The semiclassical limit of a quantum Zeno dynamics

Motivated by a quantum Zeno dynamics in a cavity quantum electrodynamics setting, we study the asymptotics of a family of symbols corresponding to a truncated momentum operator, in the semiclassical limit of vanishing Planck constant ħ→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar \rightarrow 0$$\end{document} and large quantum number N→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\rightarrow \infty $$\end{document}, with ħN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar N$$\end{document} kept fixed. In a suitable topology, the limit is the discontinuous symbol pχD(x,p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\chi _D(x,p)$$\end{document} where χD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _D$$\end{document} is the characteristic function of the classically permitted region D in phase space. A refined analysis shows that the symbol is asymptotically close to the function pχD(N)(x,p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\chi _D^{(N)}(x,p)$$\end{document}, where χD(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _D^{(N)}$$\end{document} is a smooth version of χD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _D$$\end{document} related to the integrated Airy function. We also discuss the limit from a dynamical point of view.


Introduction
In the quantum Zeno effect, frequent projective measurements can slow down the evolution of a quantum system and eventually hinder any transition to states different from the initial one.The situation is much richer when the measurement does not confine the system in a single state, but rather in a multidimensional subspace of its Hilbert space.This gives rise to a quantum Zeno dynamics (QZD): the system evolves in the projected subspace under the action of its projected Hamiltonian.This phenomenon, first considered by Beskow and Nilsson [4] in their study of the decay of unstable systems, was dubbed quantum Zeno effect (QZE) by Misra and Sudarshan [38] who suggested a parallelism with the paradox of the 'flying arrow at rest' by the philosopher Zeno of Elea.Since then, QZE has received constant attention by physicists and mathematicians, who explored different aspects of the phenomenon.
From the mathematical point of view, QZD is related to the limit of a product formula obtained by intertwining the dynamical time evolution group with the orthogonal projection associated with the measurements performed on the system.It can be viewed as a generalization of Trotter-Kato product formulas [8,32,51,52] to more singular objects in which one semigroup is replaced by a projection.The structure of the QZD product formula has been thoroughly investigated and has been well characterized under quite general assumptions [17-21, 25-27, 36, 44, 45].QZE has been observed experimentally in a variety of systems, on experiments involving photons, nuclear spins, ions, optical pumping, photons in a cavity, ultracold atoms, and Bose-Einstein condensates, see [22] and references therein.In all the abovementioned implementations, the quantum system is forced to remain in its initial state through a measurement associated with a one-dimensional projection.The present study is inspired by a proposal by Raimond et al. [40,41] for generating a multidimensional QZD in a cavity quantum electrodynamics experiment.We briefly describe the proposal, skipping most of the non-mathematical details.
The mode of the quantized electromagnetic field in a cavity can be conveniently described in the Fock space representation.The Hamiltonian of the quantized field is that of a harmonic oscillator (with angular frequency  = 1) where x is the position operator and p = −ℏ    is the momentum operator on  2 (R).The operators x, p, and  h.o. are essentially self-adjoint on the common core (R), the Schwartz space of rapidly decreasing functions.The eigenfunction   of  h.o.represents a cavity state with  photons ( = 0, 1, 2, . . . ) and energy   = ℏ( + 1/2).
The cavity field undergoes a stroboscopic evolution alternating a short continuous time evolution  −  ℏ p given by a displacement operator, that without loss of generality is taken to be generated by p, and an instantaneous interaction (1.2) with atoms injected into the cavity to ascertain whether in the cavity there are less than  photons ( ≥ 1 is a chosen maximal photon number).
The quantum Zeno dynamics consists in performing a series of  < -measurements in a fixed time interval [0, ] at times   =  ,  = 0, . . ., , with period  = /.The intertwining of the continuous time evolutions and the projective measurements corresponds to the evolution operator   () =  <  −  ℏ p  <  .
Observe that since Ran  < ⊂  ( p) =  1 (R), we have [21] lim →∞   () =  <  −   /ℏ , in the strong operator topology, uniformly for  in compact subsets of R, where the Zeno Hamiltonian   is a rank- truncation of p: Hence the QZD establishes a sort of 'hard wall' in the Hilbert space: the state of the system evolves unitarily within the -dimensional Zeno subspace spanned by states with at most ( − 1) photons,  0 , . . .,   −1 .This hard wall prevents the state to escape from the Zeno subspace and induces remarkable features in the quantum evolution [40,41].
The question addressed in this paper is: What is the semiclassical limit of the Zeno Hamiltonian   and of its corresponding quantum dynamics? of operators is mapped in a twisted convolution product of symbols (called Moyal product), see e.g.[24,42].
If we describe the QZD in the phase space, in the semiclassical limit  → ∞, ℏ → 0 with the product ℏ =  kept fixed, we expect the motion to be confined in the classically allowed region.The level sets of the classical harmonic oscillator are circles centered at the origin of the phase space R  × R  .In qualitative terms, the hard wall can be viewed in the phase space as a circle with a radius ∝ √ ℏ.In the limit, the corresponding classically allowed region is the disk is the circle of radius √ 2.This is what Raimond et al. [40,41] called the 'exclusion circle': it separates  from the classically forbidden region where  h.o.> .
Let   (, ) =  (−∞, )  h.o.(, ) be the characteristic function of the disk .The first main result of the paper is the identification of the limit of the Weyl symbols  ℏ  < (, ) and  ℏ   (, ) of the projection operator  < and the Zeno Hamiltonian   , respectively.(The definition of the Weyl symbol of an operator is given in Definition 1.) Table 1.Summary of the operators and their semiclassical limits.

Remark 1.
Here A is the space of test functions introduced by Lions and Paul [33] as the completion of the smooth functions of compact support in the phase space  ∞  (R  ×R  ) under the norm Remark 2. Theorem 1 makes precise the heuristic expectation that the symbol  ℏ  < (, ) of the projection operator converges to the characteristic function   (, ) of the classically allowed region, and the symbol of the Zeno Hamiltonian  ℏ   (, ) converges to    (, ).The content of Theorem 1 is schematically summarised in Table 1.Remark 3. A plot of the Weyl symbol exhibits pronounced oscillations, also known as quantum ripples [6], in  = { h.o.(, ) < }.If the oscillations are smoothed out, then the graph of  ℏ   is asymptotically close to a 'tilted coin'.See Fig. 1 and Fig. 2.
We see that at the boundary  the symbols  We can now state our second main result.
Theorem 2 (Weak asymptotics at the boundary).Fix  > 0. For all  ∈  ∞  (R), and Remark 4. In order to zoom at , we need to integrate the symbols  ℏ  < and  ℏ   against (sequences of) compactly supported test functions that concentrate around .
Since  is invariant under rotations, without loss of generality we consider test functions that are also rotational symmetric.The idea is to consider, for  ∈  ∞  (R), the rescaling  −2 ( −2 ( 2 +  2 − 2)) that is nonzero in a region of order O() within the boundary .The blow-up scale that gives rise to a nontrivial limit is  = ℏ 1 3 .The reason for this choice will emerge in the following (see Section 3).Note that the space of test functions A in Theorem 1 does not depend on the details of the model.On the contrary, in Theorem 2 we integrate the symbols  ℏ  < and  ℏ   against test functions that concentrate around  in a suitable way.
Remark 5.The limits in Theorems 1 and 2 do not hold pointwise, in general.For instance, it is easy to show (by using the parity of the harmonic oscillator eigenfunctions) that The reader is invited to have a glance at Fig. 2. Inside the disk , the symbols oscillate with frequency of order O(), while in the classically forbidden region   = (R  × R  ) \  the symbols are exponentially suppressed.The monotonic behaviour outside the disk suggests that for (, ) ∈   the convergence to the limits may hold in a stronger sense.In fact, a slight adaptation of the proof of Theorem 2 shows that outside the disk,  ℏ  < and  ℏ   converge pointwise to the limit symbols.  and the Moyal bracket (that does depend on ℏ) [24,42].Hence, the semiclassical limit of the dynamics should encompass a simultaneous ℏ → 0 limit of the symbol (the generator of the dynamics) and the Moyal structure.
By Theorem 1, the symbol  ℏ   of the Zeno Hamiltonian converges as ℏ → 0,  → ∞, with ℏ =  > 0, to    (, ).Moreover, the Moyal bracket has an asymptotic expansion in powers of ℏ whose leading term (zero-th order) is the classical Poisson bracket.Hence, it is reasonable to expect that the limiting dynamics is well described by the Hamiltonian evolution (i.e.Poisson) in phase space where the Hamiltonian is the limit symbol    (, ).
However, in this naïve approach we immediately face an obstruction: the symbol    (, ) is not smooth, and hence it is not possible to write Hamilton's equations of motion!If we insist in writing, formally, Hamilton's equations, we get where  = √︁  2 +  2 .The Dirac delta  √ 2 () arises as distributional derivative of the step function.We stress again that the above expressions are formal: the Hamiltonian is discontinuous at , and its vector field in (♦) is singular.We can now look at the corresponding phase portrait.First, the Hamiltonian vector field is zero outside the closure of the disk .Thus, all points there are equilibrium points.If the particle is in , then the equation of motions are  = 1,  = 0, and the particle moves with constant momentum It is thus proceeding at a constant velocity along the -axis.When it hits the boundary , the evolution is given by the singular contributions, proportional to the delta functions: Heuristically, these equations would correspond to a field tangential to the boundary of  that yields a motion along the circle  at 'infinite' speed.The particle reappears on the other side of the boundary (with the same momentum  =  0 ) and resumes its motion along the -axis at a constant velocity.The collision at the edge  thus realizes, in this semiclassical picture, a reflection around the -axis of the phase space, transforming An interesting interpretation of the semiclassical limit of the Zeno dynamics is as follows.In the limit dynamics, the points (, ), (−, ) on the cirle  ⊂ R  × R  are identified.Hence, one can think of the  → ∞, ℏ → 0 limit, with ℏ = , as yielding a change of topology: the dynamics on the disk becomes a motion on the sphere!We emphasise again that all this is formal, although very close to what was observed in [40,41], and called 'phase inversion mechanism'.The function    (, ) is not smooth and therefore it is not the generator of a classical Hamiltonian dynamics.
We know, however, by Theorem 2, that the symbol  ℏ   (, ) is asymptotically close to a smoothed version For each , it makes sense to consider the Hamiltonian system generated by   (  )  (, ), (1.17) This is a family of well-defined Hamilton equations and we can expect, for large , the solutions of (♣) to be 'close' to the sought semiclassical limiting dynamics.
We give here a sketch of an argument showing that for large , the solutions of (♣) behave as the formal solutions of the singular problem (♦).The equations of motions from (♣) are where (1.19) This can be seen, in Fourier space, from the identity We conclude that the Hamiltonian vector field generated by   (  )  (, ) converges to the singular vector field generated by    (, ).The component of the field containing 3 ) and generates a motion at speed ∝  2 3 , which becomes 'infinite' in the singular limit.Fig. 3 shows a comparison of the phase portraits of the Hamiltonian dynamics generated by the Weyl symbol  ℏ   (, ), the smooth Hamiltonian   (  )  (, ) and the discontinuous function    (, ).Note the effective change of topology in the limit singular case that results from the instantaneous motion along the circle .

Spectral analysis of the
. This is a Jacobi matrix about which we have very precise spectral information (characteristic polynomial, eigenvalues and their counting measure) for all .
Proposition 1.For all  ≥ 1,  are the zeros of  ℏ  (), we define the eigenvalues counting measure   of the Zeno Hamiltonian   to be the nonnegative measure that puts weight 1/ on each eigenvalue of   (the  (  )  's).From well-known results on Hermite polynomials [14] it follows that the measure   weakly converges to the semicircular density in the simultaneous limit ℏ → 0,  → ∞ with ℏ asymptotically fixed.See Figure 4. Proposition 2. For all continuous bounded functions  , as  → ∞, ℏ → 0, with the product ℏ converging to  > 0. Remark 6.The semicircular spectral distribution can be obtained formally from the semiclassical limit of Theorem 1. Indeed, in the limit symbol    (, ) of the Zeno Hamiltonian is concentrated on the disk  of radius √ 2.Semiclassically, the density of the eigenvalues is the fraction of the phase space volume with energy between  and  + : =   ().

Proof strategy and relations to other works.
When  is large, the symbols  ℏ  < and  ℏ   are highly oscillating smooth functions.As discussed in Remark 5, looking for a global semiclassical limit in a pointwise sense is hopeless.It turns out that the sought convergence of the symbols holds in a weak sense if the set of test functions is chosen to be A.
The proofs presented in this paper are based on the following observations: (1) The asymptotics of integrals of the Weyl symbols  ℏ  < (, ) and  ℏ   (, ) against functions on the phase space is related to the pointwise asymptotics of the Fourier transforms F 2  ℏ  < (, ) and ) is a sum of  terms (cross products of Hermite functions), see Eq. (2.13).However, thanks to the Christoffel-Darboux formula (Lemma 3) this sum can be expressed in terms of the -th and ( − 1)-th Hermite functions only.Hence, studying the large  asymptotics with ℏ =  amounts to study the large degree asymptotics of Hermite functions.This is a well-studied topic in the theory of orthogonal polynomials from which we can freely borrow explicit asymptotic formulae.So, to prove the convergence of the symbols we will show the convergence of the Christoffel-Darboux kernel along with its derivatives to the sine and Airy kernels (in the formulation presented in the book of Anderson, Guionnet and Zeitouni [2]) (3) The symbol  ℏ   (, ) is 'asymptotically close' to  ℏ  < (, ) in the dual space A ′ (Proposition 7).This is suggested by the heuristic observation that, in the limit ℏ → 0, the algebra of observables should become commutative.What we gain is that, once we know the asymptotics of  ℏ  < (, ) we can directly deduce the asymptotics of  ℏ   (, ).
The seminal paper by Lions and Paul [33] on the semiclassical limit of Wigner measures, and the more recent developments [1,3,11,23] were instrumental in our study.
We mention that the symbol  ℏ  < (, ) of the orthogonal projection  < studied in the present paper has close connection to the fuzzy approximation of two-dimensional disk proposed by Lizzi, Vitale and Zampini [34,35].A fuzzy space is an approximation of an abelian algebra of functions on an ordinary space with a sequence of finite-rank matrix algebras, which preserve the symmetries of the original space, at the price of noncommutativity.Eq. (1.5) of Theorem 1 is the precise mathematical statement behind the numerical results of [34,35].To our knowledge, the finer asymptotics of Theorem 2 is a new result, that has not been observed numerically neither.
The convergence of symbols of projection operators to the characteristic function of the classically allowed region is folklore in theoretical physics.In recent years, there has been an explosion of results on the asymptotics of the Christoffel-Darboux kernel for orthogonal polynomials on the real line, especially in connections to eigenvalue statistics of random matrices and integrable probability models [14,30,37,43].The interest to these asymptotics in the theoretical and mathematical physics community has been mostly motivated by applications to the number statistics of non-interacting fermions.The asymptotics at the 'edge' has been also investigated at various levels of rigour.See, e.g.The semiclassical structure of quite general cut quantum observables ΠΠ (with Π a spectral projection and  a pseudodifferential operator) was studied by Hernandez-Duenas and Uribe [29].Their results is consistent with Theorem 1 of the present paper.Those authors also studied the unitary dynamics generated by the cut quantum observables (the analogue of the Zeno Hamiltonian   of the present paper), and numerically found fascinating phenomena of splitting of the wave-packets and infinite propagation speed near the boundary of the classically allowed region.This is again consistent with our findings.
Given the universality results on the asymptotics of orthogonal polynomials and random matrices [14], we expect that Theorem 1 is valid for a rather large class of symbols associated to finite-rank orthogonal projections.The recent paper by Deleporte and Lambert [15] suggests that Theorem 2 would be valid as long as the gradient of the confining potential does not vanish at the points of classical inversion of motion.In any case the statement of analogues of Theorem 2 should depend on the geometry of the level sets of the corresponding classical Hamiltonian function.Further study is in progress.
1.5.Outline of the paper.The structure of the paper is as follows.In the next section we recall some preliminary background material, introduce a precise presentation of the model and provide the calculation of the symbols.In Section 3 we discuss the different scaling limits in Theorems 1 and 2. Section 4 is entirely devoted to the proofs of the main technical results, and of Theorems 1, 2 and 3.The paper includes two appendices.In Appendix A we collect known formulae on the Hermite functions.Appendix B contains the definition and a few properties of the sine and the Airy kernel.

Notation, preliminaries, Weyl symbols and kernels
We first introduce some notation and preliminary notions that we use throughout this work.For a linear operator  on  2 (R) we write   (, ) to indicate that  has kernel (, ) ∈  2 (R × R).In this paper all kernels are continuous.For ,  linear operators, we use the notation [ , ] :=  −  for the commutator.Let  be the linear operator defined, for  ∈  1 (R), by the formula (  ) () =    ().We have (2.4) where  0 (R  × R  ) is the usual space of continuous functions tending to zero at infinity.
A is a Banach algebra with the following properties (see [33]): - Let A ′ be the dual of A. From the Parseval identity it follows that (2.5) Definition 1.Given a number ℏ > 0, the Weyl symbol of the operator  (, ) is defined as Equivalently,  ℏ  is defined by the identity and, by Plancherel's theorem, We will often use the shorthand We recall that the Weyl symbol of the product of two operators ,  is not the ordinary product of the symbols  ℏ  ≠  ℏ   ℏ  , unless  and  commute.The noncommutative Moyal product ♯ is defined as the composition law that does the job: Definition 2. Given two linear operators  and  on  2 (R) with Weyl symbols  ℏ  and  ℏ  respectively, the Moyal product is defined as follows: Recall that the normalised eigenfunctions of the harmonic oscillator operator  h.o. in (1.1) are the Hermite functions (2.10) Proposition 3 (Integral kernels). (2.13) Proof.Formula (2.13) follows directly by the definition of the Hermite functions.Formula (2.14) is obtained by a direct calculation using the three-term recurrence (A.8), while (2.15) have explicit representations in terms of associated Laguerre polynomials (this is a manifestation of the so-called 'Laguerre connection' [24, §1.9]).
Proof.A consequence of the following formula by Groenewold [28] valid for all  ≤  (we write the formula as in [11, Eq. ( 30)]), (2.20) The symbols  ℏ  < and  ℏ   are rapidly decreasing functions in (R  × R  ).Notice that  ℏ  < is rotational symmetric.It may be convenient in the following to consider  ℏ  < and  ℏ   as complex-valued functions defined on the complexification C  × C  of the real phase space.They are entire functions in both variables  and .

Scaling limits
In this section we provide an heuristic explanation of the different scaling limits in Theorems 1 and 2. The following discussion is somewhat breezy.For a more careful exposition of similar ideas, see [7,9,15].
Recall that  <   (, ) and At scale ℏ, the kernel has an asymptotic limit that can be identified as follows.We start by writing the rescaled kernel in terms of the conjugation of a unitary transformation on the operator.If we conjugate the projection  < by the scaling unitary  ,ℏ in (2.2), we get that the kernel of the rescaled projection is the rescaled kernel: The action of the rescaled harmonic oscillator operator on a function  in its domain is 2  2 , for ℏ → 0,  → ∞, with ℏ = .We recall the following result adapted from [7,Lemma A.5].

), and its unique self-adjoint extension has only absolutely continuous spectrum
where  sine is the sine kernel (B.3).
From (3.1), we see that a rescaling ℏ in the Hilbert space  2 (R) corresponds to zooming at scale ℏ 0 in the phase space.The precise statement of (3.3) is Proposition 5 in Section 4.
To explain how a different asymptotics arises at the boundary , we need to study the rescaled harmonic oscillator operator in a neighbourhood of the classical turning points  = ± √ 2.Let us zoom at scale ℏ  , with  > 0 an exponent to be determined: If we choose  = 2 3 we then expect that for ℏ → 0,  → ∞, with ℏ = , where û is the position operator and   = 2 1 2  1 6 is a constant given in (B.2).Thus, the limit at the edge is related to the Airy differential operator for which we have the following spectral result, adapted from [7,Lemma A.7].
Lemma 2. The operator −  2  2 +  3  û is essentially self-adjoint on  ∞  (R), and its selfadjoint extension has only absolutely continuous spectrum where  Ai is the Airy kernel (B.4).
The precise statement of (3.5) is Proposition 6 in Section 4.
From (3.1), we see that a rescaling ℏ 2 3 at the edge in the Hilbert space  2 (R) corresponds to zooming at scale ℏ around the boundary  in the phase space.This explains the rescaling in Theorem 2.

Proofs
The proofs presented in this section are based on the following three observations: (1) The asymptotics of the Weyl symbols  ℏ  < ,  ℏ   is related (by Fourier transform in the second variable F 2 ) to the asymptotics of the integral kernels   (, ) and   (, ).
(2) The kernel   (, ) is a sum of  terms (cross products of Hermite functions), see Eq. (2.13).However, thanks to Christoffel-Darboux formula this sum can be expressed in terms of the -th and ( − 1)-th Hermite functions only.Hence, studying the large  asymptotics with ℏ ∼  amounts to study the large degree asymptotics of the Hermite functions.(3)  ℏ   (, ) is 'asymptotically close' to  ℏ  < (, ) for  → ∞, ℏ → 0 with ℏ =  > 0, see Proposition 7, therefore once we know the asymptotics of   (and hence of  ℏ  < ) we can directly deduce the asymptotics of  ℏ   .

Asymptotics of the kernels.
By telescoping the sum in (2.13) and using the three-term relation (A.7), we get the celebrated Christoffel-Darboux formula [46].Lemma 3.For all ,  ∈ R, Thus, the large- asymptotics of the Christoffel-Darboux kernel   (, ) boils down to the classical subject of large degree asymptotics of orthogonal polynomials.A consequence of the Plancherel-Rotach asymptotics for  ℏ  () (Equations (A.10)-(A.12))are the following asymptotic behaviours of the kernel   (, ).
Proposition 5 (Bulk asymptotics of the Christoffel-Darboux kernel).Suppose that ℏ = ℏ  is the sequence defined by the condition ℏ = .Then, for any compact sets  ⋐ R and  ⋐ R 2 , and for any ,  ∈ {0, 1}, there exists a constant  > 0 such that (4.Proposition 6 (Edge asymptotics of the Christoffel-Darboux kernel).Suppose that ℏ = ℏ  is the sequence defined by the condition ℏ = .For any compact set  ⋐ C 2 , and for any ,  ∈ {0, 1}, there exists a constant  > 0 such that where   is given in (B.2), and  Ai is the Airy kernel (B.4).
The scaling limits of the Christoffel-Darboux kernel to the sine and Airy kernel are well-known results.It is perhaps less known that the local uniform convergence can be promoted to their derivatives as well.We outline here a proof, adapting the presentation of the book by Anderson, Guionnet and Zeitouni [2,Chap. 3].
Notation.From now on, (ℏ  )  ≥1 is the positive sequence such that product ℏ   = , where  is a fixed positive number.We will write ℏ instead of ℏ  for short, when no confusion arises.We will also use the following shorthand   ,  0 , (, ) :=   ( 0 + ,  0 + ).It is useful to get rid of the removable singularity  =  in   ,,ℏ .Toward this end, noting that for any differentiable functions  ,  on R, we deduce that

𝑑𝜆
where we used relation (A.9) in the last equality.
We can now insert the uniform Plancherel-Rotach asymptotics (A.10)-(A.11),perform the integrals and use elementary trigonometric identities to conclude the proof of (4.5).
To prove the  1 -local uniform convergence, we start by taking the derivative(s) of the Christoffel-Darboux kernel         ,,ℏ (, ).This entails computing the derivatives of Hermite functions.Now the trick is to write the derivative  ℏ  ′ as a combination of Hermite functions (not differentiated) using again formula (A.9).Hence, the local uniform asymptotics of  ℏ  ′ can be read off from the Plancherel-Rotach asymptotics (A.10)-(A.11) of  ℏ  .The proof of the  1 -convergence is therefore a simple modification of the proof of (4.5).
To prove (4.6), we use again (A.9) to write the kernel as If we set (4.7) By the Plancherel-Rotach asymptotics (A.12), for any compact set  ⋐ C, (4.9) lim Since the functions Ψ ℏ  are entire, the above locally uniform convergence entails the uniform convergence of Ψ ℏ  ′ to Ai ′ on compact subsets of C (a standard application of Cauchy's integral formula).
By the very same argument, each finite- kernel   , 3 is analytic, and hence their derivatives converge to the derivatives of the Airy kernel.The proof is complete.□ Remark 9. Since  ℏ  < ∈ (R  × R  ), we have that the Fourier transform are rapidly decreasing functions too, F 2  ℏ  < ∈ (R  × R  ).On the contrary, is not integrable in R  × R  and this tells that we cannot get in (4.2) a convergence stronger than uniform on compact subsets.
In order to prove Theorem 2 we will also need to show that the Airy kernel on the antidiagonal is dominated by and integrable function., for all  ∈ R.
(For the first we use known bounds [31] on the largest zero of the Hermite polynomial of degree ; the second is true because Ψ ℏ  is a polynomial times a Gaussian.)The above mentioned theorem of Sonin and Polya (see the formulation in [2, Lemma 3.9.31])allows to conclude that log Hence, , for all  ≥ .A short calculation shows that 0 <  < 1 2 ℏ 1 3 (2) . Since Ψ ℏ  () → Ai() pointwise, with different constants we have , for  ≥ 0.
We can now estimate, for  > 0, .
where ,  denote different constants in each line.In the second to last step we used the uniform bound (4.12) on R and the integrable bound (4.13) on [0, ∞).
Proof of Proposition 7. Let  ∈ A. From Plancherel's theorem We can estimate Similarly, The convergent sequence ℏ is bounded from above.The proof of the uniform boundedness of the symbols is complete.
With the help of Lemma 5, similar calculations are used in the proof of (4.15),We can now prove Theorem 2.

1. 1 .Figure 1 .
Figure 1.Plot of the symbol  ℏ   in the phase plane (, ).Here  = 17 and  = 2. Already for such a small value of , the graph of the symbol resembles a (rippled) tilted coin in the disk  and zero outside.

Figure 3 .
Figure 3. Phase portraits for the Hamiltonian dynamics.The red solid line is the boundary  of the disk.Here  = 2.

Figure 5 .
Figure 5.The kernel at the edge   ,
ℏ  < and  ℏ   develop a jump, for large .The second main result of the paper concerns a finer asymptotics of  ℏ  < and  ℏ

1.2. Semiclassical limit of the quantum Zeno dynamics. The
quantum dynamics in phase space is ruled by two elements: the Weyl symbol of the Zeno Hamiltonian  ℏ Zeno Hamiltonian   .The matrix representation of   (in the Hermite basis  ℏ   ∈N , see Appendix A) is the  ×  complex Hermitian matrix Figure 4. Illustration of Proposition 2. The histogram of the eigenvalues of the Zeno Hamiltonian   for  = 2000, and ℏ =  = 2 is compared with the semicircular density   () = 1 the two operators have equal characteristic polynomial.The claim now follows from a result for general orthogonal polynomials on the real line due to Simon [47, Prop.2.2].□ + of Eq. (B.1).where ℎ  is the Hermite polynomial of degree , see Appendix A. In particular, the eigenvalues of   are the  (simple and real) zeros of the Hermite function  ℏ  ().Proof.is unitarily equivalent to  < x < (see equations (A.7)-(A.8)),andso(  ) 2      − 2is the -th Hermite polynomials, see Appendix A. Consider the orthogonal projection < =  (−∞,ℏ ) ( h.o. )   =  < p < = p < − [ p,  < ] < .