QQuantum footprints of Liouville integrablesystems
V ˜u Ngo. c
San ∗ Abstract
We discuss the problem of recovering geometric objects from thespectrum of a quantum integrable system. In the case of one degreeof freedom, precise results exist. In the general case, we report on therecent notion of good labellings of asymptotic lattices.
Keywords :
Liouville integrable system, quantization, inverse spectral problem, MorseHamiltonian, semiclassical analysis, asymptotic lattice, good labelling
MS Classification :
Reviews in Mathematical Physics , DOI: 10.1142/S0129055X20600144
In this paper, we use for convenience the vocabulary of classical and quantummechanics, but one should keep in mind that inverse problems can be statedin a more universal way. Our general question is: “What footprints does aclassical system leave on its quantum counterparts, and are they sufficient torecognize the classical system that produced these footprints?”Imagine you’re walking in a snowy landscape, trying to take a good pho-tograph of a wild animal. A silhouette appears in the distance, you have noidea what beast it can be; you seize your camera, look at the small screen. . . ,and the ghost has just disappeared. Just as if it knew you wanted to captureit. The same goes for quantum particles, they want to delocalize when theyare observed. Yet we know they live there. On the other hand, the footprintson the snow, they are real, and stable. You can take your time and studythem, until, maybe, by clever induction, you find out what kind of animalwas standing there.What we have just described is an inverse problem: from the observationof a signal emitted by some device, can we recognize the device that has emit-ted the signal? If we hear the sound produced by various instruments playingthe same note C, can we tell the instrument without looking? This question ∗ Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France a r X i v : . [ m a t h . SP ] S e p an easily be turned into a mathematical problem, up to some simplifica-tions, and was, for instance, popularized by the “Can one hear the shape ofa drum” paper by Kac [18]. The “sound” is the superposition of all possible“frequencies” of a drum, i.e. the spectrum of the Laplace operator on a Eu-clidean domain Ω ⊂ R n , and the question is whether the shape of Ω (whichis the equivalence class of Ω under the action of the orthogonal group O ( n ) and translations) can be determined by the spectrum of the Dirichlet Lapla-cian. Natural variants of this question exist. One can consider Riemannianmetrics on a compact manifold X : can one recover the metric from the (dis-crete) spectrum of the corresponding Laplace-Beltrami operator? Or, backto quantum mechanics, we may consider a Schrödinger operator − (cid:126) ∆ + V on R n , and ask whether its spectrum determines the electric potential V .Both questions gave rise to an important literature, see for instance the ref-erences cited in [35, 7] and [13, 14], and have applications in non-quantumsituations, for instance in seismology, see [6, 9].In this paper we consider the so-called “symplectic case”. Given an ar-bitrary “semiclassical operator” ˆ H (cid:126) depending on a small parameter (cid:126) > (see the next section) and its symbol H , which is a smooth function on asymplectic manifold ( M, ω ) , can you recover the triple ( M, ω, H ) from the (cid:126) -family of spectra of ˆ H (cid:126) , where (cid:126) varies in a set accumulating at zero? Thisnatural question can be found for instance in [17]. Actually, Kac’s problem,in disguise, is such a symplectic inverse problem; in this case indeed, thesemiclassical Laplacian is simply (cid:126) ∆ , and the Hamiltonian H is the metricon the cotangent bundle M = T ∗ X induced by g . Semiclassical analysis is a general framework for obtaining a “geometric limit”from PDEs with highly oscillating solutions; the name “semiclassical” comesfrom the model situation where classical mechanics can be seen as a (singu-lar) limit of quantum mechanics, as Planck’s constant (cid:126) tends to zero. Ourconventions in this text are the following:By
Classical observables , or classical Hamiltonians, we mean smoothfunctions H ∈ C ∞ ( M ) on a symplectic manifold ( M, ω ) , the classical phasespace. For instance, M = R n with the canonical symplectic structure ω = d ξ ∧ d x . Each function H defines an evolution equation, the flow ofthe associated Hamiltonian vector field X H defined by ι X H ω = − d H .By Quantum observables , or quantum Hamiltonians, we mean a selfad-joint operator ˆ H on a Hilbert space H , and the Hilbert space itself must bethe quantization of a classical phase space M . Each quantum Hamiltoniangives rise to the evolution governed by the Schrödinger equation (cid:126) i ∂ t ψ = ˆ Hψ, ψ ( t ) = e it (cid:126) ˆ H ψ (0) . Stationary states are solutions of the form ψ ( t ) = e iλt (cid:126) u , where u ∈ H and ˆ Hu = λu . It is a fascinating subject tounderstand the relationships between the classical Hamiltonian flow and thequantum Schrödinger evolution.Two rigorous quantization schemes allow us to realise the above pic-ture: when M is a cotangent bundle, M = T ∗ X , one can use pseudo-differential quantization, see for instance [22]. When M is a prequantiz-able Kähler manifold, one can use Berezin-Toeplitz quantization, see for in-stance [20] (Berezin-Toeplitz quantization was later extended to general sym-plectic manifolds, see [3]). Berezin-Toeplitz and pseudo-differential quanti-zation are similar in many respects, and both benefit from the power andflexibility of microlocal analysis à la Maslov, Hörmander, etc..
Let ( M, ω ) be a 2-dimensional symplectic manifold. Let H : M → R a properMorse function. Following the usual Morse approach we will be interested inthe (singular) foliation of M by level sets of H . An important object, the Reeb graph G , is the set of leaves, i.e. connected components of level sets of H ; in a neighbourhood of a regular level set, G is a smooth, one-dimensionalmanifold. The smooth parts are the edges of the graph. Each critical pointof H contributes to a graph vertex, its degree is one for elliptic singularities(the vertex is then a leaf), and three for hyperbolic singularities.The Reeb graph turns out to be essential in the description of the spec-trum of a 1D quantum Hamiltonian. Let ˆ H := Op (cid:126) ( H (cid:126) ) be the quantiza-tion of a symbol H h := H + (cid:126) H + (cid:126) H + · · · on M . Let I ⊂ R be aclosed, bounded interval. Since H is proper, H − ( I ) is compact. In the caseof pseudo-differential quantization, we assume that H (cid:126) belongs to a symbolclass and is elliptic at infinity, see for instance [22]. Then the spectrum of ˆ H in any compact subset of int I is discrete. In this case, the inverse spectraltheory is well understood, and summarized by the following statement. Theorem 3.1 ([34, 19])
Let M I := H − ( int I ) . Suppose that H (cid:22) M I is asimple Morse function. Assume that the graphs of the periods of all trajec-tories of the Hamiltonian flow defined by H (cid:22) M I , as functions of the energy,intersect generically.Then the knowledge of the spectrum σ ( ˆ H ) ∩ I + O ( (cid:126) ) determines ( M I , ω, H ) . The proof of this theorem, like many of its kind, is divided in two steps.The first one is to recover the Reeb graph G of ( M I , H I ) from the spectrum.The second step is to prove that G , decorated with appropriate numericalinvariants that we can also recover from the spectrum, completely determinesthe classical system ( M I , ω, H ) . The last step was proven by Dufour-Toulet-Molino [11]. The first step was established in the pseudo-differential case3n [34], and in the case of Berezin-Toeplitz operators by Le Floch [19]. Itinvolved microlocal analysis in the (time/energy) phase space to be able toseparate the various connected components of G contributing to the sameregion of the spectrum.More recently, a new interpretation of this result has been proposed byseveral mathematicians, in particular Leonid Polterovich and the author.Suppose you add a generic non-selfadjoint perturbation to the quantum op-erator ˆ H . Then, the connected components of G , instead of leading to over-lapping parts of the spectrum — and hence potentially difficult to tell apart— should instead give rise to different complex branches of the spectrumof the non-selfadjoint operator. Thanks to the recent result by Rouby [30]explaining the non-selfadjoint version of Bohr-Sommerfeld quantization con-ditions, we believe that this conjectural interpretation should produce newrigorous results. Theorem 3.2 (Rouby [30])
Let P (cid:15) be an analytic pseudodifferential oper-ator on R or S of the form P (cid:15) = ˆ H + i(cid:15)Q , where ˆ H is selfadjoint withdiscrete spectrum, and Q is ˆ H -bounded.Then, near any regular value of the symbol H , with connected fibers, thespectrum of P (cid:15) is given by { g ( (cid:126) m ; (cid:15) ); m ∈ Z } , where g : C → C is holomor-phic and g ∼ g + (cid:126) g + (cid:126) g + · · · Moreover, g is the inverse of the action variable, and g ∼ H + i(cid:15) (cid:104) q (cid:105) + O ( (cid:15) ) . Rouby’s theorem is technically quite involved, because one has to takeadvantage of analyticity to fight non-selfadjoint instability (pseudo-spectraleffects), and usual C ∞ microlocal analysis is not strong enough for this. Noanalogue of this result for Berezin-Toeplitz quantization exists yet. However,very recent advances on the analyticity of the Bergman projection give somehope, see [31, 10, 4]. In view of Rouby’s theorem, one can notice that a particular case where an-alyticity is not required occurs when P (cid:15) is normal , i.e. the non-selfadjointperturbation Q commutes with the selfadjoint part ˆ H . More generally, anumber of results exist in the presence of a completely integrable quantumsystem , by which we mean the data of n pairwise commuting selfadjoint op-erators P , . . . , P n , when the phase space M is n -dimensional. In fact, even4or operators that are not quantum integrable but still have a completely in-tegrable classical limit, quite precise results can be obtained, for both directand inverse problems; see [16, 15], and references therein.This notion of quantum integrability parallels the usual Liouville integra-bility of classical Hamiltonians, where we dispose of n independent Poisson-commuting functions f , . . . , f n on M . Note that, near a regular level set ofthe joint map F := ( f , . . . , f n ) : M → R n , one has action-angle coordinates,due to the celebrated Liouville-Mineur-Arnold theorem, but Liouville inte-grability is more general : it allows for singularities where the action-angletheorem cannot apply.The natural multi-dimensional generalization of the Reeb graph is theleaf space of the “moment map” F , which is equipped with a natural integralaffine structure (see for instance [32]). The quantum analogue of this singularintegral affine manifold is the joint spectrum of the commuting operators P , . . . , P n . Hence, we are naturally lead to the following inverse problem:given an (cid:126) -family of joint spectra, can one recover the triple ( M, ω, F ) ?A first approach to this question is to restrict oneself to Hamiltonian sys-tems with many compact symmetries; namely the toric and semitoric cases.See [29] for a description of a general program of study, and conjectures. Having in mind the general inverse problem for quantum integrable systems,another angle of attack is to consider the regular part of the integral affinestructure, and exploit the lattice structure of the joint spectrum, which wasalready established by Colin de Verdière [5]. This leads to the notion of asymptotic lattices , whose systematic study was initiated recently in [8]. Al-though the initial goal of that paper was to recover from the quantum spec-trum a specific classical invariant, the rotation number , we believe that thegeneral setup should help understanding all invariants related to the integralaffine structure. In particular, we hope that it will allow to finally obtain acomplete result on the inverse theory of semitoric systems.Let B ⊂ R n be a simply connected bounded open set. Let L (cid:126) ⊂ B be adiscrete subset of B depending on the small parameter (cid:126) ∈ I , where I ⊂ R ∗ + is a set of positive real numbers admitting as an accumulation point. Hereis a slightly imprecise definition of asymptotic lattices (we don’t delve intomultiplicity issues and the details of the ( (cid:126) ∞ ) topology). Definition 5.1 (Asymptotic lattice [33, 8])
We say that ( L (cid:126) , I , B ) is an asymptotic lattice if L (cid:126) = G (cid:126) ( (cid:126)Z n ∩ U ) + O ( (cid:126) ∞ ) with G (cid:126) = G + (cid:126) G + (cid:126) G + · · · n the C ∞ ( U ) topology, where G : U → R n is a diffeomorphism on its image. The definition is motivated by the following older result.
Theorem 5.2 ([5, 1, 2])
Let P := ( P , . . . , P n ) be a Quantum integrablesystem. Let c ∈ R n be a regular value of the classical moment map F withconnected fiber. Then the joint spectrum of P near c is an asymptotic lattice . In order to recover the integral affine structure from the spectrum, one needsto recover the map G in the previous theorem. In order to do this, we claimthat it is enough to find a “good labelling” of all joint eigenvalues. By thiswe mean, to assign to each joint eigenvalue λ a n -uple of integers ( k , . . . , k n ) such that λ = G (cid:126) ( (cid:126) k , . . . , (cid:126) k n ) + O ( (cid:126) ∞ ) . In [8], we investigated the case of two degrees of freedom, n = 2 . To oursurprise, this question turned out to be more intricate (and more interesting)than what we initially thought.On the other hand, the process of finding a good labelling is elementary,and can be described algorithmically, which is important for the followingreason. The informal question “can one hear the shape of a drum” has twopossible interpretations. The minimal one is to prove injectivity of the mapsending a classical system to its quantum spectrum. In this case, the classicalsystem is determined by the quantum spectrum in a weak sense: two differentclassical systems cannot give rise to the same quantum spectrum. A strongerresult would be to obtain the classical system that produced the spectrum ina constructive way. Writing an algorithm contributes to the latter.The algorithm will be constructed in two steps. In the first one, thevalue (cid:126) is fixed, and the algorithm returns a candidate labelling λ (cid:55)→ ( k , k ) .However, this candidate does not have the required continuity property inthe variable (cid:126) . Hence we perform a second step where we consider now a fullsequence of values (cid:126) i → , and we correct the discontinuity by an inductivealgorithm in the variable i ∈ N . We don’t know whether a direct approach,in one step, would be possible. When constructing a good labelling, anotherdifficulty comes from the choice of a valid “origin” for the lattice. For thispurpose, the set of values of (cid:126) must be “dense enough” when accumulatingat zero. Values of the form (cid:126) = k for k ∈ N \ { } do not fulfill this require-ment, which is an issue for Berezin-Toeplitz quantization. However, in manyapplications, the choice of the lattice origin is irrelevant. Introducing the no-tion of “linear labelling” as a good labelling “modulo its origin”, we have thefollowing result. 6igure 1: The labelling algorithm Theorem 6.1
There exists an explicit algorithm such that the followingholds. Let ( L (cid:126) , I , B ) be an asymptotic lattice, where B ⊂ R . Let (cid:126) j ∈ I , j ≥ , be a decreasing sequence tending to . Then, from this data, the algo-rithm produces a linear labelling of the asymptotic lattice ( L (cid:126) , I (cid:48) , B ) , where I (cid:48) = { (cid:126) j , j ∈ N ∗ } . Below is the complete description of the first step. If ( k , k ) is a labelfor a point λ ∈ L (cid:126) , we shall denote this point λ = λ k ,k . The completealgorithm is as follows [8], and pictured in Figure 1.1. Choose an open subset B (cid:98) B , and fix c ∈ B .2. Choose a closest point to c . Label it as (0 , .3. Choose a closest point to λ , (in the set L (cid:126) \ { λ , } ). Label it as (1 , .4. Choose a closest point to λ , − λ , = λ , + ( λ , − λ , ) and label itas (2 , .Continuing in this fashion (if λ k − , is chosen, take λ k , to be a closestpoint to λ k − , + ( λ k − , − λ k − , ) ), label points λ k , , k > , untilthe next point lies outside of B .Label λ k , for negative k in the same way, starting from the closestpoint to λ , − λ , .5. Choose a closest point to λ , not already labeled and label it as (0 , .6. Label a closest point to λ , + ( λ , − λ , ) as (1 , .7. Use the points λ , , λ , to repeat the process in step 4, labelling asmany points λ k , as possible (if λ k − , is chosen, take λ k , to be aclosest point to λ k − , + ( λ k − , − λ k − , ) ).8. Label a closest point to λ , − λ , as λ , . Repeat steps 6-7 to labelall points λ k , .9. Continuing as above, label all points λ k ,k , k > which lie in thegiven neighborhood.10. Label a closest point to λ , − λ , as (0 , − .11. Repeat steps 6,7,8,9 with negative k indices.12. Finally, if the determinant of the vectors ( λ , − λ , , λ , − λ , ) isnegative, switch the labelling λ k ,k (cid:55)→ λ − k ,k (in order to make itoriented).This algorithm should be useful in several inverse spectral problems. Forinstance, in [8], Theorem 6.1 was used to prove that the classical rotationnumber of any Liouville torus can be recovered from the joint spectrum.From a quite different perspective, it could also be interesting to investigatethe proximity of our approach with topological data analysis and manifoldlearning . The detection of good labellings should allow the complete reconstruction ofthe integral affine structure, at least for its regular part. The next step wouldbe to globalize the notion of asymptotic lattice, and include singularities , toobtain quantized integral affine structures with singularities . For instance,the singular limit of the rotation number, as explained in [12], should be afeature of asymptotic lattices with focus-focus singularities.As far as inverse spectral theory is concerned, we certainly hope to usethe notion of asymptotic lattice to advance towards the
Spectral semitoricconjecture [28, 21]: can you detect the five symplectic invariants of a semitoricsystem on the joint spectrum (or asymptotic lattice)? In an ongoing workwith Le Floch, which intially focussed on the reconstruction of the twistingindex invariant, we finally expect to obtain not only the injectivity of the“semiclassical joint spectrum map” for simple semitoric systems, but also afull reconstruction procedure. A more general result should include multi-pinched tori (see [27, 24, 23]), and systems with non-proper circle momentmap [25, 26]. 8 eferences [1] A.-M. Charbonnel. Comportement semi-classique du spectre conjointd’opérateurs pseudo-différentiels qui commutent.
Asymptotic Analysis ,1:227–261, 1988.[2] L. Charles. Quasimodes and Bohr-Sommerfeld conditions for theToeplitz operators.
Comm. Partial Differential Equations , 28(9-10):1527–1566, 2003.[3] L. Charles. Quantization of compact symplectic manifolds.
TheJournal of Geometric Analysis , 26(4):2664–2710, Oct 2016. See also arxiv:1409.8507 (2017).[4] L. Charles. Analytic Berezin-Toeplitz operators. arXiv: 1912.06819,2019.[5] Y. Colin de Verdière. Spectre conjoint d’opérateurs pseudo-différentielsqui commutent II.
Math. Z. , 171:51–73, 1980.[6] Y. Colin de Verdière. A semi-classical inverse problem II: reconstruc-tion of the potential. In
Geometric aspects of analysis and mechanics ,volume 292 of
Progr. Math. , pages 97–119. Birkhäuser/Springer, NewYork, 2011.[7] K. Datchev and H. Hezari. Inverse problems in spectral geometry. In
In-verse problems and applications: inside out. II , volume 60 of
Math. Sci.Res. Inst. Publ. , pages 455–485. Cambridge Univ. Press, Cambridge,2013.[8] M. Dauge, M. Hall, and S. V˜u Ngo. c. The rotation number for quantumintegrable systems. Preprint hal-02104892, 2019.[9] M. V. de Hoop, A. Iantchenko, R. D. van der Hilst, and J. Zhai. Semi-classical inverse spectral problem for seismic surface waves in isotropicmedia: part I. Love waves.
Inverse Problems , 36(7):075015, 27, 2020.[10] A. Deleporte. Toeplitz operators with analytic symbols. Preprint hal-01957594 .[11] J.-P. Dufour, P. Molino, and A. Toulet. Classification des systèmes in-tégrables en dimension 2 et invariants des modèles de Fomenko.
C. R.Acad. Sci. Paris Sér. I Math. , 318:949–952, 1994.[12] H. Dullin and S. V˜u Ngo. c. Vanishing twist near focus-focus points.
Nonlinearity , 17(5):1777–1785, 2004.[13] V. Guillemin. Wave-trace invariants.
Duke Math. J. , 83(2):287–352,1996. 914] V. Guillemin, T. Paul, and A. Uribe. “Bottom of the well” semi-classicaltrace invariants.
Math. Res. Lett. , 14(4):711–719, 2007.[15] M. A. Hall. Diophantine tori and non-selfadjoint inverse spectral prob-lems.
Math. Res. Lett. , 20(2):255–271, 2013.[16] M. Hitrik, J. Sjöstrand, and S. V˜u Ngo. c. Diophantine tori and spectralasymptotics for non-selfadjoint operators.
Amer. J. Math. , 169(1):105–182, 2007.[17] A. Iantchenko, J. Sjöstrand, and M. Zworski. Birkhoff normal forms insemi-classical inverse problems.
Math. Res. Lett. , 9(2-3):337–362, 2002.[18] M. Kac. Can one hear the shape of a drum ?
The American Math.Monthly , 73(4):1–23, 1966.[19] Y. Le Floch.
Théorie spectrale inverse pour les opérateurs de Toeplitz1D . PhD thesis, Université Rennes 1, 2014. Thèse de doctorat, Mathé-matiques et applications,.[20] Y. Le Floch.
A brief introduction to Berezin-Toeplitz operators on com-pact Kähler manifolds . CRM Short Courses. Springer, Cham, 2018.[21] Y. Le Floch, Á. Pelayo, and S. V˜u Ngo. c. Inverse spectral theory for semi-classical Jaynes-Cummings systems.
Math. Ann. , 364(3):1393–1413,2016.[22] A. Martinez.
An introduction to semiclassical and microlocal analysis .Universitext. Springer-Verlag, New York, 2002.[23] A. D. Meulenaere and S. Hohloch. A family of semitoric systemswith four focus-focus singularities and two double pinched tori. arXiv:1911.11883, 2019.[24] J. Palmer, Á. Pelayo, and X. Tang. Semitoric systems of non-simpletype. arXiv:1909.03501, 2019.[25] Á. Pelayo, T. Ratiu, and S. V˜u Ngo. c. Fiber connectivity and bifurcationdiagrams for almost toric systems.
Journal of Symplectic Geometry ,13(2):343–386, 2015.[26] Á. Pelayo, T. Ratiu, and S. V˜u Ngo. c. The affine invariant of generalizedsemitoric systems.
Nonlinearity , 30(11):3993–4028, 2017.[27] Á. Pelayo and X. Tang. Vu Ngoc’s conjecture on focus-focus singularfibers with multiple pinched points. arXiv:1803.00998.[28] Á. Pelayo and S. V˜u Ngo. c. Symplectic theory of completely integrableHamiltonian systems.
Bull. Amer. Math. Soc. (N.S.) , 48(3):409–455,2011. 1029] Á. Pelayo and S. V˜u Ngo. c. First steps in symplectic and spectral theoryof integrable systems.
Discrete Contin. Dyn. Syst. , 32(10):3325–3377,2012.[30] O. Rouby. Bohr–Sommerfeld Quantization Conditions for Non-Selfadjoint Perturbations of Selfadjoint Operators in Dimension One.
Int. Math. Res. Not. , 2018(7):2156–2207, 01 2017.[31] O. Rouby, J. Sjöstrand, and S. V˜u Ngo. c. Analytic Bergman operatorsin the semiclassical limit. Preprint arXiv 1808.00199; to appear in
DukeMath. J. [32] D. Sepe and S. V˜u Ngo. c. Integrable systems, symmetries, and quanti-zation.
Lett. Math. Phys. , 108(3):499–571, 2017.[33] S. V˜u Ngo. c. Quantum monodromy in integrable systems.
Commun.Math. Phys. , 203(2):465–479, 1999.[34] S. V˜u Ngo. c. Symplectic inverse spectral theory for pseudodifferentialoperators. In
Geometric aspects of analysis and mechanics , volume 292of
Progr. Math. , pages 353–372. Birkhäuser/Springer, New York, 2011.[35] S. Zelditch. The inverse spectral problem. In