Formulation and picture of quantum phase

Based on the concept of classical phase, we formulate a new explanation for the quantum phase from the quantum mechanical point of view. The quantum phase is the canonically conjugate variable of an angular momentum operator, which corresponds to the angular position φ in an actual physical space with a classical reference frame, but it takes a complex exponential form e ^iφ≡cosφ+i sinφ in the abstract Hilbert space of a quantum reference frame. This formulation is simply the famous Euler formula in a complex number field. In particular, when φ = π/2, the correlative quantum phase is a unitary pure imaginary number e ^iπ/2≡cos(π/2)+i sin(π/2) ≡ i. By using a photon state-vector function that is the general solution of photon Schr?dinger equation and can completely describe a photon’s behavior, we discuss the relationship between the angular momentum of a photon and the phase of the photon; we also analyze the intrinsic relationship between the macroscopic light wave phase and the microscopic photon phase.     

Lattice Dislocations in a 1-Dimensional Model

The spectral properties of the Schr?dinger operator T(t)=−d ²/dx ²+q(x,t) in L ²(ℝ) are studied, where the potential q is defined by q=p(x+t), x>0, and q=p(x), x<0; p is a 1-periodic potential and t∈ℝ is the dislocation parameter. For each t the absolutely continuous spectrum σ _ac (T(t))=σ _ac (T(0)) consists of intervals, which are separated by the gaps γ _n (T(t))=γ _n (T(0))=(α _n ⁻,α _n ⁺), n≥1. We prove: in each gap γ _n ≠?, n≥ 1 there exist two unique “states” (an eigenvalue and a resonance) λ _n ^±(t) of the dislocation operator, such that λ _n ^±(0)=α _n ^± and the point λ _n ^±(t) runs clockwise around the gap γ _n changing the energy sheet whenever it hits α _n ^±, making n/2 complete revolutions in unit time. On the first sheet λ _n ^±(t) is an eigenvalue and on the second sheet λ _n ^±(t) is a resonance. In general, these motions are not monotonic. There exists a unique state λ₀(t) in the basic gap γ₀(T(t))=γ₀(T(0))=(−∞ ,α₀ ⁺). The asymptotics of λ _n ^±(t) as n→∞ is determined. Received: 5 April 1999 / Accepted: 3 March 2000     

Time Quasi-Periodic Unbounded Perturbations¶of Schrödinger Operators and KAM Methods

We eliminate by KAM methods the time dependence in a class of linear differential equations in ℓ² subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator H ₀+εP(ωt) for ε small. Here H ₀ is the one-dimensional Schr?dinger operator p ²+V, V(x)∼|x|^α, α <2 for |x|→∞, the time quasi-periodic perturbation P may grow as |x|^β, β <(α−2)/2, and the frequency vector ω is non resonant. The proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on Kuksin's estimate of solutions of homological equations with non-constant coefficients. Received: 3 October 2000 / Accepted: 20 December 2000     

Growth of Sobolev Norms in Linear Schrödinger Equations with Quasi-Periodic Potential

In this paper, we consider the following problem. Let iu _t +Δu+V(x,t)u= 0 be a linear Schr?dinger equation ( periodic boundary conditions) where V is a real, bounded, real analytic potential which is periodic in x and quasi periodic in t with diophantine frequency vector λ. Denote S(t) the corresponding flow map. Thus S(t) preserves the L ²-norm and our aim is to study its behaviour on H ^s (T ^D ), s> 0. Our main result is the growth in time is at most logarithmic; thus if φ∈H ^s , then

Cuspons and Smooth Solitons of the Degasperis–Procesi Equation Under Inhomogeneous Boundary Condition

This paper is contributed to explore all possible single peakon solutions for the Degasperis–Procesi (DP) equation m _t  + m _x u + 3mu _x  = 0, m = u − u _xx . Our procedure shows that the DP equation either has cusp soliton and smooth soliton solutions only under the inhomogeneous boundary condition lim_|x|→ ∞ u =A ≠0, or possesses the regular peakon solutions ce ^{− |x − ct|} ∈ H ¹ (c is the wave speed) only when lim_|x|→ ∞ u = 0 (see Theorem 4.1). In particular, we first time obtain the stationary cuspon solution of the DP equation. Moreover we present new cusp solitons (in the space of ) and smooth soliton solutions in an explicit form. Asymptotic analysis and numerical simulations are provided for smooth solitons and cusp solitons of the DP equation.      

Validity of the equivalent-photon approximation for the production of massive spin-1 particles

We point out that the equivalent-photon approximation (EPA) for processes with massive spin-1 particles in the final state would have validity in a more restricted kinematic domain than for processes where it is commonly applied, viz., those with spin-1/2 or spin-0 particles in the final state. We obtain the criterion for the validity ofEPA for the two-photon production of a pair of charged, massive, point-like spin-1 particlesV ^±, each of massM and with a standard magnetic moment (κ=1). In a process in which one of the photons is real and the other virtual with four-momentumq, the condition for the validity ofEPA is |q ²|≪M ², in addition to the usual condition |q ²|≪W ²,W being theV ⁺ V ⁻ invariant mass. In a process in which both photons are virtual (with four-momentaq andq′), our condition is |q ²||q′²|W ⁴ ≪ 16M ⁸, in addition to |q ²| ≪M ², |q′²| ≪M ² and |q ²| ≪W ², |q′²| ≪W ². Even when these extra conditions permitting the use ofEPA are not fulfilled, convenient approximate expressions may still be obtained assuming merely |q ²| ≪W ² and |q′²| ≪W ². We also discuss how the extra conditions are altered when the vector bosons are incorporated in a spontaneously broken gauge theory. Examples ofW boson production in Weinberg-Salam model are considered for which the condition |q ²||q′²|W ⁴ ≪ 16M ⁸ is shown to be removed.     

Bulk singularities at critical end points: a field-theory analysis

A class of continuum models with a critical end point is considered whose Hamiltonian [φ,ψ] involves two densities: a primary order-parameter field, φ, and a secondary (noncritical) one, ψ. Field-theoretic methods (renormalization group results in conjunction with functional methods) are used to give a systematic derivation of singularities occurring at critical end points. Specifically, the thermal singularity ∼ | t|^{2 - α} of the first-order line on which the disordered or ordered phase coexists with the noncritical spectator phase, and the coexistence singularity ∼ | t|^{1 - α} or ∼ | t|^β of the secondary density <ψ> are derived. It is clarified how the renormalization group (RG) scenario found in position-space RG calculations, in which the critical end point and the critical line are mapped onto two separate fixed points _CEP ^* and _λ ^*, translates into field theory. The critical RG eigenexponents of _CEP ^* and _λ ^* are shown to match. _CEP ^* is demonstrated to have a discontinuity eigenperturbation (with eigenvalue y = d), tangent to the unstable trajectory that emanates from _CEP ^* and leads to _λ ^*. The nature and origin of this eigenperturbation as well as the role redundant operators play are elucidated. The results validate that the critical behavior at the end point is the same as on the critical line. Received 18 January 2001     

Isospin violations in largeP T pion inclusive processes in perturbative quantum chromodynamics

We investigate the asymmetries arising due to electromagnetic interactions in largeP _T pion inclusive processes. The hardqcd processes that contribute to such asymmetries areq+g→q+γ,q+q→g+γ etc. which are suspected to be substantial, as indicated by theqcd predictions for a significant and increasingγ/π ^o ratio at largeP _T. We calculate the expected isospin related asymmetries and propose tests that might detect them. Our estimates indicate that the effects are much smaller than may be naively expected. We also observe a remarkable scaling of asymmetries in the variableP _T/(s)^1/2.     

Improvement of the pinning properties of (Bi/Pb)-2223 textured ceramics by columnar defects introduced by 6 GeV Pb ions

Pinning properties of (Bi/Pb)-2223 textured ceramics have been improved by 6 GeV Pb ions irradiation. Two samples were irradiated at fluencesφ _t=5 × 10¹⁰ ions/cm² and 2 × 10¹¹ ions/cm², respectively. The magnetic irreversibility is clearly improved in the intermediate range of temperature: (30–35) K<T<60 K for magnetic fields lower than the “fluence equivalent field”B _φ =φ _t.φ ₀. Moreover the irreversibility line is shifted towards higher fields. The results reported here show that the introduction of columnar defects or other extended defects is a route to improve the screening properties of tubes or cans based on (Bi/Pb)-2223 material and operating in liquid nitrogen.     

On certain correspondences among various coupled-cluster theories for closed-shell systems

In this paper certain correspondences have been shown among various formulations of coupled-cluster theories for many electron closed-shell systems. Specifically it is shown that (i) the energy functional using unitary ansatz of the form exp (T−T ⁺) is exactly sameorder by order inT with the size-consistent energy functional 〈ψ|H|ψ〉/〈ψ|ψ〉 recently obtained by us in coupled-cluster framework; (ii) in the framework of unitary ansatz of the form exp (T−T ⁺), both non-variational and variational approaches lead to identical equations upto any given order inT andT ⁺ inT∼T ₂ approximation; (iii) variational procedure using our size-consistent energy functional or using the functional obtained in the framework of unitary ansatz (as envisaged by Kutzelnigg) leads to energy in both cases, inT∼T ₂ approximation, for a total of quadratic powers inT andT ⁺, same as Cizek’s linearised coupled pair many electron theory energy; (iv) in case of practical calculation of the energy through the variational procedures using our size-consistent energy functional and the functional in unitary ansatz framework, there is a loss of upper bound.     

Metastable Behavior for Bootstrap Percolation on Regular Trees

We examine bootstrap percolation on a regular (b+1)-ary tree with initial law given by Bernoulli(p). The sites are updated according to the usual rule: a vacant site becomes occupied if it has at least θ occupied neighbors, occupied sites remain occupied forever. It is known that, when b>θ≥2, the limiting density q=q(p) of occupied sites exhibits a jump at some p _T=p _T(b,θ)∈(0,1) from q _T:=q(p _T)<1 to q(p)=1 when p>p _T. We investigate the metastable behavior associated with this transition. Explicitly, we pick p=p _T+h with h>0 and show that, as h ↓0, the system lingers around the “critical” state for time order h ^−1/2 and then passes to fully occupied state in time O(1). The law of the entire configuration observed when the occupation density is q∈(q _T,1) converges, as h ↓0, to a well-defined measure.     

Fractional Non-Linear,Linear and Sublinear Death Processes

This paper is devoted to the study of a fractional version of non-linear \mathfrakMⁿ(t)\mathfrak{M}^{\nu}(t), t>0 death processes. Fractionality is introduced by replacing the usual integer-order derivative in the difference-differential equations governing the state probabilities, with the fractional derivative understood in the sense of Dzhrbashyan–Caputo. We derive explicitly the state probabilities of the three death processes and examine the related probability generating functions and mean values. A useful subordination relation is also proved, allowing us to express the death processes as compositions of their classical counterparts with the random time process T _2ν(t), t>0. This random time has one-dimensional distribution which is the folded solution to a Cauchy problem of the fractional diffusion equation.     

Integrable Equations of the Form q t =L 1(x,t,q,q x,q xx )q xxx +L 2(x,t,q,q x,q xx )

Integrable equations of the form q _t =L ₁(x,t,q,q _x ,q _xx )q _xxx +L ₂(x,t,q,q _x ,q _xx ) are considered using linearization. A new type of integrable equations which are the generalization of the integrable equations of Fokas and Ibragimov and Shabat are given.     

Electronic plus phonon-exchange mechanism for high-temperature superconductivity in layered crystals

A general mathematical formulation is developed for calculating the effective electron-electron interaction in layered crystals like YBa₂Cu₃O_7−δ, and for finding the resulting superconducting transition temperatureT _c in such systems within the framework of the conventional BCS pairing arising from various possible excitations in the medium. This differs considerably from the usual case of an effective three-dimensional homogeneous system, and should be relevant in the calculation ofT _c for the new class of high-T _c perovskites in which oxygen deficiencies in Cu-O layers and their distribution in the crystal play a crucial role. The explicit form of the effective interactionV _jj(q _t,ω) in a given layerj in the unit cell of the crystal is found to be determined not only by the true polarization functionπ _j(q _t,ω) of that layer, but also of other layers. The exchange of electronic excitations of a nearby insulating layer by carriers in a conducting layer thus becomes possible to get highT _c, with or without the usual phonon exchange.     

Dynamics in molecular crystals: An application of proton NMR to selectively protonated biphenyl. Do the rings flip together or independently?

The synthesis of a specific isotopomer, C₆D₄H(ortho)-H(ortho)D₄C₆ of biphenyl is reported. The intramolecular dipolar coupling of the protons leads to a well-resolved single-crystal proton nuclear magnetic resonance (NMR) spectrum and allows one to study the dynamics of the phenyl rings in a unique way. At room temperature and above, the most conspicuous dynamical mode consists of 180° ring flips. The present data together with previous measurements of the total flip rate allow us to conclude that the rings flip almost exclusively independently of each other. Between the incommensurate (IC) phase transition of biphenyl at 38 K andT=250 K, the prominent namical mode consists of oscillatory twists ϕ(t) of the two rings. The data allow us to infer the mean square, (φ²), of these twists. (φ²) is found to grow linearly withT for 50<T<200 K. From the slope of (φ²) vs.T the frequency (the wave number[(v)\tilde]\tilde v) is derived. The result is[(v)\tilde] = 20\tilde v = 20 cm⁻¹. ForT<38 K, the spectra give direct evidence of the IC phase transition and its nature (stripelike rather than quiltlike). The temperature dependence of the magnitude of the order parameter of the IC phase is obtained.     

Studies of the quantum Hall to quantum Hall insulator transition in InSb-based 2DESs

The temperature dependence of ρ_xx is studied in the vicinity of the quantum Hall to quantum Hall insulator transition (ν=1→0) in InSb/InAlSb based 2DESs. ρ_xx displays a symmetric temperature dependence about the transition with on the QH side and on the insulating side. A plot of 1/T₀ for successive ν displays power-law divergence with 1/T₀∝|ν−ν_c|^−γ,² with γ=2.2±0.3. This critical behavior in addition to the behavior expected of the quantum transport regime confirms that the QH/QHI transition is indeed a good quantum phase transition.     

On the phase boundaries of the integer quantum Hall effect. Part II

It has been shown that the observation of the transitions between the dielectric phase and the integer-quantum-Hall-effect phases with the quantized Hall conductivity σ _xy ^q ≥ 3e ²/h announced in a number of works is unjustified. In these works, the crossing points of the magnetic-field dependence of the diagonal resistivity ρ _xx at different temperatures T and ω_cτ = 1 have been misidentified as the critical points of the phase transitions. In fact, these crossing points are due to the sign change of the derivative dρ _xx /dT owing to the quantum corrections to the conductivity. Here, ω_c = eB/m is the cyclotron frequency, τ is the transport relaxation time, and m is the effective electron mass.     

A Pregenerator for Burgers Equation¶Forced by Conservative Noise

We consider the stationarity of a Burgers equation with an external random force of gradient type in one space dimension. The expected stationary measure is the white noise measure on the space of tempered distributions. As a consequence, the nonlinearity of the formal equation u _t +λu u _x =νu _xx +η _x is ill-defined. Introducing a pregenerator we can formulate a generalized martingale problem leading to a meaningful version of the formal equation which was an open problem. Received: 9 March 2001 / Accepted: 10 October 2001     

One Dimensional Behavior of Singular N Dimensional Solutions of Semilinear Heat Equations

We consider u(x,t) a solution of u _t =Δu+|u| ^{p − 1} u that blows up at time T, where u:ℝ ^N ×[0, T)→ℝ, p>1, (N−2)p<N+2 and either u(0)≥ 0 or (3N−4)p<3N+8. We are concerned with the behavior of the solution near a non isolated blow-up point, as T−t→ 0. Under a non-degeneracy condition and assuming that the blow-up set is locally continuous and N−1 dimensional, we escape logarithmic scales of the variable T−t and give a sharper expansion of the solution with the much smaller error term (T−t)^{1, 1/2−η} for any η>0. In particular, if in addition p>3, then the solution is very close to a superposition of one dimensional solutions as functions of the distance to the blow-up set. Finally, we prove that the mere hypothesis that the blow-up set is continuous implies that it is C ^{1, 1/2−η} for any η>0. Received: 20 June 2001 / Accepted: 6 October 2001     

Hilbert Modules and Stochastic Dilation of a Quantum Dynamical Semigroup on a von Neumann Algebra

A general theory for constructing a weak Markov dilation of a uniformly continuous quantum dynamical semigroup T _t on a von Neumann algebra ? with respect to the Fock filtration is developed with the aid of a coordinate-free quantum stochastic calculus. Starting with the structure of the generator of T _t , existence of canonical structure maps (in the sense of Evans and Hudson) is deduced and a quantum stochastic dilation of T _t is obtained through solving a canonical flow equation for maps on the right Fock module ?⊗Γ(L ²(ℝ₊,k ₀)), where k ₀ is some Hilbert space arising from a representation of ?^′. This gives rise to a *-homomorphism j _t of ?. Moreover, it is shown that every such flow is implemented by a partial isometry-valued process. This leads to a natural construction of a weak Markov process (in the sense of [B-P]) with respect to Fock filtration. Received: 15 June 1998/ Accepted: 4 March 1999