On the pseudospectra of Schr\"odinger operators on Zoll manifolds

TL;DR

This paper studies the pseudospectra of non-self-adjoint Schr"odinger operators on Zoll manifolds, providing asymptotic descriptions of their pseudo-spectrum and numerical range.

Contribution

It offers new asymptotic results on the pseudo-spectrum and numerical range for Schr"odinger operators on Zoll manifolds, extending understanding of their spectral properties.

Findings

Asymptotic descriptions of the pseudo-spectrum

Analysis of the numerical range of these operators

Extension of spectral theory to Zoll manifolds

Abstract

We consider non-self-adjoint Schr\"odinger operators $\Delta+V$ where $\Delta$ is the Laplace-Beltrami operator on a Zoll manifold $X$ and $V\in C^\infty(X,\mathbb C)$. We obtain asymptotic results on the pseudo-spectrum and numerical range of such operators.

Equations

$$\Vert (P-\lambda I{)}^{-1}\Vert \ge \frac{1}{ϵ},$$

$$\frac{\Vert (P-\lambda I)\psi \Vert }{\Vert \psi \Vert }\le ϵ.$$

$$\frac{\Vert (\Delta +M_V-{\lambda }_{k}I){\psi }_k\Vert }{\Vert {\psi }_k\Vert }=O(k^{-\infty })\text{and}\underset{k\to \infty }{\lim }|{\lambda }_k|=+\infty .$$

$${\Lambda }_k=\frac{4{\pi }^2}{T^2}{\left(k+\frac{\beta }{4}\right)}^2,k=1,2,\dots ,$$

$$\exists C>0\text{such that}\sigma (\Delta )\subset \bigcup _{k\in \mathbb{N}}[{\Lambda }_k-C,{\Lambda }_k+C],$$

$$\Delta =\frac{4{\pi }^2}{T^2}{\left(A+\frac{\beta }{4}\right)}^2+Q_0.$$

$$E_k:=\text{eigenspace of }A\text{corresponding to the eigenvalue }k,$$

$$\beta =2(n-1)\text{and}{Q}_0=-\frac{(n-1{)}^2}{4}I.$$

$$\Delta +M_V=\frac{4{\pi }^2}{T^2}{\left(A+\frac{\beta }{4}\right)}^2+Q_0+M_V.$$

$$\bigcup _{k=0}^{\infty }{D}_k,D_k:={\Lambda }_k+D.$$

$$\tilde{V}(\gamma )=\frac{1}{T}{\int }_{\gamma }Vds,$$

$$\{\Re \tilde{V}+q_0,\Im \tilde{V}\}(\gamma )<0.$$

$$\{\Re \tilde{V}+q_0,\Im \tilde{V}\}(\gamma )>0,$$

$$\left\{{\lambda }_k+\tilde{V}(\gamma );\{\Re \tilde{V}+q_0,\Im \tilde{V}\}(\gamma )\ne 0\right\}$$

$$\tilde{\tilde{V}}=\frac{1}{4}\widetilde{V^2}-\frac{1}{8\pi }{\int }_0^{2\pi }dt{\int }_0^t\{{\varphi }_t^{\ast }V,{\varphi }_s^{\ast }V\}ds$$

$$\mathcal{R}(\hat{H})=\left\{\frac{⟨\hat{H}(\psi ),\psi ⟩}{⟨\psi ,\psi ⟩};\psi \in \mathcal{D}(\hat{H})\right\}.$$

$$\left\{\frac{⟨\hat{H}(\psi ),\psi ⟩}{⟨\psi ,\psi ⟩};\psi \in E_k\right\}.$$

$${\mathcal{R}}_k=\left\{\frac{⟨(\Delta +M_V)\psi ,\psi ⟩}{⟨\psi ,\psi ⟩}-{\Lambda }_k;\psi \in E_k\setminus \{0\}\right\}$$

$$\forall z\in {\mathcal{C}}_k,d(z,\sigma (\Delta ))\ge g(k)/3\text{ and }d(z,\sigma (\hat{H}))\ge g(k)/3.$$

$$P_k=\frac{1}{2\pi ı}{\oint }_{{\mathcal{C}}_k}{\left(z-H\right)}^{-1}dz$$

$${\Pi }_k=\frac{1}{2\pi ı}{\oint }_{{\mathcal{C}}_k}{\left(z-\Delta \right)}^{-1}dz.$$

$$\Vert {\left(z-H\right)}^{-1}\Vert$$

$$\le$$

$$\Vert P_k-{\Pi }_k\Vert$$

$$e^{F}\bigl{(}\Delta+M_{V}\bigr{)}e^{-F}=e^{F}\bigl{(}\frac{4\pi^{2}}{T^{2}}(A+\frac{\beta}{4})^{2}+Q_{0}+M_{V}\bigr{)}e^{-F}=\frac{4\pi^{2}}{T^{2}}(A+\frac{\beta}{4})^{2}+B+S,$$

$$\tilde{A}=\frac{2\pi }{T}(A+\frac{\beta }{4});$$

$$[F_1,{\tilde{A}}^2]=B_0-(M_V+Q_0)+\text{order }(-1),$$

$$B_0=\frac{1}{T}{\int }_0^T{e}^{it\tilde{A}}(M_V+Q_0)e^{-it\tilde{A}}dt.$$

$$f_1:=-\frac{H_0^{-1}}{2T}{\int }_0^{T}dt{\int }_0^t{\varphi }_s^{\ast }(V+q_0)ds$$

$$\{f_{1}\,,\,H_{0}^{2}\}=2H_{0}\{f_{1}\,,\,H_{0}\}=-\frac{1}{T}\int_{0}^{T}\bigl{(}\phi_{t}^{*}(V+q_{0})-(V+q_{0})\bigr{)}\,dt=V-V^{\text{\tiny ave}},$$

$$e^{F_{1}}\bigl{(}\widetilde{A}^{2}+Q_{0}+M_{V}\bigr{)}e^{-F_{1}}=\widetilde{A}^{2}+B_{0}+R_{-1},$$

Full text

D. Sher supported in part by NSF grant EMSW21-RTG 1045119.

C. Villegas-Blas partially supported by projects PAPIIT-UNAM IN105718 and CONACYT 000000000283531

1. Introduction and statement of results

Let $\Delta$ denote the Laplace-Beltrami operator on a manifold $X$ and let $V\in C^{\infty}(X,\mathbb{C})$. We will study asymptotic properties of the pseudospectrum and numerical range of the Schrödinger operator $\Delta+M_{V}$, where $M_{V}$ is the operator of multiplication by $V$. Here we are taking $\Delta$ as a self-adjoint operator with a suitable domain in $L^{2}(X)$.

Recall that, given an operator $P$ densely defined on some Hilbert space ${\mathcal{H}}$, and $\epsilon>0$, the $\epsilon$-pseudospectrum of $P$ is the set of complex numbers $\lambda$ such that

[TABLE]

where, by definition, the norm on the left-hand side is infinite if $\lambda$ is an eigenvalue of $P$. Equivalently, $\lambda$ is in the $\epsilon$-pseudospectrum of $P$ iff there exists $\psi\in{\mathcal{H}}\setminus\{0\}$ such that

[TABLE]

This motivates the following definition:

Definition 1.1**.**

We say that a sequence of complex numbers $\{\lambda_{k}\}$ is in the asymptotic pseudospectrum of $\Delta+M_{V}$ if there exists a sequence $\{\psi_{k}\}$ of functions in $L^{2}(X)$ such that as $k\to\infty$,

[TABLE]

The sequence $\{\psi_{k}\}$ is referred to as an associated pseudo mode.

Note that (1.1) implies that, $\forall\epsilon>0$, $\lambda_{k}$ will be in the $\epsilon$-pseudospectrum of $\Delta+M_{V}$ for all sufficiently large $k$. The previous definition has an analogue in the semi-classical setting, see for example [18] §12.5 (and references therein). In the present case the role of Planck’s constant is played by $\hbar=1/k$.

It is well-known that the spectrum of a non-self-adjoint operator is very unstable. The pseudospectrum, however, is much more stable, which makes it a natural object to study. It is also known that points in the pseudospectrum do not have to be close to the spectrum, even in the semi-classical limit.

In this paper we consider the case where $X$ is a closed, connected Zoll manifold. More specifically, we make the assumption there is a $T>0$ such that each geodesic on $X$ has minimal period $T$ (we rule out the existence of exceptional short geodesics). Such manifolds are often called $C_{l}$-manifolds [1]. The simplest examples are the spheres with the round metric. It is known that the spectrum of the Laplacian $\Delta$ on a Zoll manifold consists of clusters of eigenvalues of uniformly bounded width centered at the points

[TABLE]

where $\beta$ is the Morse index of one periodic geodesic (hence all periodic geodesics). Specifically

[TABLE]

where $\sigma(\Delta)$ denotes the spectrum of $\Delta$. Note that the distance between the clusters $\Lambda_{k}$ and $\Lambda_{k+1}$ is linear in $k$, which allows us to separate the eigenvalues of the Laplacian unambiguously into clusters, at least for large $k$.

More precisely, as proved in [4], there exist self-adjoint, commuting pseudo-differential operators $A$ of order one and $Q_{0}$ of order zero such that the spectrum of $A$ consists of the eigenvalues $k=1,2,\ldots$ and

[TABLE]

In particular, we can take $C$ equals to the operator norm $\|Q_{0}\|$ with $Q_{0}\in\mathcal{B}(L^{2}(X))$. Moreover, if for each $k$ we let

[TABLE]

the eigenvalues in the $k$-th cluster consist of the eigenvalues of the restriction $Q_{0}|_{E_{k}}$, shifted by $\Lambda_{k}$.

The operator $(A+\frac{\beta}{4})^{2}$ has spectrum contained in $(\mathbb{N}+\frac{\beta}{4})^{2}$, and has the same principal and sub principal symbols as $\Delta$ up to the factor $\frac{4\pi^{2}}{T^{2}}$. For example, in case $X=S^{n}$, the unit $n$-dimensional sphere, $T=2\pi$ and the eigenvalues of the Laplacian are $k(k+n-1)=(k+\frac{n-1}{2})^{2}-\frac{(n-1)^{2}}{4}$. Therefore in this case

[TABLE]

For the round sphere the principal symbol of $Q_{0}$ is constant. In the general Zoll case the symbol of $Q_{0}$ is not constant (see [17], Theorem 3, for an expression for it), but it is always constant along geodesics since $[Q_{0},\Delta]=0$. The fact that the symbol if $Q_{0}$ is not constant corresponds to the non-trivial clustering phenomenon of the Laplace eigenvalues seen in Zoll metrics, as the asymptotic behavior of the eigenvalue clusters is given to first order by the principal symbol of $Q_{0}$ via a Szegö limit theorem.

In conclusion, our Schrödinger operator $\Delta+M_{V}$ has the form

[TABLE]

Thus we can think of $M_{V}$ as a non-self-adjoint perturbation added to $Q_{0}$.

1.1. Statements of our results

We can now state the main results of this paper.

1.1.1. On the spectrum

First we will show that the spectrum of $\Delta+M_{V}$ also has a cluster structure (at least if we are sufficiently far from the origin) generalizing (1.2) (see also [5]). Let us denote by $D$ the closed disk around the origin with radius $||Q_{0}+M_{V}||$.

Theorem 1.2**.**

Let $V:X\to{\mathbb{C}}$ be a bounded measurable function. Then the spectrum of $H:=\Delta+M_{V}$ is contained in the union

[TABLE]

The spectrum of $H$ consists entirely of isolated eigenvalues with finite algebraic multiplicities and, for $k$ sufficiently large, the spectrum of $H$ in the the disk $D_{k}$ consists of eigenvalues with total algebraic multiplicity equal to the dimension of $E_{k}$. Moreover, the generalized eigenspaces of $\Delta+M_{V}$ span $L^{2}(X)$.

Note that the sets ${D}_{k}$ are pairwise disjoint for large $k$.

1.1.2. Results on the asymptotic pseudospectrum

We now turn to the statement of our results on the asymptotic pseudospectrum of $\Delta+M_{V}$.

Let $H_{0}(x,\xi)=\lVert\xi\rVert$, $H_{0}\in C^{\infty}(T^{*}X\setminus\{0\})$. This Hamiltonian generates geodesic flow re-parametrized by arc length. In what follows we shall think of oriented geodesics as trajectories of the Hamiltonian $H_{0}$ on the unit tangent bundle of $X$. Let ${\mathcal{O}}$ be the manifold of oriented geodesics on $X$. This is the symplectic manifold obtained by symplectic reduction of $T^{*}X\setminus\{0\}$ under the circle action generated by $H_{0}$. Given any smooth function $V$ on $X$, one can define its Radon transform to be the function $\widetilde{V}$ on ${\mathcal{O}}$ defined by

[TABLE]

where $s$ denotes arc length along the oriented geodesic $\gamma\in{\mathcal{O}}$, and we are abusing the notation and continue to denote by $V$ the pull-back of the potential to the unit tangent bundle. Note that ${\mathcal{O}}$ has an involution corresponding to reversing the orientation of the geodesics. This involution is anti-symplectic, and $\widetilde{V}$ is even with respect to it.

Now recall that $Q_{0}$ commutes with $\Delta$ (and with $(A+\frac{\beta}{4})$). Therefore its principal symbol $q_{0}$ is invariant along geodesics and may thus be viewed as a function on ${\mathcal{O}}$, also denoted $q_{0}$. We reviewed above that, in case $X$ is the unit sphere, $q_{0}=-\frac{(n-1)^{2}}{4}$ (constant). With this notation, our result is as follows:

Theorem 1.3**.**

(1) Let $\mu=(\widetilde{V}+q_{0})(\gamma)$, where $\gamma\in{\mathcal{O}}$ is such that

[TABLE]

Then $\{\Lambda_{k}+\mu\}$ is in the asymptotic pseudospectrum of $\Delta+M_{V}$. In fact, for each $m\in\gamma$, there exists a pseudo mode $\{\psi_{k}\}$ such that the sum $\sum_{k=1}^{\infty}\psi_{k}$ is a distribution whose wave-front set is the ray in $T^{*}X\setminus\{0\}$ through $m$.

(2) If $\{\Lambda_{k}+\mu\}$ is in the asymptotic pseudospectrum with $\mu=(\widetilde{V}+q_{0})(\gamma)$, and

[TABLE]

then for any associated pseudo mode $\{\psi_{k}\}$, the wave-front set of $\sum_{k}\psi_{k}$ is disjoint from the conic set in $T^{*}X\setminus\{0\}$ generated by $\gamma$.

Note that although the potential $\widetilde{V}$ is even under the involution of reversing the orientation of $\gamma$, the Poisson bracket $\{\Re\widetilde{V}+q_{0}\,,\,\Im\widetilde{V}\}(\gamma)$ is odd under the same involution. Together with part (1) of the theorem, this implies that the set of sequences of the form

[TABLE]

is contained in the asymptotic pseudospectrum. However, in general this set does not equal the asymptotic pseudospectrum. For example, in the case of spheres, if $V$ is odd (with respect to the antipodal map) then $\widetilde{V}$ is identically zero. In that case, however, there is a substitute function for $\widetilde{V}$, namely the function

[TABLE]

where $\phi_{t}$ is geodesic flow, see [14] (Theorem 3.2). More subtly we expect that one can have that $\{\Re\widetilde{V}+q_{0}\,,\,\Im\widetilde{V}\}(\gamma)=0$ and have pseudo modes concentrated at $\gamma$ with pseudo eigenvalue $\widetilde{V}(\gamma)$, in the spirit of Theorem 2.1 in [13], where non-vanishing of higher-order derivatives is assumed.

Theorem 1.3 is analogous to Theorem 12.8 in [18] (originally due to Davies), where a similar Poisson bracket condition appears. It is also analogous in spirit to the main result in [2].

1.1.3. Results on the numerical range

Recall that the numerical range of an operator $\hat{H}$ on $L^{2}(X)$ with domain ${\mathcal{D}}(\hat{H})$ is the subset of the complex plane defined by:

[TABLE]

We are interested in the numerical range of $\hat{H}=\Delta+M_{V}$. We will only consider the asymptotics of the sets of values

[TABLE]

To state our result we shift these sets by $\Lambda_{k}$:

Theorem 1.4**.**

In case $X=S^{n}$ or $X$ a Zoll surface, the limit ${\mathcal{R}}_{\infty}$ of the sets

[TABLE]

as $k$ tends to infinity is the convex hull of the image of $\widetilde{V}+q_{0}:{\mathcal{O}}\to{\mathbb{C}}$.

For a similar result for Berezin-Toeplitz operators see [2].

2. Proof of theorem 1.2

Let $D$ be the operator $\frac{4\pi^{2}}{T^{2}}\left(A+\frac{\beta}{4}\right)^{2}$. Let $z$ be in the resolvent set $\rho(D)$ of the operator $D$, with the distance $d(z,\sigma(D))$ from $z$ to the spectrum $\sigma(D)$ of $D$ greater than the norm $||Q_{0}+M_{V}||$. Since $D$ is self adjoint then $||\left(D-z\right)^{-1}||=1/d(z,\sigma(D))$ which implies $||\left(Q_{0}+M_{V}\right)\left(D-z\right)^{-1}||<1$. Therefore the operator $\left(I+(Q_{0}+M_{V})\left(D-z\right)^{-1}\right)^{-1}$ is a bounded operator which in turn implies that $(\Delta+M_{V}-z)^{-1}$ exists and it is a bounded operator, i.e. $z$ is in the resolvent set $\rho(H)$ of $H$. Thus the spectrum $\sigma(H)$ of H must be contained in the union of the closed disks $D_{k}$ indicated in (1.6).

Consider $z\in{\mathbb{C}}$ such that $d(z,\sigma(\Delta))>2\|M_{V}\|$. Since $\Delta$ has compact resolvent then writing $H-z=\left(I+M_{V}(\Delta-z)^{-1}\right)(\Delta-z)$ we see that $(H-z)^{-1}$ exists and it must be compact (note that since $\Delta$ is self adjoint then $\|(\Delta-z)^{-1}\|=1/d(z,\sigma(\Delta))$). Thus we have that the spectrum of $H$ consists entirely of eigenvalues with finite algebraic multiplicities, see reference [12], theorem III, section 6.8.

Let $g(k)=\Lambda_{k}-\Lambda_{k-1}=O(k)$ and let ${\mathcal{C}}_{k}$ be the circle of radius $g(k)/2$ and center $\Lambda_{k}$. Note that, for k sufficiently large, both the interval of radius $||Q_{0}||$ around $\Lambda_{k}$ and the disk $D_{k}$ are inside ${\mathcal{C}}_{k}$. Moreover,

[TABLE]

In particular, ${\mathcal{C}}_{k}$ is contained in both $\rho(\Delta)$ and $\rho({H})$. Thus the following two projections are well defined:

[TABLE]

and

[TABLE]

Since $\Delta$ is self adjoint then $\|(z-\Delta)^{-1}\|=1/d(z,\sigma(\Delta))=O(k^{-1})$ uniformly in ${\mathcal{C}}_{k}$. This implies for k sufficiently large that $\|M_{V}(z-\Delta)^{-1}\|\leq 1/2$ which in turn implies, using a Neumann series, that

[TABLE]

uniformly in ${\mathcal{C}}_{k}$.

Thus we obtain

[TABLE]

Therefore, for $k$ sufficiently large, the norm $\|P_{k}-\Pi_{k}\|$ is less than one which in turn implies that the dimension of the range of $P_{k}$, $\text{dim}(\text{Ran}(P_{k}))$, and the dimension of the range of $\Pi_{k}$, $\text{dim}(\text{Ran}(\Pi_{k}))$, must be the same, see reference [12], chapter I, section 4.6. Moreover, $\text{dim}(\text{Ran}(P_{k}))$ must be finite which implies that the spectrum of $H$ inside ${\mathcal{C}}_{k}$ consists only of eigenvalues, see reference [12], chapter III, section 6.5. We conclude that the total algebraic multiplicity of the eigenvalues of $H$ inside $D_{k}$ must be the same as the initial geometric multiplicity of the cluster of eigenvalues of $\Delta$ in the interval $[\Lambda_{k}-\|Q_{0}\|,\Lambda_{k}+\|Q_{0}\|]$, namely the dimension of $E_{k}$.

Finally, for $1<p<\infty$, let us denote by ${\mathcal{S}}_{p}$ the space of all completely continuous operators $A$ for which $\sum_{j=1}^{\infty}s_{j}^{p}(A)<\infty$ where $s_{j}(A)$ are the eigenvalues of $|A|\equiv\sqrt{A^{*}A}$. The fact that the generalized eigenspaces of $\Delta+M_{V}$ span $L^{2}(X)$ is a consequence of theorem 10.1 in reference [5] noting that $\Delta$ has discrete spectrum and for all $\lambda$ in the resolvent set of $\Delta$, $(\Delta-\lambda)^{-1}V(\Delta-\lambda)^{-1}$ belongs to a Schatten class ${\mathcal{S}}_{p}$ for some $1<p<\infty$. This last fact is a consequence of (i) the property that ${\mathcal{S}}_{p}$ is an ideal in ${\mathcal{B}}(L^{2}(X))$, (ii) $V(\Delta-\lambda)^{-1}$ is a bounded operator and (iii) $(\Delta-\lambda)^{-1}$ belongs to ${\mathcal{S}}_{p}$ for some $1<p<\infty$ (the multiplicity of the eigenvalues of $\Delta$ in the interval $[\Lambda_{k}-C\,,\,\Lambda_{k}+C]$ is given by a polynomial in $k$ for $k$ sufficiently large, see reference [4]).

3. Proof of theorem 1.3

3.1. The averaging method

We begin by recalling the “averaging method” ([16, 8]):

Proposition 3.1**.**

If $V:X\to{\mathbb{C}}$ is smooth, there exists a pseudo-differential operator of order $(-1)$, $F$, such that

[TABLE]

where $B$ is a pseudo-differential operator of order zero such that $[A,B]=0$, and $S$ is a smoothing operator. Moreover, the symbol of $B$ is $\widetilde{V}+q_{0}$, regarded as a function on $T^{*}X^{n}\setminus\{0\}$.

Proof.

We’ll sketch the proof, for completeness. Let

[TABLE]

then $\widetilde{A}^{2}+Q_{0}=\Delta$, and $\widetilde{A}$ takes the value $\sqrt{\Lambda_{k}}$ on the space $E_{k}$. We proceed inductively, with $F_{1}$ an operator or order $(-1)$ solving the equation

[TABLE]

where $B_{0}$ is the operator

[TABLE]

It is easy to see that $B_{0}$ commutes with $\widetilde{A}$ and hence with $A$.

For $F_{1}$ we take an operator with principal symbol

[TABLE]

where $H_{0}(x,\xi)=\lVert\xi\rVert$ and $\phi_{s}$ is its Hamilton flow. Note that the principal symbol of $\widetilde{A}^{2}$ is the same as the principal symbol of $\Delta$, i.e. $H_{0}^{2}$. The symbol of $[F_{1},\widetilde{A}^{2}]$ is

[TABLE]

where $V^{\text{\tiny ave}}+q_{0}=\frac{1}{T}\int_{0}^{T}\phi_{t}^{*}(V)\,dt+q_{0}$ is the principal symbol of $B_{0}$. Note that the factors of $q_{0}$ cancel since $\phi_{t}^{*}(q_{0})=q_{0}$ (i.e. $q_{0}$ is already averaged over the flow). It follows that

[TABLE]

where $R_{-1}$ is an operator of order $(-1)$.

Using the same constructions, one can modify $F_{1}$ by an operator $G_{1}$ of order $(-2)$ so that, if $F_{2}=F_{1}+G_{1}$,

[TABLE]

where $R_{-2}$ is now an operator of order $(-2)$ and both $B_{0}$ and $B_{-1}$ commute with $A$. This process can be continued indefinitely, and after a Borel summation we can find an operator $F_{\infty}$, with the same principal symbol as $F_{1}$, such that

[TABLE]

where $[A,B]=0$ and $S$ is a smoothing operator, and where the principal symbol of $B$ is the same as that of $B_{0}$. ∎

3.2. Constructing the pseudo mode

We now proceed to construct a pseudo mode for $\Delta+M_{V}$, under the hypotheses of part (1) of Theorem 1.3. We will do this by first constructing a pseudo mode for $\widetilde{A}^{2}+B$, then showing that it is also a pseudo mode for $\tilde{A}^{2}+B+S$, and finally applying the averaging lemma.

If $S^{*}X$ denotes the unit cosphere bundle of $X$, one has a diagram

[TABLE]

and $\pi^{*}\widetilde{V}$ is the symbol of $B$ restricted to $S^{*}M$. Let us assume the hypotheses of part (1) of the theorem, taking without loss of generality $\sigma_{B}(m)=\mu=0$. Thus, if $m\in\gamma$

[TABLE]

Let ${\mathcal{R}}\subset T^{*}X\setminus\{0\}$ be the ray through $m$. ${\mathcal{R}}$ is a conic isotropic submanifold of $T^{*}X$. Associated to any such manifold are spaces of distributions $J^{k}(X,{\mathcal{R}})$, the Hermite distributions of Boutet de Monvel and Guillemin [3].

Lemma 3.2**.**

For every $\ell$ there exists $\psi\in J^{\ell}(X,{\mathcal{R}})$ (with non-zero principal symbol) such that $B(\psi)\in C^{\infty}(X)$.

Proof.

The construction is symbolic. The symbol of an element $\psi\in J^{\ell}(X,{\mathcal{R}})$ at $m\in{\mathcal{R}}$ is an object in the space

[TABLE]

where:

1. (1)

${\mathcal{R}}_{m}=T_{m}{\mathcal{R}}$ and $\wedge^{1/2}$ stands for half forms on it, 2. (2)

$\Sigma_{m}={\mathcal{R}}_{m}^{\circ}/{\mathcal{R}}_{m}$ is the symplectic normal space to ${\mathcal{R}}_{m}$ (here ${\mathcal{R}}_{m}^{\circ}$ is the symplectic orthogonal to ${\mathcal{R}}_{m}$ inside $T_{m}S^{*}X$). 3. (3)

${\mathcal{S}}(\Sigma_{m})$ is the space of smooth vectors for the representation of the Heisenberg group of $\Sigma_{m}$ in a quantization of $\Sigma_{m}$.

Given that ${\mathcal{R}}$ is one-dimensional and the symbol is homogeneous, the half form portion of the symbol is trivial: It is $dr^{\ell/2}$. The interesting part of the symbol is the vector in ${\mathcal{S}}(\Sigma_{m})$.

We first note that $\Sigma_{m}$ is naturally isomorphic with $T_{\gamma}{\mathcal{O}}$. The isomorphism arises from the fact that $\omega(\partial_{r},\Xi)=1$, where $\Xi$ is the Hamilton field of $\lVert\xi\rVert$ and $\partial_{r}$ is the radial field on $T^{*}X$. Therefore the intersection ${\mathcal{R}}_{m}^{\circ}\cap T_{m}(S^{*}X)$ is both naturally isomorphic to $\Sigma_{m}$ and to $T_{\gamma}{\mathcal{O}}$.

The argument is now identical to that in [2], §3. To find the symbol of $\psi$, we solve the equation $B(\psi)=0$ mod $C^{\infty}$ using the symbol calculus of Hermite distributions. For any $\psi\in J^{\ell}(X,{\mathcal{R}})$ the symbol of $B(\psi)$ as an element in $J^{\ell}(X,{\mathcal{R}})$ is zero, because $\sigma_{B}(m)=0$. Therefore $B(\psi)\in J^{\ell-1/2}(X,{\mathcal{R}})$. To find the symbol of $B(\psi)$ as an element of this space we need to use the first transport equation for Hermite distributions (Theorem 10.2 in [3]). The symbol of $B(\psi)$ is ${\mathcal{L}}(\sigma_{m})$, where ${\mathcal{L}}:{\mathcal{S}}(\Sigma_{m})\to{\mathcal{S}}(\Sigma_{m})$ is the operator in the infinitesimal representation of the Heisenberg group of $\Sigma_{m}$ associated to the Hamilton field of $\sigma_{B}$ at $m$ (which is tangent to ${\mathcal{R}}$). By Lemma 3.1 in [2], the condition (3.2) on the Poisson bracket implies that ${\mathcal{L}}$ is surjective with non-trivial kernel. The symbol of $\psi$ is chosen in the kernel of ${\mathcal{L}}$. If $\psi^{(1)}\in J^{\ell}(X,{\mathcal{R}})$ has such a symbol, then $B(\psi^{(1)})\in J^{\ell-1}(X,{\mathcal{R}})$. We then modify $\psi^{(1)}$ to $\psi^{(2)}=\psi^{(1)}+\chi^{(1)}$ where $\chi^{(1)}\in J^{\ell-1/2}(X,{\mathcal{R}})$ is chosen so that $B(\psi^{(2)})\in J^{\ell-3/2}(X,{\mathcal{R}})$. This will be the case if the symbol, $\tau$, of $\chi^{(1)}$ is such that ${\mathcal{L}}(\tau)$ cancels the symbol of $B(\psi^{(1)})$. Such a symbol $\tau$ exists because the operator ${\mathcal{L}}$ is surjective. Continuing in this fashion indefinitely (and doing a Borel-type summation) gives the result. ∎

Let $\psi$ be as in the lemma and $\psi=\sum_{k=1}^{\infty}\psi_{k}$ be its decomposition into sums of eigenfunctions corresponding to the eigenvalue clusters. Since $[\Delta,B]=0$, it follows that

[TABLE]

is the cluster decomposition of $B(\psi)$, and therefore

[TABLE]

since $B(\psi)$ is smooth. Since $\psi\in J^{\ell}(X,{\mathcal{R}})$, proceeding similarly as in [2], one can prove that the sequence of norms $\{\psi_{k}\}$ has an asymptotic expansion as $k\to\infty$ in decreasing powers of $k$ with non-trivial leading term. It follows that $\{\psi_{k}\}$ is a pseudo mode for $\Delta+B$ with pseudo-eigenvalues $\lambda_{k}=\Lambda_{k}$.

At this point, we should have a pseudo mode for $\widetilde{A}^{2}+B$ with $\psi_{k}$ an eigenfunction of $\widetilde{A}^{2}$ for each $k$, with eigenvalue precisely $\Lambda_{k}$.

Of course, the averaging lemma related $\Delta+M_{V}$ to $\widetilde{A}^{2}+B+S$, not to $\widetilde{A}^{2}+B$. However:

Lemma 3.3**.**

$\frac{||S\psi_{k}||}{||\psi_{k}||}=\mathcal{O}(k^{-\infty})$.

Proof.

For any $N\in\mathbb{N}$, the operator $S\widetilde{A}^{N}$ is smoothing (since $S$ is) and hence is bounded on $L^{2}$ by some constant $C_{N}$. Then $||S\widetilde{A}^{N}\psi_{k}||\leq C_{N}||\psi_{k}||$. On the other hand, $||S\widetilde{A}^{N}\psi_{k}||=||S\Lambda_{k}^{N/2}\psi_{k}||=\Lambda_{k}^{N/2}||S\psi_{k}||$. We conclude that $||S\psi_{k}||\leq C_{N}\Lambda_{k}^{-N/2}||\psi_{k}||$; however, $\Lambda_{k}$ is a quadratic polynomial in $k$ with positive leading coefficient. Since $N$ is arbitrary, this completes the proof. ∎

Corollary 3.4**.**

The sequence $\{\psi_{k}\}$ is also a pseudo mode for $\widetilde{A}^{2}+B+S$.

Proof.

We may as well assume $\psi_{k}$ are $L^{2}$-normalized; the proof then follows immediately from the previous lemma. ∎

We now finish the proof of the first part of Theorem 1.3. Let us define $\varphi_{k}:=e^{-F}\psi_{k}$ where $\{\psi_{k}\}$ is a pseudo mode as in the previous corollary such that, without loss of generality, $\lVert\psi_{k}\rVert\equiv 1$. Then

[TABLE]

since $e^{-F}$ is a bounded operator and and $\{\psi_{k}\}$ is a pseudomode for $\tilde{A}^{2}+B+S$, this is $\mathcal{O}(k^{-\infty})$. On the other hand, since $F$ is of order $(-1)$,

[TABLE]

where $T$ is an operator of order $(-1)$. Therefore

[TABLE]

Since the second term on the right-hand side tends to zero as $k\to\infty$, $\{\varphi_{k}\}$ is a pseudo mode for $\Delta+M_{V}$ with pseudo-eigenvalues $\lambda_{k}=\Lambda_{k}$. (Recall that we assumed without loss of generality that $\mu=0$.) This proves the first part of Theorem 1.3.

3.3. Proof of part 2

Let us assume that $\{\Lambda_{k}+\mu\}$, where $\mu=(\widetilde{V}+q_{0})(\gamma)$, is in the asymptotic pseudospectrum, By definition, there exists a pseudo-mode $\{\psi_{k}\in E_{k}\}$, a sequence that satisfies $\lVert\psi_{k}\rVert=1$ for all $k$ and

[TABLE]

Let $\psi$ be the distribution $\psi=\sum_{k=1}^{\infty}\psi_{k}$. This is clearly non-smooth, while

[TABLE]

by the Sobolev embedding theorem.

Assume now that

[TABLE]

By Theorem 27.1.11 of [11], the operator $M_{V}^{\text{\tiny ave}}+Q_{0}-\mu I$ is microlocally subelliptic on the cone over the geodesic $\gamma\subset S^{*}X$, with loss of $1/2$ derivatives. Given (3.3), it follows that the wave-front set of $\psi$ must be disjoint from this cone.

4. Proof of Theorem 1.4

Recall our notation: $X$ is a Zoll manifold, and let $A$ be the operator appearing in (1.5). $A$ is a first-order pseudo-differential operator with symbol $\sigma_{A}(x,\xi)=\lVert\xi\rVert_{x}$ and spectrum the eigenvalues $k=0,1,\ldots$. Let $L^{2}(X)=\oplus_{k=0}^{\infty}E_{k}$ be the decomposition of $L^{2}(X)$ into eigenspaces of $A$.

Referring to (1.5) note first that, for each $k$,

[TABLE]

that is, ${\mathcal{R}}_{k}$ is the numerical range of the operator $Q_{0}+M_{V}+R_{-1}$ restricted to the finite-dimensional subspace $E_{k}$. It is clear that ${\mathcal{R}}_{k}$ is closed for each $k$ and, by the Toeplitz-Hausdorff theorem, ${\mathcal{R}}_{k}$ is convex. An elementary argument shows that the limit ${\mathcal{R}}_{\infty}$ is also closed and convex.

4.1. Existence of modes with microsupport on geodesics

The purpose of this section is to prove:

Proposition 4.1**.**

Let $X$ be either a standard $S^{n}$ or a Zoll surface, and let $\gamma\subset S^{*}X$ be a geodesic. Then there exist sequences $\{u_{k}\}$ of functions such that:

1. (1)

$\forall k\ u_{k}\in E_{k}$* and $\lVert u_{k}\rVert=1$,* 2. (2)

the semi-classical wave front set of the sequence is equal to $\gamma$, and 3. (3)

For all pseudo-differential operators $Q$ of order zero on $X$ one has

[TABLE]

where $\sigma_{Q}$ is the principal symbol of $Q$.

Proof.

First suppose $X$ is a sphere. Since the construction is SO$(n+1)$ equivariant, without loss of generality $\gamma$ corresponds to the intersection $\gamma_{0}$ of $S^{n}\subset{\mathbb{R}}^{n+1}$ with the $x_{1}x_{2}$ plane, where $(x_{1},\ldots,x_{n+1})$ are the standard coordinates in ${\mathbb{R}}^{n+1}$. Then one can take

[TABLE]

where $a_{k}\in{\mathbb{R}}$ is chosen so that the $L^{2}$ norm of $u_{k}$ is equal to one. It is known that these functions have the required properties (see Proposition 7.6 in [7]).

Now assume that $X$ is a Zoll surface, which we normalize so that its geodesics have length $2\pi$. Then it is known that there exists an invertible Fourier integral operator $U:L^{2}(X)\to L^{2}(S^{2})$ such that

[TABLE]

where $\Delta_{0}$ is the standard Laplacian on $S^{2}$ and $R$ is a pseudo-differential operator of order zero. (See [15].) The FIO $U$ is associated to a homogeneous canonical transformation $T:T^{*}S^{2}\setminus\{0\}\to T^{*}X\setminus\{0\}$ that intertwines the geodesic flows. Applying once again the averaging method (more precisely Lemma 1 in [8]) to $\Delta_{0}+R$, one can assume without loss of generality that $[\Delta_{0},R]=0$. (The result cited says that any operator of the form $\Delta_{0}+R$ with $R$ self-adjoint pseudo-differential of order zero can be conjugated, this time by a unitary pseudo-differential operator, to an operator of the same form but now such that $[\Delta_{0},R]=0$.) For each $k$, the operator $U$ maps the space of spherical harmonics of degree $k$ onto $E_{k}$.

Pre-composing $U$ with a rotation, we can also assume that $T$ maps the geodesic $\gamma_{0}$ to $\gamma$. We then define $u_{k}$ to be the result of applying the operator $U$ to the right-hand side of (4.2). Since the desired properties are equivariant with respect to actions of unitary FIOs, we are done.

∎

This existence result immediately implies:

Corollary 4.2**.**

The image of $\widetilde{V}+q_{0}$, and therefore its convex hull, are contained in ${\mathcal{R}}_{\infty}$.

Proof.

Given $\gamma\in{\mathcal{O}}$, let $\{u_{k}\}$ be as in the previous Proposition. By (4.1), the limit of the matrix coefficients

[TABLE]

as $k\to\infty$ is precisely $(q_{0}+\widetilde{V})(\gamma)$. ∎

4.2. The converse

We now prove that ${\mathcal{R}}_{\infty}$ is contained in the convex hull of the image of $\widetilde{V}+q_{0}$. (The proof is modeled on the proof of [2, Prop. 5.2].) To show this, we will show that for all lines in $\mathbb{C}$ such that one of the half planes cut out by the line contains the range of $\widetilde{V}+q_{0}$, every element of ${\mathcal{R}}_{\infty}$ is contained in the same half plane. This will show that ${\mathcal{R}}_{\infty}$ is contained in, hence equal to, the convex hull in question.

We will use the following Lemma (closely related to the sharp Gårding inequality):

Lemma 4.3**.**

Let $Q$ be a zeroth order, self-adjoint pseudo-differential operator on $S^{n}$. Assume that its principal symbol $q\in C^{\infty}(T^{*}S^{n}\setminus\{0\})$ is non-negative: $q\geq 0$. Let $\{\psi_{k}\in E_{k}\}$ be a sequence of spherical harmonics such that $\lVert\psi_{k}\rVert=1$ for all $k$. Assume furthermore that

[TABLE]

Then $\ell\geq 0$.

Proof.

By the sharp Gårding inequality (see [10], Exercise 4.9), there exists a pseudo-differential operator $B$ of order $-1$ such that $\forall u\in L^{2}(S^{n})$

[TABLE]

As before, let $A$ be the operator which is equal to multiplication by $k$ when restricted to $E_{k}$. Recall that $A$ is first-order pseudo-differential, so that $BA$ is of order zero and therefore bounded in $L^{2}$. Therefore $\exists C>0$ such that, with $\psi_{k}$ as in the hypotheses of the Lemma, $\forall k$

[TABLE]

Therefore $\forall k$

[TABLE]

Taking limits as $k\to\infty$ yields the desired result. ∎

To finish the proof of the converse, consider a line

[TABLE]

in $\mathbb{C}$, and assume that the range of $\widetilde{V}+q_{0}$ is contained in the region $au+bv\geq c$. Let $u_{0}+iv_{0}\in{\mathcal{R}}_{\infty}$. We need to show that $au_{0}+bv_{0}\geq c$. By definition of ${\mathcal{R}}_{\infty}$, there exists a sequence $\left\{\psi_{k}\in E_{k}\right\}$ such that $\forall k\ \lVert\psi_{k}\rVert=1$ and such that the sequence of complex numbers

[TABLE]

converges to $u_{0}+iv_{0}$ as $k$ goes to infinity. Consider now the operator

[TABLE]

This is a self-adjoint pseudo-differential operator of order zero with non-negative symbol, and

[TABLE]

By the Lemma this limit is non-negative, and the proof is complete.

5. Examples on $S^{2}$

5.1. Preliminaries on the averaging operator

We present here preliminary results on the operator $V\mapsto\widetilde{V}$ on the two sphere $S^{2}\subset{\mathbb{R}}^{3}$ (see also the appendix in [6]). The space, ${\mathcal{O}}$, of oriented geodesics on $S^{2}$ can be identified with a copy of $S^{2}$. Indeed an oriented great circle on $S^{2}$ is the intersection of $S^{2}$ with an oriented plane through the origin:

[TABLE]

The identification ${\mathcal{O}}\cong S^{2}$ is via the map $\gamma\mapsto$ the unit normal vector to $\pi_{\gamma}$ defining the orientation. Clearly this identification is equivariant with respect to the action of SO$(3)$. We can therefore regard the averaging operator as an operator on $C^{\infty}(S^{2})$:

[TABLE]

where $\quad\widetilde{V}(\gamma)=\frac{1}{2\pi}\int_{\gamma}V\,ds$. This operator is obviously SO$(3)$ equivariant, and therefore it maps each space of spherical harmonics, $E_{k}$, into itself. Furthermore, by Schur’s lemma, it is a constant $c_{k}$ times the identity on $E_{k}$.

To proceed further, let $(x,y,z)$ denote the ambient (${\mathbb{R}}^{3}$) coordinates and let $\zeta=x+iy$. Note that, for each $k$,

[TABLE]

(we will abuse the notation and denote the restrictions of $x,y,z$ to the sphere by the same letters). Indeed $(x+iy)^{k}$ is a homogeneous polynomial of degree $k$ and it is harmonic, by the Cauchy-Riemann equations. Following the identification ${\mathcal{O}}\cong S^{2}$ one finds that

[TABLE]

Using Wallis’ formula for $\pi$ it follows that $c_{2l}\sim(-1)^{l}(l\pi)^{-1/2}$ .

5.2. Analytic functions

Our first example is actually a class of examples, namely potentials of the form

[TABLE]

where the series is assumed to have a radius of convergence $>1$. In §5 of [9] it was shown that the eigenvalues of $\Delta+M_{V}$ are exactly the same as those of $\Delta$ (note that there is no constant term in the series), the intuition being that the average of the operator $M_{V}$, restricted to ${E_{k}}$, is nilpotent for each $k$.

To see what Theorem 1.3 says about the asymptotic pseudospectrum in this example, we need to investigate the Poisson bracket condition on $\widetilde{V}$. Note that, by the previous discussion,

[TABLE]

which shows that $\widetilde{V}$ is of the same form as $V$.

Lemma 5.1**.**

Let $\widetilde{V}=F+iG$ with $F$ and $G$ real-valued. Then

[TABLE]

Proof.

Denote by $(\theta,z)$ toric coordinates on ${\mathcal{O}}\cong S^{2}$, so that the symplectic form is $dz\wedge d\theta$. Then

[TABLE]

[TABLE]

Using the Cauchy-Riemann equations and simplifying one obtains

[TABLE]

[TABLE]

But $\{x\,,\,y\}=z$. ∎

As the pre-image under $\widetilde{V}$ of any complex number contains points whose $z$ coordinates differ by a sign, we get:

Corollary 5.2**.**

For each $\displaystyle{\mu\in\{\widetilde{f}(\zeta)\;;\;|\zeta|<1\ \text{and}\ \widetilde{f}^{\prime}(\zeta)\not=0\}}$ the sequence $\{\Lambda_{k}+\mu\}$ is in the asymptotic pseudospectrum of $\Delta+M_{V}$.

5.3. Quadratic examples

In this section we take $V$ of the form $V=(ax+iy)^{2}$, where $a$ is a real constant not equal to either zero or one. If $\widetilde{\Delta}$ denotes the (negative) Laplacian on ${\mathbb{R}}^{3}$, then $\widetilde{\Delta}V=2(a^{2}-1).$ It follows that

[TABLE]

is a harmonic, homogeneous polynomial on ${\mathbb{R}}^{3}$. Therefore, the decomposition of $V$ into spherical harmonics is

[TABLE]

This allows us to compute $\widetilde{V}$, which is, up to constants, basically $V$ itself:

[TABLE]

Therefore

[TABLE]

[TABLE]

Corollary 5.3**.**

For each $\displaystyle{\mu\in\{\widetilde{V}(\zeta)\;;\;0<|\zeta|<1\}}$ the sequence $\{\Lambda_{k}+\mu\}$ is in the asymptotic pseudospectrum of $\Delta+M_{(ax+iy)^{2}}$.