Improvement of Pointwise Bounds for Eigenfunctions in the Quantum Completely Integrable System

Abstract

On a compact n-dimensional Riemannian manifold without boundary $(M,g)$, it is well-known that the $L^2$-normalized Laplace eigenfunctions with semiclassical parameter h satisfy the universal $L^{\infty }$ growth bound of $O(h^{{\displaystyle \frac{1-n}{2}}})\mathrm{as}h\to 0^{+}.$ In the context of a quantum completely integrable system on M, which consists of n commuting self-adjoint pseudodifferential operators $P_1\left(h\right),\dots ,P_n\left(h\right)$, where $P_1\left(h\right)=-h^2{\mathrm{\Delta }}_g+V\left(x\right)$, Galkowski-Toth showed polynomial improvements over the standard $O(h^{{\displaystyle \frac{1-n}{2}}})$ bounds for typical points. Specifically, in the two-dimensional case, such an improved upper bound is $O(h^{-1/4})$. In this study, we aim to further enhance this bound to $O\left(\right|lnh{|}^{1/2})$ at the points where a strictly monotonic condition is satisfied.

1. Introduction

Let $(M,g)$ be a smooth compact n-dimensional Riemannian manifold without a boundary, and let $u_h$ be an $L^2$-normalized eigenfunction solving

$$-h^2{\mathrm{\Delta }}_g{u}_h=u_h\left(x\right),x\in M,$$

(1)

where $h>0$ is the small semiclassical parameter and ${\mathrm{\Delta }}_g$ is the Laplace–Beltrami operator associated with the metric g.

Beginning in the 1950’s, the works of Levitan [1], Avakumović [2], and Hörmander [3] proved the estimate

$$\parallel u_h{\parallel }_{L^{\infty }}=O(h^{{\displaystyle \frac{1-n}{2}}}),$$

(2)

as $h\to 0$ and (2) is saturated on the round sphere. This bound was improved to $o(h^{{\displaystyle \frac{1-n}{2}}})$ by Safarov, Sogge, Toth, Zelditch, and Galkowski [4,5,6,7,8,9,10] under various dynamical assumptions at x. When $(M,g)$ has no conjugate points, a quantitative improvement

$$\parallel u_h{\parallel }_{L^{\infty }}=O(h^{{\displaystyle \frac{1-n}{2}}}/\sqrt{|logh|}),$$

(3)

as $h\to 0$ has been known since the classical work of Bérard [11,12,13]. In recent times, Canzani and Galkowski [14,15] developed the tool of geodesic beams to study the quantitative improvements without global geometric assumptions on $(M,g)$.

Now let us turn our attention to the joint eigenfunctions of quantum completely integrable systems. We begin by considering a self-adjoint semiclassical pseudodifferential operator $P_1\left(h\right):C^{\infty }\left(M\right)\to C^{\infty }\left(M\right)$ of order m in the classical sense. This means that the symbol $p_1(x,\xi )$ of $P_1\left(h\right)$ satisfies the inequality $|p_1{(x,\xi )|\ge c|\xi |}^m-C$. We say that $P_1\left(h\right)$ is quantum completely integrable (QCI) if there exist functionally independent h-pseudodifferential operators $P_2\left(h\right),\dots ,P_n\left(h\right)$ with properties that

$$[P_i\left(h\right),P_j\left(h\right)]=0;i,j=1,\dots ,n.$$

Let $E\left(h\right)=(E_1\left(h\right),\dots ,E_n\left(h\right))\in {\mathbb{R}}^n$ denote the joint eigenvalues of the operators $P_1\left(h\right),\dots ,P_n\left(h\right)$. We denote the $L^2$-normalized joint eigenfunction with joint eigenvalue $E\left(h\right)$ by $u_{E,h}$. When the joint energy value E is clear from the context, we will simplify the notation and write $u_h$ instead of $u_{E,h}$.

Galkowski-Toth [16] investigated the pointwise bounds of $u_h$ for the Schrödinger operator $P_1\left(h\right)$. More explicitly, in the two-dimensional case and when $E_1$ is a regular value of $p_1$ and the system is of the Morse type at point x, they established the following bound:

$$|u_h\left(x\right)|=O\left(h^{-1/4}\right).$$

(4)

In this study, we aim to show an improvement of the pointwise bounds (Equation (4)) under a strictly monotonic condition. In fact, it can be observed that the highest-weight spherical harmonics at the equator saturate the upper bound (Equation (4)) (refer to the examples in Section 3). Therefore, additional assumptions regarding the point are necessary to improve the bound (Equation (4)).

We say that $p_1$ is of the real principal type on the hypersurface $p_1^{-1}\left(E_1\right)$ if $E_1>0$ is a regular value of $p_1$; for any $(x,\xi )\in p_1^{-1}\left(E_1\right)$, the following inequality holds:

$${\partial }_{\xi }{p}_1(x,\xi )\ne 0.$$

(5)

One sets

$$C_x:=\{\xi \in T_x^{*}M;p_1(x,\xi )=E_1\}.$$

(6)

Now we can explain the strictly monotonic condition.

Definition 1.

We say that the ${\mathbb{R}}^2$-integrable system with moment map $\mathcal{P}=(p_1,p_2)$ is strictly monotonic at $x\in M$ if following condition holds:

$${p_2|}_{C_x\cap \left\{\right|p_2(x,\xi )-E_2|<\epsilon \}}is strictly mnotonic$$

(7)

for some sufficiently small positive constant ε.

Then one can state the main theorem of this note,

Theorem 1.

Let $\left\{u_h\right\}$ be the $L^2$-normalized joint eigenfunctions of commuting operators $P_1\left(h\right)=-h^2{\mathrm{\Delta }}_g+V$, where $V\in C^{\infty }(M,\mathbb{R})$ and $P_2\left(h\right)$ on a compact smooth Riemannian surface $(M,g)$ with $E_1\left(h\right)=E_1+O\left(h\right)$ and joint eigenvalues $\left(E_1\left(h\right),E_2\left(h\right)\right)\in \mathit{Spec}{P}_1\left(h\right)\times \mathit{Spec}{P}_2\left(h\right)$. Also let $p_1$ be a real principle type on the hypersurface $p_1^{-1}\left(E_1\right)$. Suppose that the QCI system is strictly monotonic at $x\in M$. Then for $h\in (0,h_0]$ with a sufficiently small positive constant $h_0$, one has the following upper bound:

$$|u_h{\left(x\right)|=O(|lnh|}^{{\displaystyle \frac{1}{2}}}).$$

(8)

Remark 1.

The estimate (Equation (8)) in Theorem 1 gives an explicit polynomial improvement over the bound (Equation (4)), and the above estimate is uniform over all energy values $\{E_2\in \mathbb{R};(E_1,E_2)\in \mathcal{P}\left(T^{*}M\right)\}$. From the examples constructed in Section 3, one can see that the strictly monotonic assumption is crucial.

2. Proof of Theorem 1

Let us take a real-valued function $\chi \in \mathcal{S}\left(\mathbb{R}\right)$ satisfying

$$\chi \left(0\right)=1\mathrm{and}\widehat{\chi }\left(t\right)=0,\left|t\right|\ge \epsilon ,$$

(9)

where $\epsilon$ is a small positive constant. Since $\chi \left(h^{-1}[P_1\left(h\right)-E_1\left(h\right)]\right)u_h=u_h$, in order to prove (8), it suffices to show that

$$\left(\chi \left(h^{-1}[P_1\left(h\right)-E_1\left(h\right)]\right)u_h\right)\left(x\right)\le C{|lnh|}^{{\displaystyle \frac{1}{2}}}.$$

(10)

The joint spectrum of $P_1\left(h\right)$ (resp. $P_2\left(h\right)$) will be denoted by ${\lambda }_j^{\left(1\right)}\left(h\right)$ (resp. ${\lambda }_j^{\left(2\right)}\left(h\right)$) with $j=1,2,3,\dots$. The kernel of the operator $\chi \left(h^{-1}[P_1\left(h\right)-E_1\left(h\right)]\right)$ is given by

$$\chi \left(h^{-1}[P_1\left(h\right)-E_1\left(h\right)]\right)(x,y)=\sum _j\chi \left(h^{-1}[{\lambda }_j^{\left(1\right)}\left(h\right)-E_1\left(h\right)]\right)u_j^h\left(x\right)\overline{u_j^h\left(y\right)},$$

where $u_j^h,j=1,2,3,\dots$ are the corresponding $L^2$-normalized joint eigenfunctions.

By the Cauchy-Schwarz inequality and using the orthogonality of $\left\{u_j^h\right\}$, one can show that

$$\begin{array}{cc}\hfill & |\left(\chi \left(h^{-1}[P_1\left(h\right)-E_1\left(h\right)]\right)u_h\right)\left(x\right){|}^2\hfill \\ \hfill =& |\int \chi \left(h^{-1}[P_1\left(h\right)-E_1\left(h\right)]\right)(x,y)u_h\left(y\right)dy{|}^2\hfill \\ \hfill \le & \int |\chi \left(h^{-1}[P_1\left(h\right)-E_1\left(h\right)]\right)(x,y){|}^{2}dy{\parallel u_h\left(y\right)\parallel }_{L^2\left(M\right)}^2\hfill \\ \hfill =& \int \sum _j\chi \left(h^{-1}[{\lambda }_j^{\left(1\right)}\left(h\right)-E_1\left(h\right)]\right)u_j^h\left(x\right)\overline{u_j^h\left(y\right)}\sum _k\chi \left(h^{-1}[{\lambda }_k^{\left(1\right)}\left(h\right)-E_1\left(h\right)]\right)\overline{u_k^h\left(x\right)}{u}_j^h\left(y\right)dy\hfill \\ \hfill =& \sum _j\rho \left(h^{-1}[{\lambda }_j^{\left(1\right)}\left(h\right)-E_1\left(h\right)]\right)u_j^h\left(x\right)\overline{u_j^h\left(x\right)},\hfill \end{array}$$

(11)

with the setting $\rho \left(t\right)={\left(\chi \left(t\right)\right)}^2$.

Hence proving (10) is equivalent to showing that

$$\left|\sum _j\rho \left(h^{-1}[{\lambda }_j^{\left(1\right)}\left(h\right)-E_1\left(h\right)]\right)u_j^h\left(x\right)\overline{u_j^h\left(x\right)}\right|\le C|lnh|.$$

(12)

We claim that we need to show that

$$\begin{array}{c}\hfill \underset{\{E_2;(E_1,E_2)\in \mathcal{P}\left(T^{*}M\right)\}}{sup}\sum _j\rho \left(h^{-1}[{\lambda }_j^{\left(1\right)}\left(h\right)-E_1]\right)\rho \left(h^{-1}[{\lambda }_j^{\left(2\right)}\left(h\right)-E_2]\right)u_j^h\left(x\right)\overline{u_j^h\left(x\right)}\le C|lnh|.\end{array}$$

(13)

Indeed from [17] (Section 2), one knows that there exists a constant $C_2>0$ (independent of j and h) such that for any $h\in (0,h_0]$ and $({\lambda }_j^{\left(1\right)}\left(h\right),{\lambda }_j^{\left(2\right)}\left(h\right))\in \mathrm{Spec}(P_1\left(h\right),P_2\left(h\right))$, with $|{\lambda }_j^{\left(1\right)}\left(h\right)-E_1|\le C_{1}h$,

$$\underset{E_2}{inf}|{\lambda }_j^{\left(2\right)}\left(h\right)-E_2|\le C_{2}h.$$

So, once $\epsilon$ in (9) is sufficiently small, there exists a constant $C_3>0$ (independent of j and h) such that for all $j\ge 1$ and $h\in (0,h_0]$,

$$\underset{\{E_2;(E_1,E_2)\in \mathcal{P}\left(T^{*}M\right),|{\lambda }_j^{\left(1\right)}\left(h\right)-E_1|\le C_{1}h\}}{sup}\rho \left(h^{-1}({\lambda }_j^{\left(2\right)}\left(h\right)-E_2)\right)\ge C_3>0.$$

Since the sum on the left-hand side of (13) has non-negative terms, by restricting to $\{j;|{\lambda }_j^{\left(1\right)}\left(h\right)-E_1|\le C_1\}$ and (after taking $\epsilon$ once it is small enough) using $\rho \left(h^{-1}[{\lambda }_j^{\left(1\right)}\left(h\right)-E_1]\right)\ge C_4>0$ for these eigenvalues, one obtains the following:

$$\sum _{\{j;|{\lambda }_j^{\left(1\right)}\left(h\right)-E_1|=O\left(h\right)\}}{u}_j^h\left(x\right)\overline{u_j^h\left(x\right)}=O\left(\right|lnh\left|\right),$$

which can deduce (12).

Now we are going to prove (13). First note that

$$\begin{array}{cc}\hfill & \sum _j\rho \left(h^{-1}[{\lambda }_j^{\left(1\right)}\left(h\right)-E_1]\right)\rho \left(h^{-1}[{\lambda }_j^{\left(2\right)}\left(h\right)-E_2]\right)u_j^h\left(x\right)\overline{u_j^h\left(x\right)}\hfill \\ \hfill =& \left(\rho \left(h^{-1}[P_2\left(h\right)-E_2]\right)\circ \rho \left(h^{-1}[P_1\left(h\right)-E_1]\right)\right)(x,x).\hfill \end{array}$$

(14)

Hence in order to prove (13), one needs to show that

$$\left|\left(\rho \left(h^{-1}[P_2\left(h\right)-E_2]\right)\circ \rho \left(h^{-1}[P_1\left(h\right)-E_1]\right)\right)(x,x)\right|\le C|lnh|.$$

(15)

Next, we are going to write out the kernel of the composition $\rho \left(h^{-1}[P_2\left(h\right)-E_2]\right)\circ \rho \left(h^{-1}[P_1\left(h\right)-E_1]\right)$ explicitly. The explicit form of this kernel will serve as a fundamental building block for subsequent calculations and estimations.

Note that

$$\rho \left(h^{-1}[P_1\left(h\right)-E_1]\right)={\int }_{\mathbb{R}}\widehat{\rho }\left(t\right)e^{{\displaystyle \frac{i}{h}}t(P_1\left(h\right)-E_1)}dt,$$

and the Schwartz kernel of $e^{{\displaystyle \frac{i}{h}}t(P_1\left(h\right)-E_1)}$ is of the form

$${\left(2\pi h\right)}^{-2}{\int }_{{\mathbb{R}}^2}{e}^{{\displaystyle \frac{i}{h}}[{\phi }_1(t,y,\xi )-〈x,\xi 〉-t{E}_1]}{a}_1(t,x,y,\xi ;h)d\xi +O_{C^{\infty }}\left(h^{\infty }\right),$$

where $a_1\sim {\sum }_{j=0}^{\infty }{a}_{1,j}{h}^j$, $a_{1,j}\in C^{\infty }$, $a_{1,0}\ge C>0$, and ${\phi }_1(t,y,\xi )$ solves the eikonal equation

$${\partial }_t{\phi }_1(t,y,\xi )=p_1(y,{\partial }_y{\phi }_1),{\phi }_1(0,y,\xi )=〈y,\xi 〉.$$

(16)

Also

$$\rho \left(h^{-1}[P_2\left(h\right)-E_2]\right)={\int }_{\mathbb{R}}\widehat{\rho }\left(s\right)e^{{\displaystyle \frac{i}{h}}s(P_2\left(h\right)-E_2)}ds.$$

Here $e^{{\displaystyle \frac{i}{h}}s(P_2\left(h\right)-E_2)}$ has a Schwartz kernel with the form of

$${\left(2\pi h\right)}^{-2}{\int }_{{\mathbb{R}}^2}{e}^{{\displaystyle \frac{i}{h}}[{\phi }_2(s,x,\eta )-〈y,\eta 〉-s{E}_2]}{a}_2(s,x,y,\eta ;h)d\xi +O_{C^{\infty }}\left(h^{\infty }\right),$$

where $a_2\sim {\sum }_{j=0}^{\infty }{a}_{2,j}{h}^j$, $a_{2,j}\in C^{\infty }$, $a_{2,0}\ge C>0$, and ${\phi }_2(s,x,\eta )$ solves the eikonal equation

$${\partial }_s{\phi }_2(s,x,\eta )=p_2(x,{\partial }_x{\phi }_2),{\phi }_2(0,x,\eta )=〈x,\eta 〉.$$

(17)

From (16) and (17), one can derive the following Taylor expansions for ${\phi }_1(t,y,\xi )$ (resp. ${\phi }_2(s,x,\eta )$) centered at $t=0$ (resp. $s=0$).

$$\begin{array}{cc}\hfill {\phi }_1(t,y,\xi )& =〈y,\xi 〉+t{p}_1(y,\xi )+O\left(t^2\right)\hfill \\ \hfill {\phi }_2(s,x,\eta )& =〈x,\eta 〉+s{p}_2(x,\eta )+O\left(s^2\right).\hfill \end{array}$$

In conclusion, $\left(\rho \left(h^{-1}[P_2\left(h\right)-E_2]\right)\circ \rho \left(h^{-1}[P_1\left(h\right)-E_1]\right)\right)(x,x)$ is equal to

$$h^{-4}\int \widehat{\rho }\left(s\right)\widehat{\rho }\left(t\right)exp\left[{\displaystyle \frac{i}{h}}\mathsf{\Phi }(s,t,x,y,\xi ,\eta )\right]c(s,t,x,y,\xi ,\eta ;h)dyd\xi dtd\eta ds,$$

(18)

where, $c\sim {\sum }_{j=0}^{\infty }{c}_j{h}^j$, $c_j\in C^{\infty }$, $c_0\ge C>0$ and

$$\begin{array}{cc}\hfill & \mathsf{\Phi }(s,t,x,y,\xi ,\eta )\hfill \\ \hfill =& 〈y-x,\xi -\eta 〉+t\left(p_1(y,\xi )-E_1\right)+s(p_2(x,\eta )-E_2)+O_{y,\xi }\left(t^2\right)+O_{x,\eta }\left(s^2\right).\hfill \end{array}$$

Now we set

$$c^{\prime }(s,t,x,y,\xi ,\eta ;h)=\chi \left(p_1(y,\xi )-E_1\right)\chi \left(p_2(x,\eta )-E_2\right)c(s,t,x,y,\xi ,\eta ;h).$$

Hence (18) is equal to

$$h^{-4}\int \widehat{\rho }\left(s\right)\widehat{\rho }\left(t\right)exp\left[{\displaystyle \frac{i}{h}}\mathsf{\Phi }(s,t,x,y,\xi ,\eta )\right]c^{\prime }(s,t,x,y,\xi ,\eta ;h)dyd\xi dtd\eta ds+O\left(h^{\infty }\right).$$

(19)

One can apply a stationary phase to the $(y,\xi )$ variables in (19). The critical point equations for $(y,\xi )$ are

$$\begin{array}{cc}\hfill \xi & =\eta -t{\partial }_y{p}_1(y,\xi )+O\left(t^2\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill y& =x-t{\partial }_{\xi }{p}_1(y,\xi )+O\left(t^2\right).\hfill \end{array}$$

(21)

By applying the stationary phase at the critical point $(y_c,{\xi }_c)$ and Taylor expansion, one can see that (19) equals

$$\begin{array}{cc}\hfill h^{-2}\int \widehat{\rho }\left(s\right)\widehat{\rho }\left(t\right)& exp[{\displaystyle \frac{i}{h}}\left(t(p_1(x,\eta )-E_1\right)+s\left(p_2(x,\eta )-E_2\right)\hfill \\ \hfill & +O_{x,\eta }\left(t^2\right)+O_{x,\eta }\left(s^2\right)\left)\right]c_1(s,t,x,\eta ;h)d\eta dtds=:I(x;h),\hfill \end{array}$$

here $c_1\sim {\sum }_{j=0}^{\infty }{c}_{1,j}{h}^j$ and $c_{1,j}\in C^{\infty }$.

In conclusion, we need to show that

$$\left|I\right(x;h\left)\right|\le C|lnh|.$$

(22)

2.1. Laplacian Case

If $P_1\left(h\right)=-h^2{\mathrm{\Delta }}_g$ with $E_1=1$, we use the geodesic normal coordinate about x and can make the change in variables $\eta =r\mathrm{\Theta }$, where $r=\left|\eta \right|$ and $\mathrm{\Theta }\in S^1$. One can obtain

$$\begin{array}{cc}\hfill I(x;h)& =h^{-2}\int \int \int \int \widehat{\rho }\left(s\right)\widehat{\rho }\left(t\right)exp[{\displaystyle \frac{i}{h}}\left(t(r^2-1\right)+s\left(p_2(x,r\mathrm{\Theta })-E_2\right)\hfill \\ \hfill & +O_{x,\eta }\left(t^2\right)+O_{x,\eta }\left(s^2\right)\left)\right]c_1(s,t,x,r\mathrm{\Theta };h)rdrd\mathrm{\Theta }dtds.\hfill \end{array}$$

(23)

The critical point $(r_c,t_c)$ satisfies

$$\begin{array}{cc}\hfill 2rt+s\mathrm{\Theta }·{\partial }_{\eta }{p}_2(x,r\mathrm{\Theta })+O_{x,\eta }\left(t^2\right)+O_{x,\eta }\left(s^2\right)& =0,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill (r^2-1)+O_{x,\eta }\left(t\right)& =0.\hfill \end{array}$$

(25)

Combining (24) and (25), we can obtain the following:

$$t_c=O\left(s\right).$$

(26)

Taking the derivative of $\theta$ in (24) and (25) and combining (25), one can derive the following:

$${\displaystyle \frac{\partial t_c}{\partial \theta }}=O\left(s\right),$$

(27)

while noticing that $\mathrm{\Theta }=(\cos \theta ,\sin \theta )$.

Performing the stationary phase at the critical point $\left(r_c(s,x,\theta ),t_c(s,x,\theta )\right)$ in (23) gives

$$\begin{array}{cc}\hfill I(x;h)=& h^{-1}\int \int \widehat{\rho }\left(s\right)exp\left[{\displaystyle \frac{i}{h}}\mathsf{\Phi }(s,x,\theta )\right]·c_2(s,x,\theta ;h)d\theta ds,\hfill \end{array}$$

(28)

where $c_2\sim {\sum }_{j=0}^{\infty }{c}_{2,j}{h}^j$, $c_{2,j}\in C^{\infty }$, and

$$\begin{array}{cc}\hfill & \mathsf{\Phi }(s,x,\mathrm{\Theta })\hfill \\ \hfill =& s\left(p_2(x,r_c\mathrm{\Theta })-E_2\right)+O\left(t_c^2\right)+O\left(s^2\right)\hfill \\ \hfill =& s\left(p_2(x,\mathrm{\Theta })-E_2\right)+O\left(s{t}_c\right)+O\left(s^2\right)+O\left(t_c^2\right)\hfill \end{array}$$

with the help of (25) and Taylor expansion in the last equality.

Finally, one needs to perform integration by parts for the variable $\theta$ to deal with the integration $\left|I\right(x;h\left)\right|$.

Now one can separate the integration

$$\begin{array}{cc}\hfill \left|I\right(x;h\left)\right|& \le |h^{-1}{\int }_{\left|s\right|\le h}\int \widehat{\rho }\left(s\right)exp\left[{\displaystyle \frac{i}{h}}\mathsf{\Phi }(s,x,\theta )\right]·c_2(s,x,\theta ;h)d\theta ds|\hfill \\ \hfill & +|h^{-1}{\int }_{h<\left|s\right|<1}\int \widehat{\rho }\left(s\right)exp\left[{\displaystyle \frac{i}{h}}\mathsf{\Phi }(s,x,\theta )\right]·c_2(s,x,\theta ;h)d\theta ds|:=\left|I_1(x;h)\right|+\left|I_2(x;h)\right|.\hfill \end{array}$$

It is straightforward to show that

$$\left|I_1(x;h)\right|=O\left(1\right).$$

(29)

Hence, we only need to focus on addressing the second term $I_2(x;h)$. With the help of (26) and (27), the key observation is that

$$\left|{\partial }_{\theta }\mathsf{\Phi }(s,x,\mathrm{\Theta })\right|=\left|s\right||{\partial }_{\theta }{p}_2(x,\mathrm{\Theta })+O\left(\epsilon \right)|>C\left|s\right|>0$$

(30)

due to the strictly monotonic assumption of (7) and the support property of (9).

Also with the help of (26) and (27) one has that

$$\left|{\partial }_{\theta }^2\mathsf{\Phi }(s,x,\mathrm{\Theta })\right|\le C\left|s\right|.$$

Now one can integrate by parts to obtain the following:

$$\begin{array}{cc}\hfill \left|I_2(x;h)\right|& =|{\int }_{h<\left|s\right|<1}\int {\displaystyle \frac{1}{{\partial }_{\theta }\mathsf{\Phi }(x,s,\mathrm{\Theta })}}{\partial }_{\theta }exp\left[{\displaystyle \frac{i}{h}}\mathsf{\Phi }(s,x,\mathrm{\Theta })\right]·c_2(s,x,\theta ;h)d\theta ds|\hfill \\ \hfill & \le |{\int }_{h<\left|s\right|<1}\int {\left({\partial }_{\theta }\mathsf{\Phi }\right)}^{-1}exp\left[{\displaystyle \frac{i}{h}}\mathsf{\Phi }(s,x,\mathrm{\Theta })\right]·{\partial }_{\theta }{c}_2(s,x,\theta ;h)d\theta ds|\hfill \\ \hfill & +|{\int }_{h<\left|s\right|<1}\int {\left({\partial }_{\theta }\mathsf{\Phi }\right)}^{-2}\left({\partial }_{\theta }^2\mathsf{\Phi }\right)exp\left[{\displaystyle \frac{i}{h}}\mathsf{\Phi }(s,x,\mathrm{\Theta })\right]·c_2(s,x,\theta ;h)d\theta ds|\hfill \\ \hfill & \le C\left|{\int }_{h<\left|s\right|<1}{\displaystyle \frac{1}{s}}ds\right|=C|lnh|.\hfill \end{array}$$

(31)

2.2. Schrödinger Case

To treat the more general Schrödinger case, one can work in the Jacobi metric $g_E={(E-V)}_{+}g$ instead of the Riemann metric g. Consequently, using geodesic normal coordinates in $g_E$ centered at x like what we used in (23), one can follow the similar argument as in the homogeneous case.

3. Examples

In this section we conduct a study on the convex surface of revolution.

One parametrizes convex surfaces of revolution by using geodesic polar coordinates $(t,\phi )\in [-1,1]\times [0,2\pi ]$ where

$$p_1(t,\phi ;{\xi }_t,{\xi }_{\phi })={\xi }_t^2+f^{-2}\left(t\right){\xi }_{\phi }^2,$$

and

$$p_2(t,\phi ;{\xi }_t,{\xi }_{\phi })={\xi }_{\phi }.$$

The profile function satisfying $f(-1)=f\left(1\right)=0$ and $f^2\left(t\right)$ is a non-negative Morse function with a single non-degenerate maximum at $t=t_0\in (-1,1)$.

Consider the point $x=(t_1,\phi )$, where $t_1\ne \pm 1$. One has that

$$C_x=\{(t_1,\phi ;{\xi }_t,{\xi }_{\phi });{\xi }_t^2+f^{-2}\left(t_1\right){\xi }_{\phi }^2=1\}.$$

On the set $C_x\cap \left\{\right|{\xi }_{\phi }|\le \delta \}$ where $\delta >0$ is sufficiently small, one has that ${\partial }_{{\xi }_t}{p}_1=2{\xi }_t$, which is away from zero if $\delta >0$ is sufficiently small. Hence one can use ${\xi }_{\phi }$ to parametrize $C_x$. Now

$${\partial }_{{\xi }_{\phi }}{p}_2=1$$

(32)

which means that (7) is valid.

On the other hand, on the set $C_x\cap \left\{\right|{\xi }_{\phi }|>\delta \}$ where $\delta >0$ is sufficiently small, one has that ${\partial }_{{\xi }_{\phi }}{p}_1=2f^{-2}\left(t_1\right){\xi }_{\phi }\ne 0$. Hence one can use ${\xi }_t$ to parametrize $C_x$. Now

$${\partial }_{{\xi }_t}{p}_2^2={\partial }_{{\xi }_t}\left[f^2\left(t_1\right)(1-{\xi }_t^2)\right]=-2f^2\left(t_1\right){\xi }_t$$

(33)

which can be zero if $f^{-2}\left(t_1\right){\xi }_{\phi }^2=1$. Hence (7) is valid if $f^{-2}\left(t_1\right){\xi }_{\phi }^2$ is away from 1.

Next we focus on the case of the standard sphere ${\mathbb{S}}^2$. In this case, $f\left(t\right)=\sqrt{1-t^2}$ and $t_0=0$.

Now we use the longitudinal coordinate $\theta \in [0,\pi ]$ and the latitudinal coordinates $\phi \in [0,2\pi )$ so that ${\mathbb{S}}^2\ni x=(\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta )$. One takes the zonal harmonics

$$u_k=C_k{P}_k\left(\cos \theta \right)$$

which solves

$$-\mathrm{\Delta }{u}_k=k(k+1)u_k,$$

where $C_k\sim k^{{\displaystyle \frac{1}{2}}}$ is the $L^2$ normalisation factor, $P_k$ is the associated Legendre polynomial, and the eigenfrequency $h^{-1}=\sqrt{k(k+1)}\sim k$. In this case, the corresponding h-Laplacian $P_1\left(h\right):=-h^2\mathrm{\Delta }$ with the eigenvalue $E_1\left(h\right)=1+O\left(h\right)$ is QCI after commuting $P_2\left(h\right)=h{D}_{\phi }$ with $E_2\left(h\right)=0$.

It is well known that $u_k$ saturates $O\left(h^{-1/2}\right)$ in an $O\left(h\right)$ neighborhood of the poles.

Next consider the point $x=(\sin {\theta }_0\cos \phi ,\sin {\theta }_0\sin \phi ,\cos {\theta }_0)$; here ${\theta }_0\ne 0$ and ${\theta }_0\ne \pi$, which is outside the $O\left(h\right)$ neighborhood of the poles. From the above argument involving (32), we know that (7) is valid. From [18] (Theorem 8.21.2), one has

$$P_k\left(\cos {\theta }_0\right)=\sqrt{{\displaystyle \frac{2}{\pi k\sin {\theta }_0}}}\cos \left[(k+{\displaystyle \frac{1}{2}}){\theta }_0-{\displaystyle \frac{\pi }{4}}\right]+O\left(k^{-3/2}\right).$$

One can easily obtain the following:

$$\left|u_k\right|=\left|C_k{P}_k\left(\cos {\theta }_0\right)\right|\le C{k}^{{\displaystyle \frac{1}{2}}}·k^{-{\displaystyle \frac{1}{2}}}=C,\mathrm{as}k\to +\infty .$$

This bound is consistent with (and slightly stronger than) the general $O\left(\right|\ln h{|}^{1/2})$ bound given in Theorem 1.

Next we consider the highest-weight spherical harmonics

$$u_{\lambda }\left(x\right)={\lambda }^{1/4}{(x_1+i{x}_2)}^{\lambda }={\lambda }^{1/4}{[\sin \theta \cos \phi +i\sin \theta \sin \phi ]}^{\lambda }={\lambda }^{1/4}{\left(\sin \theta \right)}^{\lambda }{e}^{i\lambda \phi },$$

where ${\int }_M{\left|u_{\lambda }\right|}^{2}d\mathrm{Vol}\sim 1$ and the eigenfrequency $h^{-1}=\lambda =n;n=1,2,3,\dots$. In this case, the corresponding h-Laplacian $P_1\left(h\right):=-h^2\mathrm{\Delta }$ is QCI after commuting $P_2\left(h\right)=h{D}_{\phi }$ with $E_2\left(h\right)=1$.

At point $x=(\sin {\theta }_0\cos \phi ,\sin {\theta }_0\sin \phi ,\cos {\theta }_0)$, the eigenfunction is as follows:

$$u_{\lambda }\left(x\right)={\lambda }^{1/4}{\left(\sin {\theta }_0\right)}^{\lambda }{e}^{i\lambda \phi },$$

where ${\theta }_0\ne 0$ and ${\theta }_0\ne \pi$. In order to make (7) valid, ${\theta }_0\ne {\displaystyle \frac{\pi }{2}}$. This is due to the above argument (Equation (33)). Hence

$$\left|u_{\lambda }\right(x\left)\right|=\left|{\lambda }^{1/4}{\left(\sin {\theta }_0\right)}^{\lambda }\right|\ll {|ln\lambda |}^{{\displaystyle \frac{1}{2}}}.$$

This bound is also consistent with (and much stronger than) the general $O\left(\right|lnh{|}^{1/2})$ bound given in Theorem 1.