda>0$ to $\lambda\in{\mathbb{C}}$ for $n\geq 3$ odd, and to the logarithmic covering space for $n$ even. This motivates the limiting absorption assumption, that

exists as a bounded operator

provided $s>1/2$. As in the free case, we allow a possible logarithmic singularity at $\lambda=0$.

The problem of “local smoothing” estimates for the Schrödinger equation has a long history. The sharpest results to date are those of Doi [Doi] and Burq [Bur]. Doi proved if $M$ is asymptotically Euclidean, then one has the estimate

for $\chi\in{\mathcal{C}}^{\infty}_{c}(M)$ if and only if there are no trapped sets. Burq’s paper showed if there is trapping due to the presence of several convex obstacles in ${\mathbb{R}}^{n}$ satisfying certain assumptions, then one has the estimate (1.5) with the $H^{1/2}$ norm replaced by $H^{1/2-\eta}$ for $\eta>0$. In [Chr3], the author considered an arbitary, single trapped hyperbolic orbit. One of the goals of this paper is to use estimates obtained by Nonnenmacher-Zworski [NoZw] for fractal hyperbolic trapped sets to obtain similar results to [Chr3] for the semilinear Schrödinger equation. To that end we have the following corollary to Theorem 1.

We remark that the assumption $P_{E}(1/2)<0$ implies the trapped set $K_{E}$ is filamentary or “thin” (see [NoZw] for definitions).

We consider the following semilinear Schrödinger equation problem:

where $I\subset{\mathbb{R}}$ is an interval containing $0$. Here the nonlinearity $F$ satisfies

and $G:{\mathbb{R}}\to{\mathbb{R}}$ is at least $C^{3}$ and satisfies

for some $\beta\geq{\frac{1}{2}}$.

In §3 we prove a family of Strichartz-type estimates which will result in the following well-posedness theorem.

Acknowledgments. This research was partially conducted during the period the author was employed by the Clay Mathematics Institute as a Liftoff Fellow.

Since we are assuming $(-\Delta-z)^{-1}$ obeys a limiting absorption principle, we have

for $0<\epsilon_{0}\leq|\tau|\leq C$. For $|\sigma|\geq C$ for some $C>0$, $\sigma\in{\mathbb{C}}$ in a neighbourhood of the real axis, write

for

Now

is a Fredholm operator for $z$ in the specified range, and hence the “gluing” techniques from [Vod] and [Chr3, §2] can be used to conclude for $s>1/2$,

has a holomorphic extension to a slightly smaller neighbourhood in $z$, and in particular,

Rescaling, we have

where (see Figure 1)

As in [Chr3] and [Bur], the following lemma follows from integration by parts and interpolation, together with the condition on $\eta$, (1.4).

Theorem 1 now follows from the standard “$TT^{*}$” argument, letting $\epsilon\to 0$ in (2.1) (see [BGT2], the references cited therein, and [Chr3]).

∎

The following Corollary uses interpolation with an $H^{2}$ estimate to replace the $H^{1/2-\eta}$ norm on the left hand side of (1.3) with $H^{1/2}$, and will be of use in §3. See [Chr3] for the details of the proof.

In this section we give several families of Strichartz-type inequalities and prove Theorem 2. The statements and proofs are mostly adaptations of similar inequalities in [BGT2], so we leave out the proofs of these in the interest of space.

If we view $M\setminus U$, where $U$ is a neighbourhood of $K_{E}$, as a manifold with non-trapping geometry, we may apply the results of [HTW] or [BoTz] to a solution of the Schrödinger equation away from the trapping region, resulting in perfect Strichartz estimates. For this section we need (1.3) only with a compact cutoff $\chi$ instead of with the more general weight $\rho_{s}$.

The proof uses a local WKB expansion localized also in time to the scale of inverse frequency, followed by summing over frequency bands (see [Chr3] and [BGT1]). The only difference here is the explicit dependence of $\delta$ on $\eta$, which is related to the growth of the function $g(h)$.