Pointwise Bounds for Joint Eigenfunctions of Quantum Completely Integrable Systems

Let (M, g) be a compact Riemannian manifold of dimension n and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_1:=-h^2\Delta _g+V(x)-E_1$$\end{document}P1:=-h2Δg+V(x)-E1 so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$dp_1\ne 0$$\end{document}dp1≠0 on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_1=0$$\end{document}p1=0. We assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_1$$\end{document}P1 is quantum completely integrable (ACI) in the sense that there exist functionally independent pseuodifferential operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_2,\dots P_n$$\end{document}P2,⋯Pn with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[P_i,P_j]=0$$\end{document}[Pi,Pj]=0, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i,j=1,\dots n$$\end{document}i,j=1,⋯n. We study the pointwise bounds for the joint eigenfunctions, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_h$$\end{document}uh of the system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{P_i\}_{i=1}^n$$\end{document}{Pi}i=1n with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_1u_h=E_1u_h+o(1)$$\end{document}P1uh=E1uh+o(1). In Theorem 1, we first give polynomial improvements over the standard Hörmander bounds for typical points in M. In two and three dimensions, these estimates agree with the Hardy exponent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^{-\frac{1-n}{4}}$$\end{document}h-1-n4 and in higher dimensions we obtain a gain of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^{\frac{1}{2}}$$\end{document}h12 over the Hörmander bound. In our second main result (Theorem 3), under a real-analyticity assumption on the QCI system, we give exponential decay estimates for joint eigenfunctions at points outside the projection of invariant Lagrangian tori; that is at points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in M$$\end{document}x∈M in the “microlocally forbidden” region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_1^{-1}(E_1)\cap \dots \cap p_n^{-1}(E_n)\cap T^*_xM=\emptyset .$$\end{document}p1-1(E1)∩⋯∩pn-1(En)∩Tx∗M=∅. These bounds are sharp locally near the projection of the invariant tori.


Introduction
Let (M n , g) be a closed, compact C ∞ manifold and P 1 (h) : C ∞ (M) → C ∞ (M) a self-adjoint semiclassical pseudodifferential operator of order m that is elliptic in the classical sense, i.e. | p 1 (x, ξ)| ≥ c|ξ | m − C. Here, h takes values in a discrete sequence (h j ) ∞ j=1 with h j → 0 + as j → ∞. We assume in addition that there exist functionally independent h-pseudodifferential operators P 2 (h), . . . , P n (h) with the property that [P i (h), P j (h)] = 0; i, j = 1, . . . , n. (1.1) In that case we say that P 1 (h) is quantum completely integrable (QCI). Given the joint eigenvalues E(h) = (E 1 (h), . . . , E n (h)) ∈ R n of P 1 (h), . . . P n (h) we denote an L 2normalized joint eigenfunction with joint eigenvalue E(h) by u E,h (here, for notational simplicity we drop the dependence of E on h in the notation) and consequently, When the joint energy value E is understood, we will sometimes abuse notation and simply write u h = u E,h . The associated classical integrable system is governed by the moment map P := ( p 1 , . . . , p n ) : where p j ∈ C ∞ (T * M); j = 1, . . . , n are the semiclassical principal symbols of P j (h); j = 1, . . . , n. For convenience, we will denote the corresponding QCI system byP := (P 1 , . . . , P n ). We assume throughout that the classical integrable system p is Liouville integrable; that is there exists an open dense subset T * M reg ⊂ T * M such that rank(dp 1 (x, ξ), . . . ., dp n (x, ξ)) = n ∀ (x, ξ) ∈ T * M reg . (1.3) Following the notation in [TZ09], we let B := P(T * M) and B reg = P(T * M reg ) denotes the set of regular values of the moment map.
Since P is proper, the Liouville-Arnold theorem determines the symplectic structure of the level sets P −1 (E) where E ∈ B reg . The level set where the k (E)'s are Lagrangian tori which are invariant under the joint bicharacteristic flow G t : T * M → T * M, t = (t 1 , . . . , t n ) ∈ R n , G t (x, ξ) = exp t 1 H p 1 • · · · • exp t n H p n (x, ξ). Here, H p j = k ∂ ξ k p j ∂ x k − ∂ x k p j ∂ ξ k is the Hamilton vector field of p j .
In this paper, we are concerned with two questions regarding the joint eigenfunctions: (i) eigenfunction supremum bounds and (ii) eigenfunction decay estimates in the microlocally forbidden region, M\π(P −1 (E)).
We say that the system is of Morse type at x ∈ M if there exists f ∈ C ∞ (R n , R) and an h-pseudodifferential operator Q(h) := f (P 1 (h), . . . , P n (h)) with the property that its principal symbol q | x,E 1 is Morse for all x ∈ M.
Our first main result is Theorem 1. Let (M n , g) be compact Riemannian manifold andP be a QCI system with quantum Hamiltonian P 1 (h) = −h 2 g + V where V ∈ C ∞ (M; R) and E 1 ∈ R is a regular value of p 1 , i.e. so that dp 1 | p −1 In the cases where n = 2 or n = 3, one gets the Hardy-type supremum bounds: (1.6) Remark. (i) In the special case of Laplace eigenfunctions, P 1 (h) = −h 2 g − 1; that is, V = 0 and E 1 = 1. (ii) The estimate (1.5) in Theorem 1 gives an explicit polynomial improvement over the well-known Hörmander bound u h L ∞ = O(h (1−n)/2 ). In dimensions n = 2, 3, modulo the logarithmic factor in the n = 3 case, both the estimates in (1.6) are consistent with the Hardy type bound u h L ∞ = O(h (1−n)/4 ). Moreover, these estimates are sharp and are also quite robust in that they apply to many QCI examples either globally (e.g. Liouville Laplacians or Neumann oscillators on tori), or locally away from isolated points (e.g. Laplacians on convex surfaces of revolution, Laplacians on asymmetric ellipsoids (n=2,3), quantum Neumann oscillators (n=2,3), quantum spherical pendulum, and quantum Euler and Kovalevsky tops). We describe how the above results apply explicity in several classical examples in Sect. 4. In the global cases, the bounds in Theorem 1 holds for all with ⊂ {V < E}. Otherwise, one must delete arbitrarily small (but fixed independent of h) balls centered at a finite number of points (e.g. the umbilic points of an triaxial ellipsoid, or the poles of an convex surface of revolution.) Finally, we point out in the case of the Laplacian, V = 0, so that the potential well is the entire manifold, M, and the corresponding sup bounds hold over all of M; that is, one can set = M in (1.6). (iii) We point out that in Theorem 1 we fix only the energy E 1 . In particular, it is a statement about all joint eigenfunctions satisfying P 1 u h = (E 1 + o(1))u h and we crucially do not require that the total energy, E ∈ B is regular; that is we do not require E ∈ B reg .
One of the quantum integrable examples where the Morse hypothesis of Theorem 1 is not satisfied at every point is that of the triaxial ellipsoid (1.7) Here, there are four exceptional points, { p j } 4 j=1 ∈ E, the umbillic points, where the integrable system is not of Morse type. Combining the proof of Theorem 1 with results from [CG18], we prove the following sup bound for the joint eigenfunctions: Theorem 2. Let E as in (1.7) and P = −h 2 g − 1.
Then there is C > 0 so that any L 2 normalized joint eigenfunction, u h of the QCI system satisfies In [Tot96], the second author showed that there are constants c, h 0 > 0 and a sequence of L 2 normalized joint eigenfunctions of the QCI system satisfying and consequently, the estimate in Theorem 2 is sharp.
1.2. Comparison with previous L ∞ estimates. In general, for normalized Laplace eigenfunctions on a compact manifold M of dimension n i.e. solving (−h 2 g − 1)u = 0, the celebrated works [Hör68,Ava56,Lev52] show that (1.8) Under certain geometric conditions on the manifold M, this bound can be improved to (1.9) These conditions include non-existence of recurrent points (see [STZ11,Gal17,CG17]), which in particular is satisfied for manifolds without conjugate points. Under a certain uniform version of the non-recurrent hypothesis [CG18] shows that this can be improved to (1.10) This non-recurrent hypothesis is in particular satisfied on manifolds without conjugate points where improved L ∞ estimates have been proved using the Hadamard parametrix in [Bér77,Bon17]. Finally, in forthcoming work [GT18], the authors give improvements of the form for some explicit δ > 0 when the manifold has integrable geodesic flow. The only other polynomial improvements that the authors are aware of occur in the case of Hecke-Maas forms on certain arithmetic surfaces [IS95].
In this paper, we assume that eigenfunctions are joint eigenfunctions of a quantum complete system of equations. In [TZ02], it is shown that if QCI Laplace eigenfunctions have sup-norms that are O(1), then the manifold is, in fact, flat. Therefore, it is natural to understand the L ∞ growth of eigenfunctions in the QCI case. We note that the QCI assumption is very rigid and allows us to give much stronger estimates than those mentioned above. Indeed, Theorem 1 achieves the so-called Hardy estimate in dimension n = 2, and n = 3 (modulo a log h −1 loss) which is expected to hold at a generic point on a generic manifold. Moreover, in any dimension n, under a generic assumption on the QCI system, we are able to give an explicit polynomial improvement over (1.8).
While this is a dramatic improvement over the bounds above, it is important to note that the assumption of quantum complete integrability is highly sensitive. First, any small perturbation of the original operator (even a lower order perturbation) will destroy the property of being quantum integrable. Furthermore, even if the Laplacian is quantum integrable, it is not clear that all eigenfunctions for the Laplacian are joint eigenfunctions of the corresponding QCI system. On the other hand, the approaches used to obtain (1.8), (1.9), (1.10) and (1.11) are robust to lower order perturbations and apply to all sequences of eigenfunctions.
Our bounds are related to those in [Sar] where Sarnak shows that on a locally symmetric space of rank r , and the generalization of this bound to joint quasimodes of r essentially commuting operators with independent fiber differentials [Tac18]. We point out that while for some specific energy levels E, there are points satisfying the independent fiber differential assumption, the only quantum integrable example we are aware of in which there is a single point x satisfying this assumption for all energy levels is that of the flat torus. We also note that our results in Theorem 1 apply in the case of many QCI systems that do not arise from isometric group actions; these include Liouville Laplacians on tori, Laplacians on asymmertric ellipsoids, quantum Neumann oscillators on spheres and quantum Kowalevsky tops, among others.
1.3. Exponential decay estimates. Our next result deals with exponential decay estimates for joint eigenfunctions in the microlocal "forbidden" region M\π( R ) with We make the additional assumption that P j (h) : j = 1, .., n are real-analytic, hdifferential operators and that the restricted canonical projection has a fold singularity along the caustic C = π −1 ( ∂π ( R (E)) ). One can complexify R to a complex submanifold,˜ , of the complexification, T * M, of the real cotangent bundle. Here,˜ is Lagrangian with respect to the canonical complex symplectic form where ω C is the complex canonical one-form on T * M. In the terminology of [Sjö82],˜ is C-Lagrangian. There is a further submanifold˜ I ⊂˜ given by˜ that is of particular interest to the study of eigenfunction decay. Roughly speaking,˜ I is subset of˜ that consists of points with real base coordinates. We also show in Sect. 3.2 (see Proposition 3.2), under the fold assumption, one can characterize the structure of I quite readily near C ; at least locally, one can writẽ Both R and I are isotropic with respect to Im C (ie. they are I -isotropic) and I locally projects to the microlocally forbidden region, M\π( R ). Moreover, I is locally a graph over M away from the projection of the caustic ∂π( R ) with where ψ is complex-valued and real-analytic. In addition, as a consequence of the fold assumption, I can be further decomposed as a union over two branches + I ∪ − I , where these branches are (locally) characterized as follows: given any local smooth curve γ ± (α 0 , α) ⊂ ± I joining α 0 ∈ C to α ∈ ± I , Im ω C ≥ 0.
In view of (1.12), there exist locally well-defined functions S ± : π( ± I ) → C that are real-analytic away from ∂π( R ) with We then define the complex action function locally to be Our main result on the exponential decay of joint eigenfunctions is: Theorem 3. Suppose that P(h) = (P 1 (h), . . . , P n (h)) is a QCI system of real-analytic, jointly elliptic, h-differential operators and E ∈ P(T * M) a regular level of the moment map. Suppose, in addition, that the caustic C is a fold. Then, there exists an h-indepedent neighbourhood, V ⊃ π( R ), such that for any open ( V \π( R ) ) and any ε > 0, there exists h 0 (ε, ) > 0 such that for h ∈ (0, h 0 (ε, )], and u h a joint eigenfunction of P(h) with energy E, As we show in Sect. 4, under the real-analyticity assumption the decay estimate in Theorem 3 is sharp and improves on results of the second author in [Tot98]. Moreover, the fold assumption is satisfied for generic joint energy levels when n ≥ 2. In the cases where there exist appropriate coordinates in terms of which the classical generating function is separable, one can show that the decay estimates in Theorem 3 are still satisfied for non-generic energy levels E ∈ B reg . The latter condition is satisfied in all cases that we know of (see remark 3.5 for more details).

Sup Bounds for QCI Eigenfunctions: Proof of Theorem 1
Proof. We assume first that n = 2 and that P 1 (h) = −h 2 g , E 1 = 1 and indicate the minor changes in the case where P 1 (h) = −h 2 g + V (x), at the end. Since we assume the QCI condition, instead of working with long-time propagators, it simplifies the analysis to use small-time joint propagators. We will also assume without loss of generality that E 1 = 0 (replacing P 1 by P 1 − 1). Suppose P 1 (h)u h = 0 and with As usual, we let ρ ∈ S(R) with ρ(0) = 1 and with ε > 0 small we choose suppρ ⊂ [ε, 2ε].
Performing stationary phase in (y, η) gives that at the critical point (y c (x, z, s), η c (x, z, s)), Then, by Cauchy-Schwarz, Now, we observe that and also note that Therefore, and In particular, Therefore, changing variables to S = t − s T = t + s, (2.5) We split the integral into two pieces . Therefore, the Morse assumption on q| S * x M allows us to perform stationary phase in z with hS −1 as a small parameter in the second integral (2.6). The result is that the latter integral is Summarizing, we have proved that (2.7) Taking square roots completes the proof in the case where P 1 (h) = −h 2 g , and E 1 = 1.

Schrödinger case.
To treat the more general Schrödinger case, we simply note that (see e.g. [CHT15]) in analogy with the homogeneous case in (2.2), where r E (x, y) is Riemannian distance in the Jacobi metric g E = (E − V ) + g which is non-singular in the allowable region {V < E}; in particular, r E (x, y) locally satisfies the eikonal equation with ε > 0 fixed sufficiently small. Consequently, using geodesic normal coordinates in g E centered at x ∈ , it follows that the composite kernel K (x, z) has exactly the same form as in (2.4). The rest of the argument follows in the same way as in the homogeneous case.

Geometric implications of the Morse condition.
The morse assumption, Definition 1.1, may at first seem artificial. However, we observe in Sect. 4 that it is satisfied in many examples and, moreover, it implies a purely geometric condition which is natural.
In particular, for the QCI systemP and x 0 ∈ M, there are n natural submanifolds for L ∞ norms: Because we work with only two propagators, we consider E x M) indeed, not even that the intersection is clean. However, it does ensure that for every energy E 2 , the volume of E x 0 small. More precisely (in dimension n = 3) it ensures that for every E 2 , Because P 1 u = E 1 u and P 2 u = E 2 u, we can see that u is localized in an h neighborhood of { p 1 = E 1 , p 2 = E 2 } and thus h is the only region on which u can have energy producing large L ∞ norm at x 0 This volume localization then gives improved L ∞ norms. The philosophy that volume concentration over E 1 x 0 ,1 , implies improved L ∞ norms can be made rigorous [CG18]. In future work [GT18], we will use the ideas there to use directly the volume of the set h to obtain a Hardy type bound for QCI eigenfunctions under a Morse type assumption on the system.

Exponential Decay Estimate for Joint Eigenfunctions in the Microlocally Forbidden Region
In this section, to prove our eigenfunction decay estimates, we will assume that (M, g) is real-analytic and the QCI system P 1 (x, h D x ), . . . , P n (x, h D x ) consists of analytic h-differential operators. To formulate and prove our results, we will now recall some basic complex geometry and h-analytic microlocal machinery that will be used later on.

Complex geometry.
In this section, we require M be a compact, closed, real-analytic manifold of dimension n and M denote a Grauert tube complex thickening of M with M a totally real submanifold. By the Bruhat-Whitney theorem, M can be identified with , which satisfies homogeneous Monge-Ampere and its level sets exhaust the complex thickening M (see [GS91] for further details).
We consider a complexification of T * M of the form where C 1 is a sufficiently large constant and T * M ⊂ T * M is then a totally-real submanifold invariant under the involution α →ᾱ.
One has a natural complex symplectic form on T * M given by Given the complex symplectic form, C , there are some natural Lagrangian submanifolds of T * M that are of particular interest to us: First, there is the C-Lagrangian submanifold˜ where P C = ( p C 1 , . . . , p C n ) and p C j denotes the holomorphic continuation of p j to T * M. When the context is clear, in the following we will sometimes simply write p for the holomorphic continuation P C . The level set is an R-Lagrangian submanifold and, as we have already pointed out, by the Liouville-Arnold theoerem, it is a finite union of R-Lagrangian tori. We recall that a complex n-dimensional submanifold, I , of T * M is said to be I-Lagrangian if it is Lagrangian with respect to where C = dα x ∧ dα ξ is the complex symplectic form on T * M. We will denote the correponding complex canonical one form by There are several examples of I -Lagrangians that will be of particular interest to us; these include, graphs over the real cotangent bundle T * M of the form where H G is the Hamilton vector field of a real-valued G ∈ C ∞ 0 (T * M; R).

Complex symplectic geometry near caustics of fold type.
There is a natural Iisotropic associated with the integrable system P = ( p 1 , . . . , p n ) and the associated C-Lagrangian˜ . To define it we let T * M ⊗ C := T * M M , the complexification of T * M in the fibre α ξ -variables only and set We will now consider the case where π : R → M has fold singularities. As we will show below, in such a case, one can describe the structure of˜ I in detail locally near the projection of the caustic set.
Definition 3.1. We define the caustic set to be the subset of the real Lagrangian R given by In addition, we say that the caustic C is of fold type if the projection π R : R → M has fold singularities along C .
It follows from an implicit function theorem argument that, under the fold assumption on the caustic set, π( R ) is a real n-dimensional stratified subset of M with boundary, and moreover, To see this, we need only show that if α ∈ R and rank R (d α ξ p 1 (α), . . . , d α ξ p n (α)) = n, then π( R ) contains a neighborhood of π(α). For this, observe that H p i , i = 1, . . . n are tangent to R . In particular, the rank condition implies that dπ H p i , i = 1, . . . n are linearly independent and hence π : R → M is a local diffeomorphism.
Remark. In general, C is a stratified space. Under the fold assumption in (i), one has a decomposition of the form C = ∪ N k=1 H k , where the H k are closed hypersurfaces (of real dimension n − 1). We note that the fold assumption above is generically satisfied in all of the QCI examples that we are aware of.
Under the fold type assumption on C , one can locally characterize the structure of I near the caustic set. To motivate the general result, it is useful to consider first the simple case of the harmonic oscillator.

Harmonic oscillator. Consider the one-dimensional harmonic oscillator with
In this case, letting z → √ z denote the principal square root function with branch cut along the negative imaginary axis, we have˜ which is a single ellipse, and The latter set clearly has 4 connected components. See Fig. 1 for a picture of these sets.
where R = {α ∈ T * M; P(α) = 0} and I ⊂˜ I . Here, both R and I are I -isotropic submanifolds of the complex Lagrangian˜ with respect to the complex symplectic form C .
In addition, I is locally a (complex) canonical graph with Remark. Here, R is, of course, also R-Lagrangian with respect to the real symptlectic form on the real cotangent bundle T * M.
Proof. The fact that C ∩ T * M ⊗C is compact follows readily from the joint ellipticity of the p j 's. Indeed, since and by joint ellipticity, for all α ∈ T * M, it follows by Taylor expansion along T * M ⊂ T * M ⊗ C and the fact that the p j 's are symbols of h-differential operators (i.e. they are polynomials in the α ξ 's) that for Since |Im α ξ | ≤ 1 C |Re α ξ |, and in view of (3.5), it follows that for aperture constant C 1 sufficiently large, the second term on the RHS of (3.4) can be absorbed in the first; the end result is that for some m ∈ Z + . Thus,˜ ∩ (T * M ⊗ C) is clearly bounded since M is compact and since it is also closed, compactness follows.
To prove the remaining results (i) and (ii) in Proposition 3.2, we will use the fold assumption and argue in several steps.
Fix a point q ∈ H k ⊂ C . Then, by assumption π R has a fold singularity and by [Hö7, Theorem C.4.2], there are coordinates y on R and x on M so that y(q) = 0 and x(π(y)) = (y 1 , . . . y n−1 , y 2 n ) (3.6) and in particular, locally, H k = {y n = 0}. Now, since π(x, ξ) = x for (x, ξ) canonical coordinates on T * M, we have that x i (y) = y i for i = 1, . . . n − 1. Clearly, ∂ y n x n | y=0 = 0 and, since R is Lagrangian, That is, ∂ y n ξ i = 0, i = 1, . . . n − 1. Since ∂ y 1 , . . . ∂ y n are linearly independent, this implies that ∂ y n ξ n | y=0 = 0. Then, since the map κ : (y 1 , . . . , y n ) → (x (y), ξ n (y)) satisfies rank dκ = n,, by the implicit function theorem, y n = y n (ξ n , x ) where x = (x , x n ). Letting b(x ) = ξ n | y n =0 , we can write using the implicit function theorem once again, Therefore, we may choose coordinates x on M so that locally in canonical coordinates (x, ξ), Here, a ∈ C ω loc (R n ), a > 0 and b ∈ C ω loc (R n−1 ). In this case, the caustic hypersurface is We note that under the projection π R , the hypersurface H k can naturally be identified with the hypersurface {(x , x n = 0) ∈ U } ⊂ M. Henceforth, we abuse notation somewhat, and denote the latter also by H k .
Write a 2 (x , η n ) = a(x , η n + b(x )), then the normal form (3.7) can be rewritten in the form Next, we make a change of coordinates which will change the smooth structure near the caustic, but leave it unchanged away from the caustic. In particular, let x n = z 2 , z ∈ C so that Note that when we want to return to the x n coordinates, we will write √ x n = z where √ x n > 0 for x n > 0 and the branch cut is taken on −i[0, ∞). Writing ζ for the complexified fibre variables, we have and by the analytic implicit function theorem, Moreover, A simple computation using (3.8), or more precisely its analytic continuation using z as a coordinate, shows that π : I → M is locally surjective onto M near the caustic hypersurface H k . That is, there exists W k a neighborhood of H k in˜ I and V k a neighborhood of π (H k ) so that π : W k → V k is surjective and, moreover, with k := W k \H k , rank C ( d ζ p 1 (x, ζ ), . . . , d ζ p n (x, ζ ) ) = n, (x, ζ ) ∈ k . (3.10) To see this, we analytically continue (3.6). In particular, analytically continuing y ∈ R to α ∈ , Thus, dπ is surjective which implies that {dπ H p i } n i=1 = d ζ p has rank n. We also note that π | : k → M can be written as a graph over the base manifold M locally near the caustic hypersurface H k up to choice of branch; more precisely, we have for some δ > 0 Remark. We note that z 2 ∈ R for z ∈ (0, δ) i(0, δ). Also, the x normal coordinates have been chosen so that x n < 0 and consequently z ∈ i(0, δ) in the microlocally forbidden region k .
To complete the proof of Proposition 3.2, we will need the following result on solving a particular initial value problem for the complex eikonal equation associated with local branches ± k of the I -isotropic manifold k .

Complex generating functions.
In this section, we construct a generating function ψ ± of ± k locally near the caustic hypersurface H k . Specifically, we seek to solve the complex eikonal boundary value problem (3.12) In practice, we will not be able to find a unique solution ψ on all of k . However, for all such solutions, we will see that S = Im ψ agrees and hence that S is well defined on k . Lemma 3.3. Under the fold assumption on the real Lagrangian R (which is also Iisotropic), there exists S ± ∈ C 1,1/2 loc ( ± k ) ∩ C ω ( ± k ) so that S = Im ψ ± for any solution ψ ± to the complex eikonal boundary value problem in (3.12). In addition, with S ± = Im ψ ± , Proof. To solve the eikonal problem, we follow the standard method of (complex) bicharacteristics. Since the caustic hypersurface H k is characteristic for the joint flow of Hamilton vector fields of p C j ; j = 1, . . . , n, one cannot expect a smooth solution to (3.12). Nevertheless, it is still possible to solve (3.12), albeit with reduced regularity at H k . In normal coordinates (x, ξ + iη), given an initial point (x , ξ ; 0) ∈ H k and (x, ζ ) ∈ ± k , we consider the "normal" curve joining these points given by When (x, ζ ) ∈ ± k , we write γ ± for γ to specify the branch. Let (3.14) Let In particular, The fact that ψ ± k solves (3.12) on ± k respectively is clear from the definition above since from (3.11) ± k is locally a graph over U k with ± k = {(x, ζ ); ζ = ∂ x ψ ± k (x))}. Here, of course, the function ψ ± U ≡ ψ ± k for x ∈ k . Finally, from the formula in (3.14) it is clear that ψ . We now show that the definition of S ± k above is intrinsically defined in the sense that: (i) it is independent of choice of initial point on H k and (ii) it is independent of the choice of curve of integration in the same smooth homotopy class.
Indeed, to prove (i), we recall that ζ = ξ + iη and note that η| H k = 0, so that if α 0 , α 1 ∈ H k and γ (α 0 , α 1 ) ⊂ H k is a C 1 -curve joining these points, then using that As for (ii), let γ 1 (α 0 , α) ⊂ k ∩ l and γ 2 (α 0 , α) ∈ k ∩ l be two homotopic smooth curves joining α 0 ∈ H k to α ∈ k ∩ l . Then, since k ⊂ I is I -isotropic and k ∩ l ⊂ T * M M it follows by Stokes formula that Remark. Note that ψ ± k may depend on the choice of initial point in H k , but we have shown that S ± k = Im ψ ± k does not. The fact that I is I -isotropic and (i) and (ii) clearly follow from Lemma 3.3 and that completes the proof of Proposition 3.2.
Definition 3.4. From now on, we will refer to S k := S + k as the action function corresponding to the caustic hypersurface H k .
We extend S k to the entire caustic C be setting so that, by definition, S k | H l = 0 for all l = 1, . . . , N .

3.2.3.
Action function corresponding to the entire caustic set C . We now define the action function S : ∪ k π( k ) → R on the entire forbidden region ∪ k π( k ). It remains to check that the S k 's correponding to the different caustic hypersurfaces H k agree on overlaps. More precisely, we claim that S k (α) = S l (α), α ∈ k ∩ l . (3.15) The compatibility condition in (3.15) is readily checked: Let α k 0 ∈ H k and α l 0 ∈ H l and γ (α k 0 , α l 0 ) ⊂ H k ∪ H l be a piecewise smooth curve inside the caustic joining α k 0 and α l 0 (which we recall is a real submanifold of T * M). Now let α ∈ k ∩ l and γ 1 (α k 0 , α) ⊂ k and γ 2 (α l 0 , α) ⊂ l be two normal curves as above, Then, γ (α k 0 , α l 0 ) ∪ γ 1 (α k 0 , α) ∪ γ 2 (α l 0 , α) bounds a domain kl ⊂ k ∩ l . Since I is I -isotropic, it follows from Stokes formula that However, since C ⊂ T * M so that η| γ (α k 0 ,α l 0 ) = 0, the first integral on the LHS of (3.16) vanishes and hence, We now set In view of the compatibility condition (3.15), the action function in (3.27) is well-defined. Also, from now on we denote the microlocally forbidden region by

Analytic psdos and FBI transforms.
Let U ⊂ T * M be open. Following [Sjö96], we say that a ∈ S m,k cla (U ) provided a ∼ h −m (a 0 + ha 1 + . . . ) in the sense that (3.19) We sometimes write S m,k cla = S m,k cla (T * M).

We say that an operator A(h) is a semiclassical analytic pseudodifferential operator of order m, k if its kernel can be written as A(x, y; h)
where χ ∈ C ∞ c (R) is 1 near 0 and a ∈ S m,k cla . We say A is h-elliptic if |a 0 (x, ξ)| > ch −m ξ k where a 0 is from (3.19). Recall also that A is classically elliptic if there is C > 0 so that if |ξ | > C, |a 0 (x, ξ)| > C −1 h −m |ξ | k . For more details on the calculus of analytic pseudodifferential operators, we refer the reader to [Sjö82].
Remark. The normalization a ∈ S 3n/4,n/4 cla appears so that T is L 2 bounded with uniform bounds as h → 0 [Sjö96].
The phase function is required to satisfy it follows by an analytic stationary phase argument [Sjö96] that one can construct an operator S(h) : Henceforth, we use the invariantly-defined FBI transform T (h) : Here, μ > 0 is a constant that will be chosen appropriately later, r (·, ·) is geodesic distance and χ(α x , y) = χ 0 (r (α x , y) where the Schwartz kernel of R (h) satisfies the same exponential decay estimates as R(x, y, h) in (3.23).
In particular, since (|S| + |H ε | + |∂ α x H ε |) ≤ Cδ 1/2 , this implies that there is δ > 0 and μ 0 > 0 so that for all μ > μ 0 , (3.38) We also note that Let χ 1, supported on χ ≡ 1 and χ 2, ≡ 1 on supp χ with χ i, ∈ C ∞ c ( ). Then, as we show in the Appendix, there is δ > 0 so that for μ > μ 0 , one can construct a left-parametrix S : with the property that for some uniform constant C > 0, (3.39) Here, we recall the exponential constant C > 0 in the remainder terms in (3.39) does not depend on the constant μ > 0 in the phase function (3.24) of the FBI transform which we now fix large enough, with Consequently from (3.34), the Cauchy Schwarz inequality and the last line of (3.39) one gets that for x ∈ , and any ε > 0, (3.41) (3.42) Consequently, it remains to bound A (h) L 2 ( )→L ∞ (M) . We note that by Lemma 3.3 under the fold assumption, we can find local coordinates x = (x , x n ) : → R n in a neighbourhood, of the caustic in terms of which By Taylor expansion, It follows that for x ∈ , and with appropriate m > 0, uniformly for x ∈ supp χ 1, . The last line follows by an application of steepest descent under the assumption (3.40) on the constant μ > 0 in the phase function ϕ(α, x). Thus, in particular, it follows that for any ⊂ M\π( R ) sufficiently close to the caustic ∂π( R ), with some m > 0. Thus, in view of (3.44) and (3.41), we have proved Theorem 3.
Remark. Many classical integrable systems (eg. geodesic flow on ellipsoids, Neumann oscillators on spheres, geodesic flow on Liouville tori), have the feature that in terms of appropriate coordinates x = (x 1 , . . . , x n ) ∈ n j=1 (α j , α j+1 ) with α 1 < α 2 < · · · α n defined in a neighbourhood, V, of π( R ) one can separate variables in the generating function S V : V → R with Moreover, one can write each S V (x j ) as a hyperelliptic integral where R E is a polynomial of degree n − 1 with with coefficients that depend on the joint energy levels E = (E 1 , . . . , E n ) ∈ B reg When n = 2 the roots of R E (s) are necessarily simple (since it is linear) and this is generically still the case in higher dimensions as well.
The proof of Theorem 3 holds in the (non-generic) case where R E (s) has multiple roots. Indeed, in the case where R E (s) has a root r k ∈ (α k , α k+1 ) of mulitiplicity 2k + 1 corresponds to a caustic hypersurface H k = {x k = r k } with k = {x k > r k }. The complex generating function near H k in the analogue of Lemma 3.3 is then locally of the form Consequently, both S| x k =r k = 0 and d S| x k =r k = 0 and also d S(x k ) = 0 when x k > r k , the reader can readily check that the analogue of Lemma 3.5 holds in this case also and the proof of Theorem 3 then follows in the same way as in the fold case where k = 0.

Examples
We begin with some relatively simple examples of QCI systems in two dimensions: Laplace eigenfunctions on convex surfaces of revolution and Liouville tori/spheres. In these special examples, one can justify separation of variables for the joint eigenfunction that allow us to verify the sharpness of both Theorems 1 and 3.

Convex surfaces of revolution. Consider a convex surface of revolution generated by rotating a curve
where k is a nonnegative integer and f (r ) < 0 for all r ∈ (−1, 1). Moreover, we will assume that f (r ) has a single isolated critical point at r = 0; in particular, f (0) = 0 and f (0) < 0.
Let M be the corresponding convex surface of revolution parametrized by Consider M endowed with the rotational Riemannian metric g given by where w(r ) = 1 + ( f (r )) 2 .

Eigenfunction decay.
To verify the fold condition, we assume that E = (1, E 2 ) ∈ B reg . From the above, we can write Since for E ∈ B reg , we have E 2 2 < max r ∈[−1,1] f 2 (r ), it is clear from (4.4) that the restricted projection π R (E) : R (E) → M is of fold type and so the decay estimates in Theorem 3 are satsified. The fact that these estimates are sharp in this case, is an immediate consequence of above separation of variables and WKB estimates applied to (4.2).

Liouville Laplacian.
Consider the two-torus M = R 2 /Z 2 with two, smooth, positive periodic functions a, b : R/Z → R + where, for convenience, we assume that min 0≤x 1 ≤1 a(x 1 ) > max 0≤x 2 ≤1 b(x 2 ). The corresponding Liouville metric is given by g = (a(x 1 ) + b(x 2 ))(dx 2 1 + dx 2 2 ) and the associated Laplacian is QCI with commutant Given (1, E 2 ) ∈ B, it is easily checked that When E 2 ∈ (max b, min a), the projection π E has no singularities and consequently, E is a Lagrangian graph. On the other hand, when either E 2 ∈ (min a, max a) ∪  (min b, max b), it is easily seen from (4.5) that π E : E → R 2 /Z 2 is of fold type. Consequently, when a, b ∈ C ω (R 2 /Z 2 ), the decay estimates in Theorem 3 hold for the joint eigenfunctions.
As for Theorem 1, we simply note that given any point (θ ) = β cos 2 θ − α sin 2 θ which is clearly Morse as a function of θ ∈ [0, 2π ] when α > β > 0. Consequently, the global Hardy bound for joint eigenfunctions in Theorem 1 is satisfied in this case. Moreover, it is well-known [Tot96,TZ03] that this bound is saturated in this case.

Liouville oscillators.
In this example, the underlying Riemannian manifold is (R 2 /Z 2 , g) where g is the above Liouville metric. Consider the Schrodinger operator One verifies that the Schrodinger operator commutes with P 1 (h). Given a regular value E 1 of p 1 , it is easy to check that It is clear from (4.6) that π E is either regular, or has fold-type singularities.
As for the Morse condition: the same reasoning as in the case of the Liouville Laplacian shows that with E 1 ,z = {(z, ξ); p 1 (z, ξ) = E 1 } the function p 2 | E 1 ,z is Morse and consequently the joint eigenfunctions satisfy the Hardy-type bounds in Theorem 1.
Both the Liouville Laplacian and oscillator extend to QCI systems on tori of arbitrary dimension [HW95] The fold assumption is satisfied for generic joint energy levels (see also Remark 3.5 below) and so is the Morse assumption in Theorem 1.
One can show (see [CdVVuN03] ) that the intrinsic Riemannian metric on E pulledback to R is locally of Liouville form where a and b are certain hyperelliptic functions that extend to real-analytic function on R. Moreover, a(kT 1 ) = a (kT 1 ) = 0, b(kT 2 ) = b (kT 2 ) = 0 and a (kT 1 ) = 0, b (kT 2 ) = 0 for all k ∈ Z. Consequently, ds 2 extends to a C ω -metric on the torus Of course, the induced metric (which we continue to denote by ds 2 ) on the torus R 2 / degenerates at the lattice points in . Let T = R 2 /2 , the torus generated by the doubled lattice 2 and σ : T → T the natural involution given by σ (z) = −z. Then, the automorphism σ has precisely four fixed points given by the vertices (0, 0), (T 1 , 0), (0, T 2 ) and (T 1 , In view of the conformal mapping , this gives an identification E ∼ = T /σ . Consequently, under this identification, the torus T is a two-sheeted covering of the ellipsoid, E with covering map : T → E; (z) = z 2 .
This covering map is ramified over the umbilic points and the Riemannian metric g on E has the property that ds 2 = * g.

Proof of Theorem 2.
Proof. Let B j ; j = 1, 2, 3, 4 be open neighbourhoods of the umbilic points p j ; j = 1, 2, 3, 4. Then, in the complement E\ ∪ j B j , one has local coordinates (x 1 , x 2 ) in terms of which the metric has the form (4.7). Then, the same argument as in the case of the Liouville torus using Theorem 1 shows that for the joint eigenfunctions of the corresponding QCI system on the ellipsoid, one gets that To prove (4.8), we split the analysis into two cases: Case (i): Suppose first that for any fixed δ = 1/4 − ε we have x ∈ B j \B j (h δ ). Using the conformal (x 1 , x 2 ) coordinates above near the umbilic point p j we have x 1 ( p j ) = x 2 ( p j ) = 0 and a(x 1 ) = C x 2 1 + O(x 3 1 ), b(x 2 ) = C x 2 2 + O(x 3 2 ), x = (x 1 , x 2 ) ∈ B\B(h δ ).
Finally, we deal with Case (ii); where x ∈ B(h δ ). To do this, consider S * p j E. We have that p j is self-conjugate with constant return time T 0 > 0. There is a hyperbolic source/sink pair ξ ± ∈ S * p j E. In particular, let U ± ⊂ S * p j E be neighborhoods of ξ ± . Then there is C U ± so that for ξ ∈ S * p j E\U ± , d(G nT 0 ( p j , ξ), ξ ∓ ) ≤ C U ± e −|n|/C U ± , ∓n ≥ 0.
Therefore, applying [CG19, Lemmas 3.1, 3.2] to both A ± := S * p j \U ± , we have, using [CG18, Theorem 2], sup x∈B j (h δ ) |u h (x)| ≤ Ch − 1 2 | log h| −1/2 . (4.10) In summary, from (4.9) and (4.10) it follows that for joint eigenfunctions on the ellipsoid, one gets the global sup bound Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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Therefore, we need only show that one can replace T q, ,μ by T ,μ . For this, we follow the construction in [Sjö96, Propoosition 6.2] (see also [GT16, Proposition 2.2]). As above, when it comes to the application of stationary phase, we rescale α ξ → μα ξ and the small parameter ish = h/μ, but derivatives of the symbol acquire powers of μ. The same arguments then complete the proof.
Proposition A.2. With T ,μ as above, there exists μ 0 > 0, so that for all N > 0 there is C N > 0 so that for all μ > μ 0 there is h 0 (μ) so that for 0 < h < h 0 , Proof. After rescaling the fiber coordinates α ξ → μα ξ and settingh := h μ , we have and it is not difficult to show that by standard Cauchy estimates The extra μ j factor in (A.1) comes from the rescaling α ξ → μα ξ and the parametrix construction above (note that each α ξ -derivative of the rescaled symbols pulls out a factor of μ). Using (A.1) and Stirling's formula it then follows that for μ ≥ μ 0 there is a uniform constant C > 0 such that