Computable upper bounds for the adiabatic approximation errors

For a given Hermitian Hamiltonian H(s)(s∈[0,1])with eigenvalues Ek(s)and the corresponding eigenstates|Ek(s)(1 k N),adiabatic evolution described by the dilated Hamiltonian HT(t):=H(t/T)(t∈[0,T])starting from any fixed eigenstate|En(0)is discussed in this paper.Under the gap-condition that|Ek(s)-En(s)|λ0 for all s∈[0,1]and all k n,computable upper bounds for the adiabatic approximation errors between the exact solution|ψT(t)and the adiabatic approximation solution|ψadi T(t)to the Schr¨odinger equation i|˙ψT(t)=HT(t)|ψT(t)with the initial condition|ψT(0)=|En(0)are given in terms of fidelity and distance,respectively.As an application,it is proved that when the total evolving time T goes to infinity,|ψT(t)-|ψadi T(t)converges uniformly to zero,which implies that|ψT(t)≈|ψadi T(t)for all t∈[0,T]provided that T is large enough.     

Quantum gravity at very high energies

The problem of time and the quantization of three-dimensional gravity in the strong coupling regime is studied following path integral methods. The time is identified with the volume of spacetime. We show that the effective action describes an infinite set of massless relativistic particles moving in a curved three-dimensional target space, i.e., a tensionless 3-brane on a curved background. If the cosmological constant is zero the target space is flat and there is no ‘‘ graviton" propagation (i.e., G[g_ij(2),g_ij(1)]=0). If the cosmological constant is different from zero, 3D gravity is both classical and quantum mechanically soluble. Indeed, we find the following results: (i) the general exact solutions of the Einstein equations are singular at t=0 showing the existence of a big-bang in this regime and (ii) the propagation amplitude between two geometries 〈g_ij(2),t₂|g_ij(1),t₁〉 vanishes as t→0, suggesting that big-bang is suppressed quantum mechanically. This result is also valid for D>3.     

The microwave spectrum of IF5 in excited vibrational states: Accidental degeneracy and avoided crossing of the kl = 3 and kl = −1 levels in the state v9(E) = 1

Measurements of the microwave spectrum of the C_4v molecule IF₅ in the excited vibrational states v₉(E) = 1 and v₅(B₁) = 1 are reported for the transitions J12 ← 11 and J13 ← 12 (65–72 GHz). The considerable spectral perturbations produced by an accidental degeneracy and avoided crossing of the ψ^? (kl = 3) and ψ^? (kl = ?1) levels of the v₉(E) = 1 state have been measured and analyzed. A spectroscopic determination of the axial rotation constant C₉ is reported and its implications for the molecular structure of IF₅ discussed.     

Intermittency in the atmospheric surface layer: Unresolved or slowly varying?

We present streamwise velocity structure functions 〈δv_L(τ)〉=〈|v(t+τ)−v(t)|^p〉 (with p=1:5) obtained in the near neutral atmospheric surface layer at the Utah SLTEST site at the highest terrestrial Reynolds number Re_τ=O(10⁶). We show that the occurrence of very large scale coherent oscillations in the streamwise velocity throughout the wall region, interpreted as genuine structural features of the canonical turbulent boundary layer, affects the scaling exponents of the p>3 order structure functions. This results in a slight alteration of the intermittent behavior of the velocity field. It was found that for positive (fast) large scale oscillation of the low-pass filtered velocity signal, deviations from the Kolmogorov K41 prediction (absence of multiscaling) are more marked, as compared to negative (slow) excursion. The results are discussed in terms of convergence of statistics from atmospheric boundary layer measurements.     

Properties of ψN scattering

Extracting ψN scattering from photoproduction requires vector meson dominance and diffractive assumptions. We observe that (i) kinematic corrections to standard vector dominance formulas are important; (ii) both threshold and high energy behaviors of σ_t(ψN) support the diffractive assumption; (iii) q²-dependence in the γψ coupling could imply larger ψN cross sections; (iv) in any case σ_t is bounded through the slope of dσ/dt and σ_el/σ_t remains small.     

Quantum extremism: effective potential and extremal paths

The reality and convexity of the effective potential in quantum field theories has been studied extensively in the context of Euclidean space-time. It has been shown that canonical and path-integral approaches may yield different results, thus resolving the convexity problem. We discuss the transferal of these treatments to Minkowskian space-time, which also necessitates a careful discussion of precisely which field configurations give the dominant contributions to the path integral. In particular, we study the effective potential for the N=1 linear sigma model.     

Nonergodic dynamics of a system of nuclear spins 1/2 with identical spin-spin coupling constants

The exact solution for the evolution of nuclear spin polarizations in a system with spin-spin coupling constant g identical for all spin pairs is obtained on the condition that only one (first) spin is polarized at zero time. It is shown that the polarization P ₁(t) of the first spin has the form of periodic time pulsations with the period 4π/g. In every period, the function P ₁(t) changes from its initial value P(0)=1 to 1/3 during the time on the order of t≈4π/Ng, if the number of spins N?1, and remains in the state P ₁(t)=1/3 virtually during the entire period. A simple classical model within the framework of mean-field theory explains the physical nature of the nonergodic dynamics of the system.     

Domain Number Distribution in the Nonequilibrium Ising Model

We study domain distributions in the one-dimensional Ising model subject to zero-temperature Glauber and Kawasaki dynamics. The survival probability of a domain, S(t)～t ^?ψ, and an unreacted domain, Q ₁(t)～t ^?δ, are characterized by two independent nontrivial exponents. We develop an independent interval approximation that provides close estimates for many characteristics of the domain length and number distributions including the scaling exponents.     

Eigenvalue comparisons for quantum systems in a box

Comparison theorems are obtained for the first even and odd solutions of Schrödinger's equation −v″ + Q(t)v = λv, −l ≤ t ≤ l with boundary conditions v(−l) = v(l) = 0. The comparison functions Q_i(t), i = 1, 2, may intersect at a finite number of points within [−l,l]. Immediate extensions are possible for a more general class of Sturm-Liouville problems, and for problems in unbounded regions.     

On the possible coexistence of itinerant-electron antiferromagnetism with the itinerant-electron ferromagnetism

It is shown that in an electron gas with coexisting charge density wave and spin density wave a spontaneous magnetization can occur. In the first paramagnetic region (T,T_t;Δ_s = Δ_t = 0) the spin susceptibility is independent of temperature (Pauli paramagnetism). In the second paramagnetic region (T_K<T<T_t,Δ_t≠0;Δ_s=0) the total susceptibility of the spin density wave is enhanced by a temperature dependent paramagnetic susceptibility with a Curie type law. This reflects the instability of the system to the formation of a charge density wave or to ferromagnetism. At sufficiently low temperatures (T～T_K) the spin susceptibility is anomalous great a fact which cannot be explained with the standard Fedders-Martin theory. Experimental results for pure chromium are in good agreement with our predictions.     

Functional integral representations of partition function without limiting procedure. Techniques of calculation of moments

A definition of the Feynman path integral which does not rest on a limiting procedure based on time-slicing has been given by DeWitt-Morette. We present in this paper a discussion of real Gaussian measures and formulate expressions for the quantum statistical partition function directly in terms of measures of integration on the topological vector space ø₀ of continuous functions defined on the time intervalT = (t _a ,t _b ), such thatx(t _a ,t _b )=0 for allx ? ø₀. We give a definition of a measure for the space ø₀ equivalent to the path integral based on the Uhlenbeck-Ornstein probability distribution. We give expressions for the partition function using the Wiener-Feynman measure and the Uhlenbeck-Ornstein measure. As an exercise in the use of the new techniques, we present calculations of moments of potential functions. The techniques will enable one to solve in a rigorous manner practical problems in quantum statistical mechanics.     

Haldane limits via Lagrangian embeddings

In the present paper we revisit the so-called Haldane limit, i.e. a particular continuum limit, which leads from a spin chain to a sigma model. We use the coherent state formulation of the path integral to reduce the problem to a semiclassical one, which leads us to the observation that the Haldane limit is closely related to a Lagrangian embedding into the classical phase space of the spin chain. Using this property, we find a spin chain whose limit produces a relativistic sigma model with target space the manifold of complete flags U(3)/U³(1). We discuss possible other future applications of Lagrangian/isotropic embeddings in this context.     

Franck-condon factors for diatomic molecules and the factorization method

By means of the factorization method, recurrence relations for the matrix elementS$\text{<ψ}{}_{\mathrm{<mtext>v\text{'}</mtext>}}\text{¦r}{}^{\mathrm{N}}\text{¦ψ}{}_{\mathrm{<mtext>v</mtext>}}\text{<}$ for an arbitrary positive integer N are obtained in the framework of α-averaging for the Morse potential. The method of calculating the matrix element$\text{<ψ}{}_{\mathrm{<mtext>v\text{'}</mtext>}}\text{¦R}{}_{\mathrm{<mtext>e</mtext>}}\text{¦ψ}{}_{\mathrm{<mtext>v</mtext>}}\text{<}$ is outlined, provided a dependence of the electronic transition moment R_e on the internuclear distance r can be approximated by the polynomial R_e(r)=a₀+a₁r+a₂r²+…+ainNr^N. Recurrence relations are also obtained for the case of the exponential dependence R_e(r).     

Construction of kernels of the nonlinear collision integral in the Boltzmann equation using laplace transformation

It is shown that the relation between kernels L _l (v, v ₁) of the linear collision integral and kernels G _l,0 ^l (v, v ₁, v ₂) of the nonlinear collision integral can be reduced to the Laplace transformation. Analytic expressions for nonlinear kernels G _0,0 ⁺⁰ (v, v ₁, v ₂) and G _1,0 ⁺¹ (v, v ₁, v ₂) are determined for hard spheres and pseudo-Maxwellian molecules.     

Charmonium production at the Tevatron and Large Hadron Collider in the Regge limit of QCD

Processes involving direct J/ψ-meson production in proton-antiproton and proton-proton collisions are studied under the assumption of gluon Reggeization in t-channel exchanges and with the aid of the formalism of nonrelativistic quantum chromodynamics. The present calculations are performed in the leading approximation in the strong coupling constant α _s and in the relative quark velocity v. The octet nonperturbative matrix elements for the transition of a c $\bar c$ pair to quarkonium are fixed in fitting the transverse-momentumspectra obtained by the CDF Collaboration for J/ψ mesons at the Tevatron collider. The spectra of J/ψ-meson production at the energies of the Large Hadron Collider are predicted, and the resulting predictions agree well with experimental data obtained by the ATLAS, CMS, and LHCb Collaborations at $\sqrt S $ = 7 TeV.     

Gamow Vectors and Borel Summability in a Class of Quantum Systems

We analyze the detailed time dependence of the wave function ψ(x,t) for one dimensional Hamiltonians \(H=-\partial_{x}^{2}+V(x)\) where V (for example modeling barriers or wells) and ψ(x,0) are compactly supported.We show that the dispersive part of ψ(x,t) is the Borel sum of its asymptotic series in powers of t ^?1/2, t→∞. The remainder, the difference between ψ and the Borel sum, i.e., the exponential part of the transseries of ψ, is a convergent expansion of the form \(\sum_{k=0}^{\infty}g_{k}\Gamma_{k}(x)e^{-\gamma_{k} t}\), where Γ _k are the Gamow vectors of H, and iγ _k are the associated resonances; generically, all g _k are nonzero. For large k, γ _k ～const?klog?k+k ² π ² i/4. The effect of the Gamow vectors is visible when time is not very large, and the decomposition defines rigorously resonances and Gamow vectors in a nonperturbative regime, in a physically relevant way.The decomposition allows for calculating ψ for moderate and large t, to any prescribed exponential accuracy, using optimal truncation of power series plus finitely many Gamow vectors contributions.The analytic structure of ψ is perhaps surprising: in general (even in simple examples such as square wells), ψ(x,t) turns out to be C ^∞ in t but nowhere analytic on ?⁺. In fact, ψ is t-analytic in a sector in the lower half plane and has the whole of ?⁺ a natural boundary. In the dual space, we analyze the resurgent structure of ψ.     

Spin frustration,charge ordering,and enhanced antiferromagnetism in TMTTF2SbF6

We theoretically investigate the effects of charge order and spin frustration on the spin ordering in TMTTF salts. Using first-principles band calculations, we find that a diagonal inter-chain transfer integral t_q1, which causes spin frustration between the inter-chain dimers in the dimer-Mott insulating state, strongly depends on the choice of anion. Within the numerical Lanczos exact diagonalization method, we show that the ferroelectric charge order changes the role of t_q1 from the spin frustration to the enhancement of the two-dimensionality in spin sector. The results indicate that t_q1 assists the cooperative behavior between charge order and antiferromagnetic state observed in TMTTF₂SbF₆.