Resonances as Viscosity Limits for Exponentially Decaying Potentials

The complex absorbing potential (CAP) method has been used as a computational tool for finding scattering resonances – see Riss–Meyer RiMe and Seideman–Miller semi for an early treatment and Jagau et al Jag for some recent developments. For potentials $V\in L^{\infty}_{\operatorname{comp}}$ the method was justified by Zworski Zw-vis . In xiong2020 the author extended it to potentials which are dilation analytic near infinity. In this paper we show that the CAP method is also valid for potentials which are exponentially decaying. While the key component of Zw-vis and xiong2020 was the method of complex scaling (see Hunziker hunziker1986 , Sjöstrand–Zworski sjostrand1991 for an account and references), here we use complex scaling on the Fourier transform side following Nakamura nakamura1990 and Kameoka–Nakamura Nakamura .

Thus, we consider the Schrödinger operator $P:=-\Delta+V$ acting on $L^{2}({\mathbb{R}}^{n})$ whose potential is exponentially decaying, this means that there exist $C,\gamma>0$ such that

Let $R_{V}(\lambda)=(P-\lambda^{2})^{-1}$ be the resolvent of $P$, initially defined for $\operatorname{Im}\lambda>0$. The exponentially weighted resolvent $\sqrt{V}R_{V}(\lambda)\sqrt{V}$ can be meromorphically continued to the strip $\operatorname{Im}\lambda>-\gamma$, see Froese froese , Gannot gannot and a review in §2. Resonances of $P$ are the poles in this meromorphic continuation.

We now introduce a regularized operator,

(We write $x^{2}:=x_{1}^{2}+\cdots+x_{n}^{2}$.) It is easy to see, with details reviewed in §4, that $P_{\varepsilon}$ is a non-normal unbounded operator on $L^{2}({\mathbb{R}}^{n})$ with a discrete spectrum. When $V\equiv 0$, $P_{\varepsilon}$ is reduced to the rescaled Davies harmonic oscillator – see §3, whose spectrum is given by

where ${\mathbb{N}}_{0}$ denotes the set of nonnegative integers. Thus we will restrict our attentions to $\arg z>-\pi/4$. Suppose that

Zworski Zw-vis proved that resonances can be defined as the limit points of $\{\lambda_{j}(\varepsilon)\}_{j=1}^{\infty}$ as $\varepsilon\to 0+$, in the case of compactly supported potentials. We generalize this result to the case of exponentially decaying potentials. More precisely, we have

In this section we will introduce a meromorphic continuation of the weighted resolvent $\sqrt{V}R_{V}(\lambda)\sqrt{V}$ from $\operatorname{Im}\lambda>0$ to the strip $\operatorname{Im}\lambda>-\gamma$ under the assumption (1.1). As in froese , we define the resonances of $P$ as the poles of this meromorphic continuation, with agreement of multiplicities. For a detailed presentation, we refer to froese .

Let $R_{0}(\lambda):=(-\Delta-\lambda^{2})^{-1}$ be the free resolvent. For $\operatorname{Im}\lambda>0$, the resolvent equation

implies

Since $R_{0}(\lambda)=\mathcal{O}(|\operatorname{Im}\lambda|^{-1}):L^{2}\to L^{2}$, then for $\operatorname{Im}\lambda$ large, $I+\sqrt{V}R_{0}(\lambda)\sqrt{V}$ is invertible by a Neumann series argument and

We will show that the right side of $\eqref{eqn:meroCont}$ has a meromorphic continuation. For that, we recall some bounds of the free resolvent with exponential weights, see gannot for details, to prove the following lemma:

From now on, we identify the resonances $\lambda_{j}$, in $\Omega$ given in (1.4), with the poles of $(I+\sqrt{V}R_{0}(\lambda)\sqrt{V})^{-1}$, with agreement of multiplicities. More precisely, the multiplicity of resonance $\lambda$ is given by

where the integral is over a positively oriented circle enclosing $\lambda$ and containing no poles other than $\lambda$.

The operator $H_{c}:=-\Delta+cx^{2},\ -\pi<\arg c\leq 0$, was used by Davies Dav to illustrate properties of non-normal differential operators. We recall some known facts about $H_{c}$ and its resolvent. As established in Dav , $H_{c}$ is an unbounded operator on $L^{2}({\mathbb{R}}^{n})$ with the discrete spectrum given by

In particular $\sigma(H_{-i\varepsilon})\subset e^{-i\pi/4}[0,\infty)$, then one can study the resolvent of $H_{-i\varepsilon}$ outside $e^{-i\pi/4}[0,\infty)$. Unlike the normal operators, there does not exist any constant $C$ such that $\|(-\Delta-i\varepsilon x^{2}-z)^{-1}\|_{L^{2}\to L^{2}}\leq C\operatorname{dist}(z,e^{-i\pi/4}[0,\infty))^{-1}$. Instead, according to Hitrik–Sjöstrand–Viola HSV , (Zw-vis, , §3) and references given there, for $\Omega\Subset\{z:-\pi/2<\arg z<0\}\setminus e^{-i\pi/4}[0,\infty)$, there exists $C=C(\Omega)$ such that

In this section we will show how exponential weights dramatically improve the bound (3.2) for $(-\Delta-i\varepsilon x^{2}-\lambda^{2})^{-1}$ in the rectangle $\Omega$ given by (1.4), which will be crucial in the proof of Theorem 1.

First, note that $-\Delta_{x}-i\varepsilon x^{2}={\mathcal{F}}^{-1}(\xi^{2}+i\varepsilon\Delta_{\xi}){\mathcal{F}}$, where ${\mathcal{F}}$ denotes the Fourier transform ${\mathcal{F}}u(\xi)=\hat{u}(\xi)=(2\pi)^{-n/2}\int e^{-ix\cdot\xi}u(x)\,dx$. Inspired by nakamura1990 and Nakamura , we introduce a family of spectral deformations in the Fourier space as follows.

For any fixed $\Omega$ given in (1.4), we choose $\rho\in{{\mathcal{C}}^{\infty}}([0,\infty);{\mathbb{R}})$ with $\rho\equiv 0$ near $0$ and $\rho(t)\equiv 1$ for $t\gg 1$ such that

and define the map

then $\psi$ is smooth with the Jacobian:

Let $A$ be an orthogonal matrix with $n$-th column $|\xi|^{-1}\xi$, then we have

For $\theta\in{\mathbb{R}}$, we consider a family of deformations:

and the corresponding unitary operators $U_{\theta},\ \theta\in{\mathbb{R}}$ defined by

Using (3.6), we can compute $\operatorname{det}D\phi_{\theta}(\xi)$ explicitly, i.e.

then by (3.3), $U_{\theta}$ is invertible as $\operatorname{det}D\phi_{\theta}(\xi)\neq 0$ for $\theta\in{\mathbb{R}}$, $|\theta|<\gamma$, the inverse is given by

Now we consider the deformed operators of $\xi^{2}+i\varepsilon\Delta_{\xi}$:

where $D_{\xi_{k}}=-i\partial_{\xi_{k}}$, $J_{\theta}(\xi)=\operatorname{det}D\phi_{\theta}(\xi)$, $J^{lj}(\xi)=[D\phi_{\theta}(\xi)^{-1}]_{jl}$. To extend $Q_{\varepsilon,\theta}$ to $\theta\in{\mathbb{C}}$, we define

In view of (3.3) and (3.9), $D\phi_{\theta}^{-1}$ and $\operatorname{det}D\phi_{\theta}$ extend analytically to $\theta\in D_{\gamma}$. Therefore, we obtain that $Q_{\varepsilon,\theta}$, given by the second equation in (3.11), extends analytically to $\theta\in D_{\gamma}$.

Then we introduce some preliminary results about the spectrum of $Q_{\varepsilon,\theta}$ :

Now we state the main result of this section: