Volume 56, Issue 2, February 2015
Index of content:

In this paper, we consider the statistics of repeated measurements on the output of a quantum Markov chain. We establish a large deviations result analogous to Sanov’s theorem for the multisite empirical measure associated to finite sequences of consecutive outcomes of a classical stochastic process. Our result relies on the construction of an extended quantum transition operator (which keeps track of previous outcomes) in terms of which we compute moment generating functions, and whose spectral radius is related to the large deviations rate function. As a corollary to this, we obtain a central limit theorem for the empirical measure. Such higher level statistics may be used to uncover critical behaviour such as dynamical phase transitions, which are not captured by lower level statistics such as the sample mean. As a step in this direction, we give an example of a finite system whose level1 (empirical mean) rate function is independent of a model parameter while the level2 (empirical measure) rate is not.
 ARTICLES

 Partial Differential Equations

On the use of normal forms in the propagation of random waves
View Description Hide DescriptionWe consider the evolution of the correlations between the Fourier coefficients of a solution of the KadomtsevPetviashvili II equation when these coefficients are initially independent random variables. We use the structure of normal forms of the equation to prove that those correlations remain small until times of order ε^{5/3} or ε^{2} depending on the quantity considered.

A soliton hierarchy associated with a new spectral problem and its Hamiltonian structure
View Description Hide DescriptionA hierarchy of soliton equations together with its Hamiltonian structure is constructed from a new spectral problem associated with the threedimensional special orthogonal real Lie algebra, so(3,ℝ). The Liouville integrability of the presented soliton hierarchy is proved, based on the Hamiltonian structure.

Multispeed solitary wave solutions for a coherently coupled nonlinear Schrödinger system
View Description Hide DescriptionExistence of multispeed solitary wave solutions for a coherently coupled system of nonlinear Schrödinger equations is proved. Such solutions behave at large time as a couple of scalar solitary waves traveling at different speeds. Compared to incoherently coupled nonlinear Schrödinger systems which have enjoyed a lot of research during the past several decades, coherently coupled nonlinear Schrödinger systems have a significant different feature that their component masses are not conserved but only the total mass is conserved. Some restriction upon the phases and velocities of the two component solitary waves is therefore imposed to fit this situation.

Conservative weak solutions of the periodic Cauchy problem for μHS equation
View Description Hide DescriptionWe prove the existence and uniqueness of conservative weak solutions of the periodic Cauchy problem for an integrable evolution equation from mathematical physics. Our method is to first prove the existence and uniqueness of solutions for all time in Lagrangian coordinates and construct the maps to carry this result to the Eulerian coordinates.

Bound states of waveguides with two rightangled bends
View Description Hide DescriptionWe study waveguides with two rightangled bends. These waveguides are in shape of letter Z or alternatively C. For both cases, we assume the semiinfinite arms of waveguides to be of unit width. These arms are connected to each other by a rectangle with side lengths H and L. We consider the Dirichlet boundary value problem for Laplacian and study the spectrum of the corresponding operator. It is shown that the total multiplicity of the discrete spectrum depends on the parameters H and L. In particular, for the width H = 1, we compare the relation between the eigenvalues of both waveguides and moreover, we observe that the monotonicity in height L of the first eigenvalue of the Zshaped waveguide is not achieved while the question of the monotonicity of the second eigenvalue remains open. The eigenvalues in the Cshaped waveguide are monotone. We construct and justify the asymptotics of the eigenvalues for the cases H = 1, L → ∞, H = 1, L → 1 + 0, and H, L → ∞.

Higher dimensional nonclassical eigenvalue asymptotics
View Description Hide DescriptionIn this article, we extend Simon’s construction and results [B. Simon, J. Funct. Anal. 53(1), 8498 (1983)] for leading order eigenvalue asymptotics to ndimensional Schrödinger operators with nonconfining potentials given by on ℝ^{ n } (n > 2), . We apply the results to also derive the leading order spectral asymptotics in the case of the Dirichlet Laplacian −Δ^{ D } on domains .

Weak concentration and wave operator for a 3D coupled nonlinear Schrödinger system
View Description Hide DescriptionReported in this paper are results concerning the Cauchy problem and the dynamics for a cubic nonlinear Schrödinger system arising in nonlinear optics. A sharp criterion is given concerned with the dichotomy global existence versus finite time blowup. When a radial solution blows up in finite time, we prove the concentration in the critical Lebesgue space. Sufficient condition for the scattering and the construction of the wave operator in the energy space is also provided.

Homogenization of the boundary value for the Neumann problem
View Description Hide DescriptionIn this paper, we study the convergence rates for homogenization problems for solutions of partial differential equations with rapidly oscillating Neumann boundary data. Such a problem raised due to its importance for higher order approximation in homogenization theory. High order approximation gives rise to the socalled boundary layer phenomenon. As a consequence, we obtain the pointwise and W ^{1,p } convergence results. Our techniques are based on Fourier analysis.
 Representation Theory and Algebraic Methods

Relative HomHopf modules and total integrals
View Description Hide DescriptionLet (H, α) be a monoidal HomHopf algebra and (A, β) a right (H, α)Homcomodule algebra. We first investigate the criterion for the existence of a total integral of (A, β) in the setting of monoidal HomHopf algebras. Also, we prove that there exists a total integral ϕ : (H, α) → (A, β) if and only if any representation of the pair (H, A) is injective in a functorial way, as a corepresentation of (H, α), which generalizes Doi’s result. Finally, we define a total quantum integral γ : H → Hom(H, A) and prove the following affineness criterion: if there exists a total quantum integral γ and the canonical map ψ : A⊗ B A → A ⊗ H, a⊗ B b ↦ β ^{−1}(a) b [0] ⊗ α(b [1]) is surjective, then the induction functor is an equivalence of categories.

The Rmatrix of quantum doubles of Nichols algebras of diagonal type
View Description Hide DescriptionLet H be the quantum double of a Nichols algebra of diagonal type. We compute the Rmatrix of 3tuples of modules for general finitedimensional highest weight modules over H. We also calculate a multiplicative formula for the universal Rmatrix when H is finite dimensional. We show the unicity of a PBW basis (or a Lusztigtype PoincaréBirkhoffWitt basis) with a given convex order.
 ManyBody and Condensed Matter Physics

Ground state energy of large polaron systems
View Description Hide DescriptionThe last unsolved problem about the manypolaron system, in the Pekar–Tomasevich approximation, is the case of bosons with the electronelectron Coulomb repulsion of strength exactly 1 (the “neutral case”). We prove that the ground state energy, for large N, goes exactly as −N ^{7/5}, and we give upper and lower bounds on the asymptotic coefficient that agree to within a factor of 2^{2/5}.
 Quantum Mechanics

Inverse scattering problem and generalized optical theorem
View Description Hide DescriptionWe present a novel solution to the inverse scattering problem. Our solution is based on a generalization of the optical theorem, and applies directly in three dimensional space. First, we derive a necessary and sufficient condition for a halfonshell Tmatrix to be physically acceptable, which turns out to be a generalization of the optical theorem. Second, we show that the inverse scattering problem, which inquires underlying potential for a given onshell Tmatrix (scattering amplitude), can be solved by looking for a halfonshell Tmatrix that satisfies the generalized optical theorem with the given onshell Tmatrix being the boundary condition. At the end, we demonstrate that the present theory works nicely using simple systems.

Semiclassical limits of quantum partition functions on infinite graphs
View Description Hide DescriptionWe prove that if H denotes the operator corresponding to the canonical Dirichlet form on a possibly locally infinite weighted graph (X, b, m), and if v : X → ℝ is such that H + v/ħ is welldefined as a form sum for all ħ > 0, then the quantum partition function tr(e^{−βħ(H+v/ħ)}) converges to as ħ → 0 +, for all β > 0, regardless of the fact whether e^{−βv } is a priori summable or not. This fact can be interpreted as a semiclassical limit, and it allows geometric Weyltype convergence results. We also prove natural generalizations of this semiclassical limit to a large class of covariant Schrödinger operators that act on sections in Hermitian vector bundle over (X, m, b), a result that particularly applies to magnetic Schrödinger operators that are defined on (X, m, b).

Bound states for two dimensional Schrödinger equation with anisotropic interactions localized on a circle
View Description Hide DescriptionBound states for two dimensional Schrödinger equation with anisotropic interactions localized on a circle of radius r are considered. λ is a global parameter with energy as dimension. ρ and φ are radial and angular coordinates. The Dirac distribution δ localizes the interaction on the circle. measures the interaction at angle φ on the circle. A general method for determination of energies, mean values of different operators, normalized wave functions both in configuration space and momentum space is given. This method is applied to two cases. First case: , λ ≠ 0. Second case: , a > 1, and λ < 0. For the first case, the following results are obtained. Let the positive zeros j ν,n > 0 of Bessel function be numbered by integer n in increasing order, starting with n = 1 for the smallest zero. Define j ν,0 = 0. Let j 1,ℓ and j 0,k be the greatest values, which are smaller than , with M the mass. Then, the dimension of the vector space generated by even bound states is ℓ + 1, and the one generated by odd bound states is k. For the second case, let k be the greatest positive or zero integer, which is smaller than . Then, the dimension of the vector space generated by even bound states is k + 1, and the one generated by odd bound states is k.

Generalized Weyl transform for operator ordering: Polynomial functions in phase space
View Description Hide DescriptionThe generalized Weyl transforms were developed from the Hermiticity condition and the ordering rules were represented by characteristic realvalued functions. The integral transforms give rise to transformation equations between Weyl quantization and differently ordered operators. The transforms also simplify evaluation of commutator and anticommutator of a set of operators following the same ordering rule.

Recurrence theorems: A unified account
View Description Hide DescriptionI discuss classical and quantum recurrence theorems in a unified manner, treating both as generalisations of the fact that a system with a finite state space only has so many places to go. Along the way, I prove versions of the recurrence theorem applicable to dynamics on linear and metric spaces and make some comments about applications of the classical recurrence theorem in the foundations of statistical mechanics.

Inverse problems for selfadjoint Schrödinger operators on the half line with compactly supported potentials
View Description Hide DescriptionFor a selfadjoint Schrödinger operator on the half line with a realvalued, integrable, and compactly supported potential, it is investigated whether the boundary parameter at the origin and the potential can uniquely be determined by the scattering matrix or by the absolute value of the Jost function known at positive energies, without having the boundstate information. It is proved that, except in one special case where the scattering matrix has no bound states and its value is +1 at zero energy, the determination by the scattering matrix is unique. In the special case, it is shown that there are exactly two distinct sets consisting of a potential and a boundary parameter yielding the same scattering matrix, and a characterization of the nonuniqueness is provided. A reconstruction from the scattering matrix is outlined yielding all the corresponding potentials and boundary parameters. The concept of “eligible resonances” is introduced, and such resonances correspond to realenergy resonances that can be converted into bound states via a Darboux transformation without changing the compact support of the potential. It is proved that the determination of the boundary parameter and the potential by the absolute value of the Jost function is unique up to the inclusion of eligible resonances. Several equivalent characterizations are provided to determine whether a resonance is eligible or ineligible. A reconstruction from the absolute value of the Jost function is given, yielding all the corresponding potentials and boundary parameters. The results obtained are illustrated with various explicit examples.

Universal lowenergy behavior in threebody systems
View Description Hide DescriptionWe consider a pairwise interacting quantum 3body system in 3dimensional space with finite masses and the interaction term V 12 + λ(V 13 + V 23), where all pair potentials are assumed to be nonpositive. The pair interaction of the particles {1, 2} is tuned to make them have a zero energy resonance and no negative energy bound states. The coupling constant λ > 0 is allowed to take the values for which the particle pairs {1, 3} and {2, 3} have no bound states with negative energy. Let λ cr denote the critical value of the coupling constant such that E(λ) → −0 for λ → λ cr , where E(λ) is the ground state energy of the 3body system. We prove the theorem, which states that near λ cr , one has E(λ) = C(λ − λ cr )[ln(λ − λ cr )]^{−1} + h.t., where C is a constant and h.t. stands for “higher terms.” This behavior of the ground state energy is universal (up to the value of the constant C), meaning that it is independent of the form of pair interactions.

Noether’s theorem for dissipative quantum dynamical semigroups
View Description Hide DescriptionNoether’s theorem on constants of the motion of dynamical systems has recently been extended to classical dissipative systems (Markovian semigroups) by Baez and Fong [J. Math. Phys. 54, 013301 (2013)]. We show how to extend these results to the fully quantum setting of quantum Markov dynamics. For finitedimensional Hilbert spaces, we construct a mapping from observables to completely positive maps that leads to the natural analogue of their criterion of commutativity with the infinitesimal generator of the Markov dynamics. Using standard results on the relaxation of states to equilibrium under quantum dynamical semigroups, we are able to characterise the constants of the motion under quantum Markov evolutions in the infinitedimensional setting under the usual assumption of existence of a stationary strictly positive density matrix. In particular, the Noether constants are identified with the fixed point of the Heisenberg picture semigroup.

Sanov and central limit theorems for output statistics of quantum Markov chains
View Description Hide DescriptionIn this paper, we consider the statistics of repeated measurements on the output of a quantum Markov chain. We establish a large deviations result analogous to Sanov’s theorem for the multisite empirical measure associated to finite sequences of consecutive outcomes of a classical stochastic process. Our result relies on the construction of an extended quantum transition operator (which keeps track of previous outcomes) in terms of which we compute moment generating functions, and whose spectral radius is related to the large deviations rate function. As a corollary to this, we obtain a central limit theorem for the empirical measure. Such higher level statistics may be used to uncover critical behaviour such as dynamical phase transitions, which are not captured by lower level statistics such as the sample mean. As a step in this direction, we give an example of a finite system whose level1 (empirical mean) rate function is independent of a model parameter while the level2 (empirical measure) rate is not.