752 research outputs found
A pseudo-differential calculus on non-standard symplectic space; spectral and regularity results in modulation spaces
The usual Weyl calculus is intimately associated with the choice of the
standard symplectic structure on . In this
paper we will show that the replacement of this structure by an arbitrary
symplectic structure leads to a pseudo-differential calculus of operators
acting on functions or distributions defined, not on but
rather on . These operators are intertwined
with the standard Weyl pseudo-differential operators using an infinite family
of partial isometries of \ indexed by . This allows
us obtain spectral and regularity results for our operators using Shubin's
symbol classes and Feichtinger's modulation spaces.Comment: 32 pages, latex file, published versio
Zitterbewegung and semiclassical observables for the Dirac equation
In a semiclassical context we investigate the Zitterbewegung of relativistic
particles with spin 1/2 moving in external fields. It is shown that the
analogue of Zitterbewegung for general observables can be removed to arbitrary
order in \hbar by projecting to dynamically almost invariant subspaces of the
quantum mechanical Hilbert space which are associated with particles and
anti-particles. This not only allows to identify observables with a
semiclassical meaning, but also to recover combined classical dynamics for the
translational and spin degrees of freedom. Finally, we discuss properties of
eigenspinors of a Dirac-Hamiltonian when these are projected to the almost
invariant subspaces, including the phenomenon of quantum ergodicity
A semiclassical Egorov theorem and quantum ergodicity for matrix valued operators
We study the semiclassical time evolution of observables given by matrix
valued pseudodifferential operators and construct a decomposition of the
Hilbert space L^2(\rz^d)\otimes\kz^n into a finite number of almost invariant
subspaces. For a certain class of observables, that is preserved by the time
evolution, we prove an Egorov theorem. We then associate with each almost
invariant subspace of L^2(\rz^d)\otimes\kz^n a classical system on a product
phase space \TRd\times\cO, where \cO is a compact symplectic manifold on
which the classical counterpart of the matrix degrees of freedom is
represented. For the projections of eigenvectors of the quantum Hamiltonian to
the almost invariant subspaces we finally prove quantum ergodicity to hold, if
the associated classical systems are ergodic
Ergodic properties of a generic non-integrable quantum many-body system in thermodynamic limit
We study a generic but simple non-integrable quantum {\em many-body} system
of {\em locally} interacting particles, namely a kicked model of spinless
fermions on 1-dim lattice (equivalent to a kicked Heisenberg XX-Z chain of 1/2
spins). Statistical properties of dynamics (quantum ergodicity and quantum
mixing) and the nature of quantum transport in {\em thermodynamic limit} are
considered as the kick parameters (which control the degree of
non-integrability) are varied. We find and demonstrate {\em ballistic}
transport and non-ergodic, non-mixing dynamics (implying infinite conductivity
at all temperatures) in the {\em integrable} regime of zero or very small kick
parameters, and more generally and important, also in {\em non-integrable}
regime of {\em intermediate} values of kicked parameters, whereas only for
sufficiently large kick parameters we recover quantum ergodicity and mixing
implying normal (diffusive) transport. We propose an order parameter (charge
stiffness ) which controls the phase transition from non-mixing/non-ergodic
dynamics (ordered phase, ) to mixing/ergodic dynamics (disordered phase,
D=0) in the thermodynamic limit. Furthermore, we find {\em exponential decay of
time-correlation function} in the regime of mixing dynamics.
The results are obtained consistently within three different numerical and
analytical approaches: (i) time evolution of a finite system and direct
computation of time correlation functions, (ii) full diagonalization of finite
systems and statistical analysis of stationary data, and (iii) algebraic
construction of quantum invariants of motion of an infinite system, in
particular the time averaged observables.Comment: 18 pages in REVTeX with 14 eps figures included, Submitted to
Physical Review
Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere
For an -variate order- tensor , define to be the maximum value taken by the
tensor on the unit sphere. It is known that for a random tensor with i.i.d entries, w.h.p. We study the
problem of efficiently certifying upper bounds on via the natural
relaxation from the Sum of Squares (SoS) hierarchy. Our results include:
- When is a random order- tensor, we prove that levels of SoS
certifies an upper bound on that satisfies Our upper bound improves a result of Montanari and Richard
(NIPS 2014) when is large.
- We show the above bound is the best possible up to lower order terms,
namely the optimum of the level- SoS relaxation is at least
- When is a random order- tensor, we prove that levels of SoS
certifies an upper bound on that satisfies For growing , this improves upon the bound
certified by constant levels of SoS. This answers in part, a question posed by
Hopkins, Shi, and Steurer (COLT 2015), who established the tight
characterization for constant levels of SoS
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