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quantum Hamiltonian H is integrable in this naive sense when there exist n ... algebraic aspects of integrability in classical mechanics to the quantum case, as.
On the classical and quantum integrability of Hamiltonians without scattering states A. Enciso∗

D. Peralta-Salas†

Depto. de F´ısica Te´ orica II, Universidad Complutense, 28040 Madrid, Spain

Abstract In this paper we establish that every quantum Hamiltonian without scattering states possesses a complete family of conserved quantities, independently of the dimension of the system. This result leads to a comparison between generic properties of classical and quantum integrable systems. Several relevant examples and an application to the statistical distribution of energies are discussed. As a spin-off, we obtain additional support for the Berry–Tabor conjecture without taking into account the semiclassical limit.

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Introduction and definitions

Integrability has always played a significant role in the qualitative study of classical and quantum dynamical systems. In the quantum case, there are two main approaches to this topic, each one corresponding to a different area of physics. In mathematical physics, one usually deals with concrete quantum models (with a possible dependence on free parameters) and tries to compute first integrals of the Hamiltonian explicitly with the aid of group theory [1, 2]. In quantum chaos one is frequently interested in the spectral properties characterizing chaotic or integrable Hamiltonians, and models for which no exact information is available are usually investigated with the help of the semiclassical approximation and numerical computations [3, 4]. In this paper we will obtain some exact information on quantum integrability by imposing conditions on the spectrum of the Hamiltonian instead of restricting the study to concrete models, thus trying to combine both viewpoints. The results obtained will also lead to an enlightening comparison between classical and quantum mechanics and to an exact study of the genericity of Poisson statistics for Hamiltonians without scattering states. A classical Hamiltonian H(x, p) is a function from a 2n-dimensional phase space into the real numbers. The concept of classical integrability was first precisely stated by Liouville, who defined that a classical Hamiltonian H(x, p) is ∗ [email protected][email protected]

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integrable when it possesses n functionally independent first integrals in involution with a certain degree of regularity, e.g., being smooth or analytic. The complexity of the dynamics defined by H(x, p), i.e., the regular or chaotic behavior of the orbits of the Hamiltonian, strongly depends upon its integrability. Particularly, the dynamics is not considered chaotic when H(x, p) is integrable. A quantum Hamiltonian H is a self-adjoint operator acting on a separable Hilbert space H. The naive definition of quantum integrability, which appears in many research papers, stems from that of classical integrability. Namely, a quantum Hamiltonian H is integrable in this naive sense when there exist n independent linear operators Ti (i = 1, . . . , n) which commute among them and with the Hamiltonian, n being the dimension of the quantum system. It is well known that several technical nuances prevent this definition from being fully satisfactory [5]. Perhaps the most serious problem is the concept of functional independence of the conserved quantities, at least in the preferred case when the first integrals are self-adjoint. In fact, a theorem of Von Neumann [6] ensures in this case that both the Hamiltonian and its first integrals can be written as functions of another self-adjoint operator A, H = f (A) , Ti = fi (A) , and thus the functional independence of these operators is impossible. This difficulty is usually overcome by a slight modification of the concept of independence entering the naive definition of quantum integrability. For instance, one may suppose that the quantum Hamiltonian H and its first integrals Ti are well defined, smooth functions of the position and momentum operators, i.e, H = g(X, P) , Ti = gi (X, P) , and impose that these functions gi be functionally independent. Many other modifications of the naive concept of quantum integrability are common in the literature. For instance, a definition of quantum integrability based on the semiclassical limit [7] is quite common in the context of quantum chaos, especially in the mathematical literature. Another possibility would be to choose a certain realization of the Hilbert space, say the space of square-integrable functions L2 (Rn ), and impose that the first integrals be algebraically independent differential operators. Note that, of course, these definitions of integrability are not fully equivalent. In this paper, we will use a similar modification of the naive concept of quantum integrability. We will say that a Hamiltonian operator H is quantum integrable when it possesses n commuting first integrals Ti which can be transformed into harmonic oscillators in the following way. Namely, there exists a unitary isomorphism between the Hilbert space H of H and the space of square-integrable functions on Euclidean space Rn which transforms the con-

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served quantities Ti into the number operators   1 2 ∂2 1 2 2 Ni = (Pi + Xi − 1) = + xi − 1 − 2 2 ∂x2i

(1)

associated to each coordinate and H, into a smooth function of the momentum and position operators given by the canonical quantization of an integrable classical Hamiltonian. We will denote this quantum Hamiltonian by H(X, P), and its classical counterpart, by H(x, p). We have chosen units in which ~ = 2m = 1. Remember that the position and momentum operators act on L2 (Rn ) as Xψ(x) = x ψ(x) , Pψ(x) = −i∇ψ(x) . In more physical terms, this definition means that a Hamiltonian is integrable when there exists an orthonormal basis of the Hilbert space in which this Hamiltonian is the canonical quantization H(X, P) of an integrable classical Hamiltonian H(x, p), where x and p take values in the Euclidean space Rn . Furthermore, this Hamiltonian can be globally taken to be in Birkhoff’s normal form, i.e., H(x, p) = f (x21 + p21 , . . . , x2n + p2n ). Of course, H and H(X, P) are physically equivalent, since they are connected by a unitary transformation. A similar definition of integrability and action-angle variables had previously appeared in the literature in the context of finite-dimensional spin dynamics [8]. In this paper we will use the above notion of quantum integrability, whereas we will use Liouville’s definition of classical integrability. Perhaps the greatest strength of the above definition of quantum integrability is that it extends the algebraic aspects of integrability in classical mechanics to the quantum case, as the following digression will show. First of all, it should be noted that the eigenfunctions of H(X, P) are also eigenfunctions of each number operator Ni because it commutes with the Hamiltonian H(X, P). Thus it is not difficult to see that every eigenfunction of H(X, P) factorizes as n Y ψ(x) = ψi (xi ) , (2) i=1

each ψi being a normalizable eigenfunction of a one-dimensional harmonic oscillator, i.e., d2 ψi (x) − + x2 ψi (x) = (2ni + 1) ψi (x) . (3) dx2 It is important to remark the necessity of writing the Hamiltonian of the system in the appropriate basis to have this factorization. It can be checked that the algebraic features of the definition of quantum integrability used in this paper closely parallel those of Liouville integrability in classical mechanics. In the classical framework, there exists a local change of coordinates (i.e., action-angle variables), which cannot be computed algorithmically, transforming the dynamics of the system into linear flows on tori, 3

that is, into decoupled harmonic oscillators. In the quantum case, a change of orthonormal basis, which cannot be obtained using an algorithmic procedure either, transforms the dynamics into that of decoupled harmonic oscillators as well, as Eqs. (2) and (3) show. Note also that in both cases the transformation of the dynamics into decoupled harmonic oscillators is associated to separability, of the Hamilton–Jacobi equation (“into sums”) in the classical framework and in the sense of partial differential equations (“into products”) in the quantum one. From this identification it follows that the number operators Ni are, as is well known, the quantum analogues of the action variables in classical mechanics. Thus expressing the operator H in terms of the number operators Ni via a unitary transformation is equivalent to expressing the quantum Hamiltonian in terms of quantum action-angle variables. It is clear that Ni are the canonical quantization of the action variables x2i +p2i of the classical Hamiltonian H(x, p); however, it is also clear that H itself cannot be obtained as the canonical quantization of this classical Hamiltonian. In fact, if H has a classical counterpart, the existence of a unitary transformation expressing H in terms of the number operators Ni does not imply that there exists a canonical transformation expressing the classical analogue of H in terms of its action variables. Actually, these action-angle variables cannot possibly exist for the classical analogue of H, as we will explicitly see later on, since there is no general correspondence between classical canonical transformations and quantum unitary isomorphisms. This topic is treated for one-dimensional systems in [9, 10]. This paper is organized as follows. In Section 2 we discuss a sufficient integrability condition based on the spectral properties of the Hamiltonian, not on its symmetries. Namely, a Hamiltonian without scattering states is integrable independently of the dimension of the system. This condition allows us to prove that the non-genericity of classical integrability does not extend mutatis mutandis to the quantum case. Partial results due to Crehan [11] and Weigert [5] will be improved, and several concrete examples will be tackled. In Section 3 we discuss the statistical distribution of the normalized differences of energies for Hamiltonians without scattering states, and the appearance of Poisson statistics. Since these Hamiltonians are integrable, further confirmation for the Berry–Tabor conjecture is provided without making use of the semiclassical limit. This approach follows that of Rela˜ no et al. [12], but the test we perform is not numerical. In addition, we confirm that the exceptions to the Berry–Tabor conjecture found by these authors are non-generic. This paper concludes with some final remarks, presented in Section 4.

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2 2.1

Integrability of Hamiltonians without scattering states Main result

Let us start recalling some concepts, which will follow the definitions in Ref. [13]. The required concepts from spectral theory shall be kept down to a minimum to simplify the exposition. It is well known that the Hilbert space H of a Hamiltonian H decomposes orthogonally as a direct sum of (the closed span of) its bound states Hpp and its scattering states Hcont . Recall that the interpretation of the eigenfunctions of the Hamiltonian as bound states and the functions corresponding to the continuous spectrum as scattering states stems from the physically relevant situation of absence of singular continuous spectrum [14]. For simplicity we will always use “scattering states” and “states in Hcont ” as synonyms. This leads to a decomposition of the spectrum of H, denoted by σ(H), in terms of (the closure of) the energies of the bound states of the Hamiltonian, which are at most a countable set and will be denoted by σpp (H), and the energies corresponding to the scattering states, σcont (H). This decomposition reads σ(H) = σpp (H) ∪ σcont (H) . When the Hamiltonian H has no scattering states (i.e., its continuous spectrum σcont (H) is empty), the spectrum of H is the closure of its eigenenergies. Then H is said to have pure point spectrum, although the name can be misleading since the spectrum could, in fact, include an arbitrary number of accumulation points or closed intervals of the real line. For instance, spectra like σ(H) = [−10, −8] ∪ [−5, −3] ∪ {i−2 }∞ i=1 are possible for Hamiltonians with pure point spectrum. In particular, it is not true that these Hamiltonians have discrete spectrum or can only accumulate at infinity. Physically relevant examples of Hamiltonians presenting similar behaviors can be found in Refs. [15, 16]. These operators describe weak perturbations of the Stark effect and the phase transition (metal to insulator) in the Anderson model. It is also known [17] that one can construct Hamiltonians (possibly with continuous spectrum) such that their point spectra are dense in any subset of R. One should recall, however, that an important example of Hamiltonians without scattering states are those Hamiltonians whose spectrum is discrete. This is the most interesting case for applications, although certainly not the general situation. To simplify the exposition, we shall assume that there are no eigenenergies of infinite degeneracy, as will certainly happen in every physically interesting situation. It is important to stress, however, the technique we shall use to prove the main result in this case can be easily modified to deal with infinite degeneracy. The main result that we shall prove in this section is that every Hamiltonian H with pure point spectrum is integrable, independently of the dimension n of

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the system. To establish this fact, let (Ei )∞ i=0 be the sequence of eigenvalues of H, where each energy appears as many times as its multiplicity, and let H be the Hilbert space of the system. We will construct a unitary isomorphism between H and the space of square-integrable functions on Rn , L2 (Rn ), that sets H into a form H(X, P) in which its integrability is obvious. As L2 (Rn ) is isomorphic to H, this can be understood as a change of orthonormal basis in H. Our approach will be based on a reconstruction of the spectrum of the Hamiltonian H using harmonic oscillators. As far as we know, this idea goes back to Weigert [5], who made use of it to point out several deficiencies in the naive definition of quantum integrability, as we have recalled in Section 1. Later on, Crehan [11] used a similar technique to study a related problem more explicitly. The main problem of his construction was that he needed to assume a certain growth law of the energy levels to guarantee the convergence of his method. Furthermore, he was forced to assume implicitly that the spectrum of H had no finite accumulation points to interpolate the sequence of energies using entire functions. Our approach will remove these two assumptions by using a less restrictive construction based on functional calculus, allowing us to use a wider class of interpolation functions. Eliminating these restrictions is essential to tackle the study of some general properties of Hamiltonians without scattering states, as will be done in the following sections. Let X and P be the position and momentum operators on L2 (Rn ) and consider the number operators Ni , cf. Eq. (1). It is clear that these number operators commute among them, since they act on different coordinates, and their spectrum is pure point and given by the set of non-negative integers N0 = {0, 1, 2 . . . }. Let φ be an arbitrary bijection mapping Nn0 = N0 × · · · × N0 (n times) onto N0 . Then one can always choose a smooth function f mapping Rn into the real numbers and verifying the equation f (n) = Eφ(n)

(4)

for all n = (n1 , . . . , nn ) in Nn0 . Details on this construction are given in the Appendix. This interpolation function f , which can actually be expressed as a convergent power series whenever the sequence of eigenvalues does not possess any accumulation points, allows us to define a Hamiltonian H(X, P) = f (N1 , . . . , Nn ) ,

(5)

Ni being the number operator associated to the i-th coordinate. This can be easily done using functional calculus. It is clear that the Hamiltonian (5) is integrable. To see it, observe that H(X, P) can be obtained from the integrable classical Hamiltonian  (6) H(x, p) = f 21 (x21 + p21 − 1), . . . , 21 (x2n + p2n − 1) via canonical quantization (x 7→ X, p 7→ P = −i∇). The classical Hamiltonian H(x, p) appears, by construction, in Birkhoff’s normal form, and its normal 6

coordinates are global and take values in R2n . The conserved quantities of the Hamiltonian (5) can be taken to be the number operators, and thus the integrability of H(X, P) follows. It is useful to observe that any smooth function f verifying Eq. (4) gives rise to the same quantum Hamiltonian, but the classical Hamiltonian (6) that this function defines is different for each choice of f . Therefore we have an uncountable family of classically integrable Hamiltonians, all whose orbits are bounded and generally dense on tori, leading to the same quantum Hamiltonian with pure point spectrum. Now we will prove that there exists a unitary transformation U mapping H onto L2 (Rn ) such that H(X, P) is the expression of H in this realization of the Hilbert space, i.e., H(X, P) = U HU † . Hence H commutes with Ti = U † Ni U , and thus the proof of the integrability of H will be complete. It is not difficult to realize that the spectrum of H(X, P) (including degeneracies) is the same as that of H. To see this, it suffices to note that, as discussed in Section 1, the eigenfunctions of H(X, P) must factorize as product of solutions of one-dimensional harmonic oscillators (2), and therefore the sequence of eigenvalues of H(X, P), including degeneracies, is given by ∞ σpp (H(X, P)) = f (n) ni =0 = (Ei )∞ i=0 . By construction, H(X, P) has no continuous spectrum, and therefore its spectrum is given by the closure of its eigenvalues, i.e., σ(H(X, P)) = σpp (H) = {Ei }∞ i=0 .

(7)

Since H has no scattering states, one can choose an orthonormal eigenbasis {φi }∞ i=0 of its Hilbert space H, φi being an eigenfunction of H of energy Ei . As H(X, P) does not have any scattering states either, one can find an orthonormal basis of its Hilbert space L2 (Rn ), which we will denote by {ϕi }∞ i=0 , such that ϕi is an eigenfunction of H(X, P) with energy Ei also. If now one defines a unitary transformation U by setting U φi = ϕi for each i = 0, 1, . . . , by construction we have H(X, P) = U HU † . This completes the proof of the claim.

2.2

General properties of quantum integrability

The physical interest of the sufficient integrability criterion discussed in the previous section is laid bare by observing that it provides exact information on the integrability of certain quantum systems without referring to their particular 7

form. This can be used to prove some properties of quantum integrability that are the counterparts of known results in classical mechanics. First of all, one should observe that the criterion above implies that integrable Hamiltonians are dense (in the resolvent norm topology), i.e., that any given Hamiltonian H may be approximated by an integrable Hamiltonian within an arbitrary small error. This stems from the fact that one can approximate H with Hamiltonians without scattering states with arbitrary precision (this is equivalent to approximating square-integrable functions with step functions), and these Hamiltonians are integrable by the main result. One should observe that, although we have established that integrable Hamiltonians are dense (i.e., topologically many), we cannot ensure that non-integrable Hamiltonians are few, in any reasonable sense. This is in strong contrast with the classical case. In classical mechanics it is well know that there are very few integrable Hamiltonians, and in fact Markus and Meyer [18] proved that “almost all” Hamiltonian systems (i.e., an open and dense subset in the strong C ∞ topology) are non-integrable, since classically integrable Hamiltonians are nowhere dense. Markus and Meyer’s construction was based on the impossibility of having local action-angle coordinates for “almost all” Hamiltonians. As we have seen, the quantum analogue of this local change of coordinates (namely, the unitary transformation U ) is definitely not restricted. Note that the main result refers to abstract Hamiltonians and does not constrain them to the simplest case of Schr¨odinger operators acting on a space of square-integrable functions. The necessity of studying the abstract case is suggested by Ref. [18], since they needed to consider general Hamiltonians to obtain significant results on the ubiquity of non-integrability, the case of natural Hamiltonians being still open. It is remarkable that they did manage, however, to obtain results on the ubiquity of non-ergodicity in classical mechanics for natural Hamiltonians. Another property of integrable Hamiltonians that can be inferred from the main result is that the spectrum of an integrable quantum system does not need to be simple at all. This contradicts the folk wisdom that usually associates integrable Hamiltonians with simple spectra, say regularly spaced eigenvalues accumulating at zero and continuous spectrum above zero as in the case of the hydrogen atom. To see this, let C be any closed subset of the real numbers and let (Ei )∞ i=0 be a dense sequence in C, which is known to exist. The construction of H(X, P) developed in the proof of the main result provides an integrable Hamiltonian whose spectrum is given by the closure of this sequence (cf. Eq. (7)), which is C by definition. Therefore we reach the physically surprising conclusion that there are integrable Hamiltonians realizing, for instance, the Cantor set as their spectrum. In particular, the main result shows that any Hamiltonian H whose spectrum is given by the imaginary parts of the zeroes on the critical line of the Riemann Zeta function must be integrable, since this set is known to be countable. This improves a result of Crehan [11], who proves that there exists an integrable 8

quantum Hamiltonian which realizes this sequence as its spectrum. Recall that this question originated in the context of quantum chaos, as this sequence is known to follow GUE statistics and this statistics is usually associated with quantum chaos. As we shall see in the next subsection, these results do not imply that the classical counterpart of H must be integrable, so this result does not prevent non-integrability from appearing within in the classical limit or when using a definition of integrability relying on semiclassical methods. However, an interesting spin-off is that there exist a (non-unique) integrable classical Hamiltonian H(x, p), defined as in Eq. (6), whose quantum counterpart realizes the zeros of the Riemann Zeta function as its spectrum. This constitutes an exception to Berry’s claim [19] asserting that these classical Hamiltonians should be chaotic. Nevertheless, Berry’s claim will certainly hold in the generic case, as the results in the following subsection suggest. Recall that another example showing that there are exceptions to the idea that Wigner–Dyson statistics are characteristic of non-integrable classical systems is that given by Benet et al. [20], who constructed an ensemble of twodimensional integrable quantum Hamiltonians having GOE or GUE statistics for all values of the parameters. In Section 3 we will prove, however, that this phenomenon is non-generic, and thus it does not contradict the Berry–Tabor conjecture.

2.3

Applications

It is common knowledge that quantum integrability does not imply classical integrability. This can be rigorously established using the main result. In fact, a striking heuristic proof is that integrable Hamiltonians are dense in the quantum case whereas classically they are nowhere dense. We will now study the classical and quantum integrability of several relevant systems. The goal is twofold. On one hand, we will obtain specific examples of Hamiltonians that are integrable as quantum operators but whose classical analogue is non-integrable. And on the other hand we will provide non-trivial examples of integrable quantum Hamiltonians which describe a wide range of physically interesting situations and whose spectra could prove worth being numerically studied in the context of quantum chaos. First, consider the movement of a free particle on a compact Riemannian manifold M . In the quantum case, the dynamics of the particle is given by the Hamiltonian H = −∆, i.e., minus the Laplacian operator of the manifold. This Hamiltonian is known to have discrete spectrum, and therefore the movement of a free quantum particle on a compact Riemannian manifold is integrable. Note that this result generalizes a result of Matveev and Topalov [21], who proved the quantum integrability of this Hamiltonian in the particular case of a manifold having non-proportional, geodesically equivalent metrics. However, in this particular case their results are stronger since they provide some information on the form of the conserved quantities. On the contrary, the classical movement of a free particle on a compact Riemannian manifold M is given by the geodesic equation, which is usually non9

integrable. Particularly, when M has negative sectional curvature Anosov [22] proved that the orbits are ergodic, and therefore the Hamiltonian cannot possess any continuous first integrals. Another interesting example is given by the movement of a particle in Euclidean n-dimensional space Rn subjected to a strongly binding potential V . More precisely, we will require the potential to be lower bounded (V (x) ≥ c) and to go to infinity as the distance to the origin grows (lim|x|→+∞ V (x) = +∞). If V satisfies certain mild technical conditions, Browder [23] proved that the Hamiltonian H = −∆ + V (x) has no scattering states, and hence the quantum movement is integrable. Nevertheless, it is well known that having a strongly binding potential is not enough for the classical Hamiltonian H(x, p) = p2 + V (x)

(8)

to be integrable, and in fact examples are known [24] in which all the requirements on the potential V are verified but (8) is non-integrable via meromorphic first integrals. The two-dimensional Kronig–Penney model in presence of a magnetic field, which has been used to mimic crystal ions in quantum solids, also shows the same phenomenon. The quantum Hamiltonian  h ∂ eBy i2 h eBx i2 2πx 2πy  ∂ + −i + V0 cos + − + cos , H = −i ∂x 2c ∂y 2c a a can easily be shown to have discrete spectrum. In this formula, a is the lattice spacing, e is the electron charge, c is the speed of light, B is the (constant) magnetic field, and V0 is a positive constant. To prove that the above Hamiltonian has discrete spectrum, it is enough to observe that H is lower bounded by the magnetic Hamiltonian h ∂ eBy i2 h eBx i2 ∂ H0 = − i + − + −i − 2 V0 , ∂x 2c ∂y 2c which is known to have discrete spectrum, and apply the min-max principle. However, its classical counterpart   eBx 2 eBy 2  2πx 2πy  + py − + V0 cos H(x, p) = px + + cos 2c 2c a a has been numerically shown to present the standard features of chaos [25], and actually fails to pass the Painlev´e test [26]. It is also worth mentioning another famous Hamiltonian which presents the same behavior. Consider a particle on the plane subjected to the potential V (x) = x2 y 2 , where x = (x, y). It is easy to see that the classical Hamiltonian H(x, p) = p2 + V (x) 10

(9)

possesses scattering orbits, for instance y = 0. However, Simon [27] proved that its quantum counterpart H = −∆ + V (x) (10) has no scattering states. This intriguing phenomenon is due to the fact that as a matter of fact, the Hamiltonian (9) has very few scattering directions. It is interesting to observe that another peculiarity of this potential is that, having no scattering states, the quantum Hamiltonian (10) is integrable, while a straightforward application of Yoshida’s criterion [28] shows that (9) is not (at least with analytic first integrals). In fact, numerical explorations of this classical Hamiltonian show a complex orbit structure. Concerning the applicability of the main result, it is convenient to remark that it also has applications in the field of solvable quantum models, especially in those situations in which the exact or quasi-exact solvability of the model is easier to verify than its integrability, and its spectrum is discrete. This situation appears, for instance, in the study of Haldane–Shastry spin chains coupled with an external field using Dunkl operator techniques [29, 30]. Finally, it should be recalled that a result of Zhang et al. [31] stated that quantum integrability implied classical integrability, their definition of quantum integrability being weaker than the one used in this paper. The reason for this apparent contradiction is that Zhang et al. did not specify any degree of regularity of the classical first integrals, such as being smooth or at least continuous. A careful look at the proof of the main result reveals that the point which breaks the integrability of the classical analogue of the Hamiltonian H is that the unitary transformation U mapping the Hilbert space H into the space of square-integrable functions on Rn does not induce a symplectomorphism from the phase space of the system into R2n . It is interesting to observe that neither does a symplectomorphism lead to a unitary transformation, in the general case [32].

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Statistical distribution of the energy levels

The study of the statistical distribution of energies in integrable and nonintegrable systems (with various definitions of integrability) is an attractive area of contemporary quantum physics. One of the landmarks in this field is the so called Berry–Tabor conjecture, which establishes a connection between the distribution of energies of a given Hamiltonian and its integrability, where the definition of integrability used is based on the semiclassical approximation. The content of Berry–Tabor’s conjecture is that if H is the canonical quantization of an integrable classical Hamiltonian H(x, p), then the statistical distribution of the differences of its normalized energies is generically Poissonian. Thus the statistical distribution of energies is related to the well defined and mathematically precise semiclassical approach to integrability. Berry and Tabor [33] provided a semi-rigorous proof of the conjecture that bears their names, and there is abundant numerical evidence [34, 35] of the validity of their claim.

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This conjecture has been recently proved rigorously in the particular case of a particle moving on a torus in presence of a uniform magnetic field [36]. A topic of current research interest is the investigation of statistical properties of the energy spectra of quantum integrable systems (possibly without any classical counterparts) without making use of the semiclassical limit. For instance, Rela˜ no et al. [12] used the exactly solvable Richardson–Gaudin models, whose quantum integrability is established without resorting to the semiclassical limit, to perform a numerical test of the Berry–Tabor conjecture. Some exceptions to the Berry–Tabor conjecture were found, but these authors numerically showed that small integrable perturbations restored the Poissonian spectrum. The claim of this section is that Poisson-distributed Hamiltonians are dense in the ensemble of Hamiltonians with pure point spectrum. Before proving it, some comments and applications are in order. First, one should explicitly remark that, of course, it does not contradict the universally accepted wisdom in quantum chaology stating that, within the class of all Hamiltonians with a well-defined classical limit, those of them which are not Poisson distributed and whose classical counterpart is non-integrable are “generic”, in certain sense. Our claim does not imply that non-Poisson distributed, non-integrable quantum Hamiltonians, or Hamiltonians with nonintegrable classical limit are few (one can think of the analogous case of the rationals and the reals), since in fact it is folk wisdom that they should greatly outnumber the integrable ones. Moreover, our approach does not take into account the classical limit. The claim of this section can be applied to show, without resorting to numerical computations, why the existence of integrable models realizing chaotic spectra showed by Benet et al. [20] and Rela˜ no et al. [12] has no bearing on the Berry–Tabor conjecture. Actually, let H be a Richardson–Gaudin Hamiltonian whose parameters lead to a Wigner distribution, or a Hamiltonian belonging to the ensemble considered by Benet et al. In both cases, H has pure point spectrum, and our claim implies that Poisson distributed Hamiltonians are dense in the set of Hamiltonians without scattering states, and integrable by the main result. Hence there exists an arbitrary small perturbation which preserves the integrability of H (although possibly not within the same family, as the result of Benet et al. establishes), and such that the statistical distribution of energies rapidly decays to a Poisson distribution. Now let us prove the claim. As the demonstration is quite lengthy, we will divide it into four steps. To rule out inessential degrees of freedom, it is convenient to identify those Hamiltonians that only differ by a change of orthonormal basis. Step 1: First, let us denote by App the set of unitary classes of Hamiltonians of dimension n with pure point spectrum, i.e., the set of Hamiltonians without scattering states defined up to a change of orthonormal basis. The unitary class defined by a Hamiltonian H will be denoted by [H]. Let C stand for the set of

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countable sequences of (possibly repeated) real numbers  C = (ci )∞ i=0 : ci ∈ R , where two sequences differing only in the ordering of its elements are to be identified. It can be seen that the unitary classes of Hamiltonians in App and the sequences of reals in C are in a one-to-one correspondence. To prove this fact, note that each class [H] in App gives rise to the sequence of its eigenvalues (Ei ), up to an ordering. And, conversely, an unordered sequence (Ei ) uniquely defines the class [H] in App whose spectrum realizes this sequence, as the following elementary argument shows. If {ϕi }∞ i=0 is an orthonormal basis of the Hilbert space, a representative H can be defined as H=

∞ X

Ei |ϕi ihϕi | .

(11)

i=0

Besides, any other Hamiltonian in App realizing the same eigenenergies ˜ = H

∞ X

Ei |ϕ˜i ihϕ˜i | ,

i=0

where {ϕ˜i }∞ i=0 is another orthonormal basis, must be equivalent to H through the unitary transformation U ϕ˜i = ϕi ˜ into the normalized eigenwhich transforms the normalized eigenfunctions of H functions of H. Thus an unordered sequence (Ei ) ∈ C determines the only class [H] ∈ App which realizes this sequence as its spectrum. Step 2: The one-to-one correspondence between App and C can be extended to a topological equivalence. Later we will use this fact to study the density of a given set in App or C indistinctly. (n) We will endow C = RN0 with its product topology, so that (ci ) converges to (n) (ci ) if and only if ci converges to ci for each i. The one-to-one correspondence between App and C now topologizes App . In fact, the inherited topology can be easily expressed in terms of the usual resolvent norm topology, since a sequence [An ] in App converges to [A] if and only if there exist representatives A˜n , A˜ of these unitary classes such that A˜n converges to A˜ in the norm resolvent sense. The “if” part is obvious, whereas the “only if” part follows from choosing representatives A˜n , A˜ with a common orthonormal basis of eigenfunctions {ϕi } and applying the formula ∞

X

  (n)

(A˜n − λ)−1 − (A˜ − λ)−1 = (ci − λ)−1 − (ci − λ)−1 |ϕi ihϕi | i=1

= sup i∈N 0

1 − , (n) ci − λ ci − λ

13

1

which holds true for each λ ∈ C with Im λ 6= 0. Thus convergence and density in C and App become equivalent. Step 3: Now we will prove that the set of Hamiltonians without scattering states (defined up to unitary transformation) whose spectrum is u.d. is dense in the aforementioned topology. Since R is unbounded, it is necessary to consider weighted measures w(x) dx R +∞ (w positive, continuous, and satisfying −∞ w(x) dx = 1) such that the measure of R becomes finite [37, 38]. Recall that a sequence of reals {ci } is uniformly distributed with respect to the weight w (w-u.d.) if Z +∞ N −1 1 X f (ci ) = f (x) w(x) dx N →∞ N −∞ i=0 lim

for every continuous function f with compact support. Note that this definition is also valid for unordered sequences, since the above limit (which exists and equals the RHS integral) does not depend on the ordering. Now the claim follows from the ergodic theorems [37],[39, Th. 2.4] ensuring that w-u.d. sequences are dense in C. From the digression in Step 2, unitary classes in App with w-u.d. spectra are also dense. Since this property holds true for each weight function w, from now on we will omit w. Step 4: Finally, we shall prove that the spacing of energy levels is generically Poisson distributed when the eigenvalues of the Hamiltonian follow the uniform distribution. Recall that there are well known exceptions in which a uniform distribution of energies does not necessarily imply that the differences of energies are Poissonian. An example of this phenomenon is given by multidimensional harmonic oscillators [40]. Fortunately, for almost all sequences of energies {Ei }, the following argument, due to Gutzwiller [41, Sect. 16.3] and going back to Porter [42], can be used to prove that it does. For generic sequences of energies, the associated spectral distribution Q and spacing of energy levels P are well defined. So let Q(x) dx be the probability of finding a normalized eigenenergy of H in the interval (E + x, E + x + dx) assuming that E is a normalized eigenvalue of H. Similarly, let P (x) dx be the probability of finding two consecutive normalized levels at a distance x. It is well known that P and Q are related via the equation P (x) = Q(x) e−

Rx 0

Q(ξ) dξ

,

(12)

which is due to Wigner. For completeness, we shall provide a proof for Eq. (12). Let g(x) be the probability of not finding a normalized level at a distance smaller that x apart from another. Then clearly one has g(0) = 1. The probability of not finding a normalized level at a distance smaller than x+dx can be expressed as g(x + dx) = g(x) (1 − Q(x) dx) , (13) since g(x) is the probability of not finding a level at distance lower than x and 1 − Q(x) dx is the probability of not finding a level at a distance d, where

14

x ≤ d ≤ x + dx. Expanding Eq. (13), one finds that dg(x) = −g(x) Q(x) , dx  Rx  and thus g(x) = exp − 0 Q(ξ) dξ . Finally, P (x) dx = g(x) Q(x) dx by definition, so Eq. (12) holds. Using Eq. (12) it is immediate to see that if the energies of H are u.d., so Q(x) = 1, then P (x) = e−x , as we wanted to prove. This completes the proof of the fact that for a dense subset of all Hamiltonians without scattering states (defined up to change of orthonormal basis), which are integrable by the main result of Section 2.1, the differences of normalized energy levels are Poisson distributed. It should be remarked that the definition of integrability used in this paper, as well as that used by Rela˜ no et al., slightly differs from the semiclassical definition in Berry and Tabor’s paper. However, the results obtained in this section do provide additional confirmation to this celebrated conjecture, as previously discussed.

4

Final remarks

In this paper we have obtained a sufficient condition for the integrability of a quantum Hamiltonian, namely, it is integrable whenever it does not have any scattering states. To the best of our knowledge, this spectral approach to quantum integrability is new in the literature and partially connects different approaches to integrability that are used in quantum physics. Recall that it was also shown that the definition of integrability used in this paper is particularly convenient because it inherits the algebraic properties of the classical concept of Liouville integrability. We believe that a detailed study of the relationship between this definition and the semiclassical one would match with interest. In Section 2.2 it was proved that, contrary to the intuition we obtain from classical physics, there are plentiful integrable quantum systems. It would be interesting to know whether this result is also valid in quantum field theory, or this phenomenon only appears within the realm of quantum mechanics. Several authors have also considered that quantum mechanics is very often integrable, although the definitions of integrability considered to some extent lacked the physical interpretation of ours. For instance, Kupershmidt [43] argued that the Hamilton equations of motion have an infinite set of conserved quantities in the sense of partial differential equations, and Cirelli and Pizzocchero [44] developed an interesting abstract definition of integrability in terms of invariant cylinders in the space of rays.

15

Additionally, we managed to establish that the spectrum of an integrable system is not necessarily simple, contrary to folk wisdom. This suggests that it would be interesting to find integrable Hamiltonians realizing the Cantor set as its spectrum, which by the main result are known to exist, and determine whether these Hamiltonians can be cast into the form of a Schr¨odinger operator. Recall that an example of Hamiltonian whose spectrum is given by the Cantor set was found by Hofstadter [45] as an effective single-band model representing a crystal electron in a uniform magnetic field. This Hamiltonian does not appear in natural form and, to our best knowledge, its integrability has never been studied. In Section 2.3 we developed several applications of the main result and provided several concrete systems that are integrable in the quantum case but are non-integrable in the classical case. Recall that the examples we provided covered several physically relevant situations, such as particles subjected to strongly binding potentials or free particles on compact manifolds. Finally, in Section 3, lead by the Berry–Tabor conjecture and following previous work on this topic, we studied the statistical distribution of the energies of Hamiltonians without scattering states. We rigorously obtained that a dense subset of this ensemble is Poisson distributed, thus providing further evidence of the validity of this celebrated conjecture.

Acknowledgements The authors wish to acknowledge the contribution of Professors F. Finkel, A. Gonz´ alez-L´ opez and M.A. Rodr´ıguez in offering valuable suggestions during the course of this work. A.E. is supported by an FPU scholarship from the Spanish Ministry of Education, and acknowledges the DGI for partial financial support under grant No. BFM2002–02646.

Appendix It is easy to provide an explicit expression for the smooth interpolation function f used in Eq. (4), Section 2.1. Let φ : Nn0 → N0 be an arbitrary bijection from Nn0 onto N0 . An example of this bijection is given by the function  φπ (n) = card m ∈ Nn0 : π m < π n , where we use the notation π n = π1n1 · · · πnnn

(14)

and π1 , . . . , πn are relatively prime positive integers. Now let the smooth bump function ϕ : R → R be given by ( 2 − 4x if |x| ≤ 12 , e 1−4x2 ϕ(x) = 0 if |x| ≥ 12 . 16

Let the function f : Rn → R be defined by X f (x) = Eφ(m) ϕ(|x − m|) . m∈Nn 0

This series is clearly pointwise convergent, and uniformly convergent on any compact set. It is also obvious that this function is infinitely differentiable, and that f (n) = Eφ(n) , thus establishing our claim. When the sequence of energies does not have accumulation points (other than infinity), the interpolation function f can be chosen analytic, and thus can be expanded in a convergent power series. To construct this analytic function, we will make use of the functions Wj : C → C defined by # " j X zk , Wj (z) = (1 − z) exp k k=1

where j ∈ N0 . One can now define the function  z  Y g(z) = Wφ(n) n , π n n∈N0

n

where the positive integer π is given by Eq. (14). Since the set {π n }n∈Nn0 does not accumulate at any finite point and does not contain 0, the above product converges and defines an analytic function on the complex plane, whose only zeros lie at z = π n (n ∈ Nn0 ) [46]. Now let h : C → C be the Mittag-Leffler function [46] with poles at {π n }n∈Nn0 , and whose principal part at each z = π n is given by Pn (z) =

Eφ(n) 0 Wφ(n) (π n )

z − π n )−1 .

Note that the derivative of Wφ(n) at π n appearing in this formula can be computed to yield  πn  Y 0 Wφ(n) (π n ) = eHφ(n) Wφ(m) m , π n n6=m∈N0

Pj

where Hj = k=1 k −1 is the j-th harmonic number. Finally, let us define the interpolation function f : Rn → R as f (x) = g(π x ) h(π x ) . Since the function

Pn

π x = π1x1 · · · πnxn = e

j=1

xj log πj

is analytic, it is obvious that so is f . By construction [46], f satisfies f (n) = Eφ(n) for all n ∈ Nn0 , and thus constitutes the required function. 17

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