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A spectral theoretic approach to quantum integrability Alberto Enciso and Daniel Peralta-Salas Departamento de Física Teórica II, Universidad Complutense math-ph/0406022

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Introduction Classical (Liouville) integrability of a Hamiltonian is defined by the existence of n functionally independent, sufficiently smooth first integrals in involution. This concept is closely related to the complexity of its orbit structure, and in fact an integrable classical Hamiltonian cannot lead to chaotic dynamics.

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Quantum integrability is usually defined as the naive extension of this classical definition. A quantum Hamiltonian is said to be integrable when there exist n `functionally independent' linear operators which commute among them and with the Hamiltonian. The definition of dimension of a quantum system has been proposed by Zhang et al. (1989). In 1990 they also studied the correspondence between classical and quantum integrability, but their results are not satisfactory. The main new result to be discussed is that every n- dimensional quantum Hamiltonian with pure point spectrum is integrable. Several applications of this theorem will be developed.

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Statement of the results and discussion We will follow the definitions and notation of Reed & Simon (1979), with the decomposition Given a sequence of real numbers we will also consider the associated set of the values taken in the sequence. Theorem. Let H be an n-dimensional Hamiltonian with pure point spectrum. Then it is integrable via self-adjoint first integrals.

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Remarks This theorem improves partial results due to Weigert (1992) and Crehan (1995) and complements the results of Matveev and Topalov (2001) on quantum integrability of Laplacians on closed manifolds with non-proportional geodesically equivalent metrics. Furthermore, it answers in the affirmative a conjecture by Percival (1973), stating that ‘regular’ spectra correspond to integrable Hamiltonians. In fact, we prove that H is integrable in a stronger sense: it is equivalent (via change of orthonormal basis) to an integrable, canonically quantized, smooth classical n- dimensional Hamiltonian over, set into Birkhoff’s normal form. Thus, in this basis we have separation of variables in the sense that every eigenfunction factorizes as

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A sketchy proof of the main theorem would be as follows. Given any sequence of real numbers there exists an integrable n-dimensional Hamiltonian A which realizes this sequence as its spectrum. If one defines the number operator associated to the i-th coordinate as, this Hamiltonian can be constructed as f being an arbitrary function such that there exists a bijection satisfying The operators A and H can be proved to be unitarily equivalent. The self-adjoint operators provide the required complete set of commuting first integrals.

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Several remarks on the theorem and its proof are in order. First of all, one should observe that the physical interest of this theorem is laid bare in the two following observations: 1. Contrary to folk wisdom and unlike the classical case, integrable Hamiltonians are dense in the set of self- adjoint operators. 2. Given any closed set (for instance, the Cantor set), there exists an integrable n-dimensional quantum Hamiltonian H such that

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A physically relevant application of this theorem is a purely quantum analogue of Berry's conjecture. This celebrated conjecture essentially states that the normalized energies of a generic quantum Hamiltonian whose classical analogue is integrable are uniformly distributed. Let stand for the set of classes of unitarily equivalent Hamiltonians with pure point spectrum. We have proved that for at least one representative of each of these classes Berry's conjecture should apply, and actually we have managed to proved the following statement: Theorem. For almost all classes of Hamiltonians in, their eigenvalues are uniformly distributed.

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The results above give raise to the question of to what extent the classical and quantum notions of integrability are related. The fact that quantum integrable Hamiltonians are dense whereas classically integrable Hamiltonians are nowhere dense do not contradict each other. Despite the theorems of Zhang et al., it is fairly obvious that quantum integrability does not imply classical integrability. The underlying reason for this is that unitary transformations in Hilbert space do not induce symplectomorphisms in the classical phase space.

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Examples: Laplacian on compact Riemannian manifolds (with strictly negative sectional curvature, C 0 ) (C ) as (and certain mild technical assumptions) (C )

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The results of this work show the ubiquity of quantum integrability, in strong contrast with classical integrability. In fact, and due to various arguments (existence of invariant cylinders in quantum phase space or of infinite conservation laws, linearity and functions of eigenprojectors... ), several authors have regarded QM as generically (super)integrable. In any case, the stronger (and physically meaningful) definition of integrability discussed above ensure the nontriviality of our results and manages to get over some usual technical problems appearing in the definition of functional independence of operators.

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Note that this definition closely resembles the classical one in several nontrivial algebraic features. Classical Mechanics Existence of local action- angle variables No algorithmic procedure to compute them Dynamics is linearized into decoupled harmonic oscillators Quantum Mechanics Existence of unitary transformation U No algorithmic procedure to compute it Dynamics is given by that of harmonic oscillators ( )

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Up to date, one major problem of quantum integrability is its lack of geometric content. In the light of this critique, the various problems which appear can be more or less understood. A geometrically significant notion of quantum integrability would probably reproduce those beliefs on this issue which belong to folk wisdom even though are not compatible with the current definition of integrability. It would come to no surprise that this definition were independent of the d.o.f. of the analogue classical system, when it exists. An important step in this direction is due to Cirelli and Pizzocchero, but most questions are still unanswered.

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A spectral theoretic approach to quantum integrability Alberto Enciso and Daniel Peralta-Salas Departamento de Física Teórica II, Universidad Complutense math-ph/0406022

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