Download A spectral theoretic approach to quantum

A spectral theoretic approach
to quantum integrability
Alberto Enciso and Daniel Peralta-Salas
Departamento de Física Teórica II,
Universidad Complutense
math-ph/0406022
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.
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 ndimensional quantum Hamiltonian with pure point
spectrum is integrable. Several applications of this theorem
will be developed.
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.
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 ndimensional 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
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.
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 selfadjoint operators.
2. Given any closed set
(for instance, the Cantor
set), there exists an integrable n-dimensional quantum
Hamiltonian H such that
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.
•
,
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.
Examples:
• Laplacian on compact Riemannian manifolds (with
strictly negative sectional curvature, C0)
•
•
(Cw)
as
(and certain mild technical assumptions) (Cw)
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.
Note that this definition closely resembles the classical one
in several nontrivial algebraic features.
Classical Mechanics
Quantum Mechanics
• Existence of local actionangle variables
• No algorithmic procedure
to compute them
• Dynamics is linearized
into decoupled harmonic
oscillators
• Existence of unitary
transformation U
• No algorithmic procedure
to compute it
• Dynamics is given by that
of harmonic oscillators
(
)
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.
A spectral theoretic approach
to quantum integrability
Alberto Enciso and Daniel Peralta-Salas
Departamento de Física Teórica II,
Universidad Complutense
math-ph/0406022