A spectral theoretic approach to quantum integrability

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A spectral theoretic approachto 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 Birkhoffs
  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, C0)
• (Cw)
• 
  as (and certain mild
  technical assumptions) (Cw)

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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 approachto quantum
integrability
Alberto Enciso and Daniel Peralta-Salas Departame
nto de Física Teórica II, Universidad
Complutense math-ph/0406022