Large order expansion in perturbation theory

We investigate the large n behavior of the perturbation coefficients E_n for the ground state energy of the anharmonic oscillator, considered as a field theory in one space-time dimension. We combine the saddle point expansion for functional integrals introduced in this context by Lipatov with the dispersion relation (in coupling constant) used by Bender and Wu. The complete Feynman rules for the expansion in $\text{1}\text{n}$ are worked out, and we compute the first two terms, which agree with those computed by Bender and Wu using the WKB approximation. One feature of our analysis is a deformation of the integration contour in function space as one analytically continues in the coupling.     

Intrinsic surface state on Be(0 0 0 1)

Using monochromatized synchrotron radiation in the range 24–30 eV, we have recorded angle-resolved photoemission spectra from a clean Be(0 0 0 1) crystal face. A surface state located in a band gap around Γ with an initial state energy of ?2.8 eV in normal emission was found. For k_∥ along the $\text{Γ}$$\text{M}$ line the surface state disperses upwards and passes E_F at about 55% of the distance to the surface Brillouin zone boundary.     

Quantum maps

We quantize area-preserving maps M of the phase plane q, p by devising a unitary operator $\text{U}$ transforming states | φ_n〉 into | φ_n+1〉. The result is a system with one degree of freedom q on which to study the quantum implications of generic classical motion, including stochasticity. We derive exact expressions for the equation iterating wavefunctions ψ_n(q), the propagator for Wigner functions W_n(q,p), the eigenstates of the discrete analog of the quantum harmonic oscillator, and general complex Gaussian wave packets iterated by a $\text{U}$ derived from a linear M. For | ψ_n〉 associated with curves $\text{L}$_n in q, p, we derive a semiclassical theory for evolving states and stationary states, analogous to the familiar WKB method. This theory breaks down when $\text{L}$_n gets so complicated as to develop convolutions of area ? or smaller. Such complication is generic; its principal morphotologies are“whorls” and “tendrils,” associated respectively with elliptic and hyperbolic fixed points of M. Under $\text{U}$, ψ_n(q) eventually transforms into a new sort of wave that no longer perceives the details of $\text{L}$_n. For all regimes, however, the smoothed | ψ_n(q)|² appears semiclassically appears to be given accurately by the smoothed projection of $\text{L}$_n onto the q axis, both smoothings being over a de Broglie wavelength. The classical, quantum, and semiclassical theory is illustrated by computations on the discrete quartic oscillator map. We display for the first time stochastic wavefunctions, dominated by dense clusters of caustics and characterized by multiple scales of oscillation.     

Bosonic construction of the Lie algebras of some non-compact groups appearing in supergravity theories and their oscillator-like unitary representations

We give a construction of the Lie algebras of the non-compact groups appearing in four dimensional supergravity theories in terms of boson operators. Our construction parallels very closely their emergence in supergravity and is an extension of the well-known construction of the Lie algebras of the non-compact groups $\text{SP}\text{(2n,}\text{R}$ and $\text{SO}\text{(2n)}{}^1$ from boson operators transforming like a fundamental representation of their maximal compact subgroup U(n). However this extension is non-trivial only for n?4 and stops at n = 8 leading to the Lei algebras of SU(4) × SU(1, 1), SU(1, 1), SU(5, 1), SO(12)¹ and E₇(7). We then give a general construction of an infinite class of unitary irreducible representations of the respective non-compact groups (except for E₇₍₇₎ and SO(12)¹ obtained from the extended construction). We illustrate our construction with the examples of SU(5, 1) and SO(12)¹.     

Perturbation theory with a tridiagonal zeroth-order hamiltonian

We generalize the Rayleigh-Schrödinger perturbation formalism to the hamiltonians H=H₀+λH₁ where the correction λH₁ is small and the unperturbed operator H₀ is represented by an infinite tridiagonal matrix. This enables us to construct the solutions E=E₀+λE₁+λ²E₂+… and |ψ〉 = |ψ₀〉+λ|ψ₁〉+λ²|ψ₂〉+… in terms of the analytic continued fractions.     

Scaling function for order-parameter correlations in expansion to order 1/n: I. Short-range interactions

The order-parameter correlation function $\text{G}\text{?}\text{(}\text{q}\text{, ξ}{}_1\text{)}$ is calculated in the critical region of momentum space $\text{q}$ in terms of a second-moment correlation length ξ₁ by means of perturbation expansion to order 1/n, for an n-vector system with short-range interactions, in zero field above T_c, for 2 < d < 4. The scaling function of the $\text{q}$ dependence is obtained in closed form with a precisely identified cutoff-dependent factor which is the amplitude of the correlation-length dependence of the susceptibility. Both the exponents and the coefficients of the expansion for fixed q as t = (T?T_c)/T_c → 0 are given explicitly and the former are shown to be in accordance with the operator product expansion. The coefficients of order 1/n in the terms associated with a t^k(1?α) dependence of the energy density, for integer k ≥ 1, are expected to be explicitly cutoff-dependent and this is verified by the detailed calculations for k = 1. The behaviour for fixed t and q → 0 is shown to be markedly different from the Ornstein-Zernike approximation. Detailed comparison is provided with the scaling function of the t dependence of the correlations appearing in parallel work.     

The T = 1, isospin triplet states in A = 30 nuclei

The level structure of ³⁰S was studied using the ${}^{28}\text{Si}\text{(τ,}\text{n}\text{p}\text{)}\text{reaction at}\text{E}{}_{\mathrm{\tau }}\text{= 9.5}\text{MeV}\text{,}\text{and}\text{J}{}^{\mathrm{\pi }}$ values were determined for some proton-emitting states by the neutron-proton angular correlation measurements. The Coulomb displacement energy of the isospin triplet state of A = 30 nuclei was analyzed using first-order perturbation theory. Large deviations from perturbation theory were found for the ground state, the 2⁺₁ state, the 3^?₁ state and the second excited 0⁺ state. These results might be explained by the effect of the pairing correlation of the two valence protons in ³⁰S under the assumption of charge symmetry for the nucleon-nucleon interaction.     

Some studies of T = 32 states in 17F

Measurements have been made of some parameters of the second and sixth $\text{T =}\text{3}\text{2}$ states in ¹⁷F. For the second state, the resonance energy was found to be E_p = 12.707 ± 0.001 MeV (E_n = 12.550±0.001 MeV), which agrees with and improves on the accuracy of earlier work. For the sixth $\text{T =}\text{3}\text{2}$ state, at E_p = 14.435 MeV, the γ-decay was determined to be predominantly γ₀ with a branch to the first excited state of Γ(γ₁)/Γ(γ₀) ≦ 0.14. Together with other work, this determines J^π to be $\text{3}\text{2}{}^{\mathrm{?}}$. The capture strength is found to be (2J + 1)Γ_pΓ_γ/Γ = 11.4 ± 2.6 eV.     

Quantizing a classically ergodic system: Sinai's billiard and the KKR method

Sinai's “billiards on a torus,” i.e., free motion of a particle in a plane amongst reflecting discs of radius R centred on points of the unit square lattice, is a classically ergodic system with two freedoms, parametrized by R. Quantal energy levels E_n are given by the vanishing of the Korringa-Kohn-Rostoker (KKR) determinant of solid state theory. This gives a rapid computational scheme for computing E_n as functions of R. Except for the integrable case R = 0, no degeneracies are found, illustrating the theorem that two parameters, not one, are required to make levels cross in a generic system. The same theorem leads to the prediction that the probability distribution of the spacings S of neighbouring levels is $\text{O}$(S) as S → 0, in good agreement with computation. The KKR determinant is transformed analytically to give the level density d(E) semiclassically (i.e., as ? → 0) as the sum of a steady contribution $\text{d}\text{?}\text{(E)}$ and an oscillatory contribution d_osc(E). $\text{d}\text{?}$ is $\text{O}$(?^?2) and is given by the Weyl “area” formula plus “edge,” “corner” and “curvature” corrections, in excellent agreement with computation. d_osc is given by a sum over classical closed orbits (all unstable). Nonisolated closed orbits (not hitting discs) contribute terms with $\text{O}$$\text{(?}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext><mtext>)</mtext>}}$ to d_osc, while isolated closed orbits (bouncing between discs) contribute terms with $\text{O}$(?^?1) to d_osc. The isolated orbits are vastly more numerous than the nonisolated orbits and their contributions cannot be neglected. As a means of calculating the individual E_n (rather than the smoothed spectrum), the KKR method is much more efficient than the classical path sum.     

Comparison of calculated photoionization cross-sections of diamond and silicon in the approximations of plane wave and orthogonalized plane wave

In the approximation of the orthogonalized plane wave the shape of X-ray photoelectron spectra of the valence band in diamond and silicon has been calculated. It is shown that consideration of orthogonality terms influences greatly the value of the ratio between photoionization cross-sections of s- and p-electrons. X-ray emission and photoelectron spectroscopy allow us to define the density of states for silicon valence electrons.X-ray photoelectron spectroscopy is at present widely used for the study of the electronic structure of solids. The photoelectron spectra of crystals clearly reveal all the variations in the density of states. However the curves for the photoelectron energy distribution may be different from the calculated density of states for valence electrons. This indicates the importance of the transition probability for the formation of a photoelectron spectrum. In refs. 1 and 2 the X-ray photoelectron spectra obtained with high resolution for diamond and silicon crystals are compared with the density of states and the conclusion made that the s-states lie higher than the p-states. Densities of states and X-ray photoelectron spectra for diamond and silicon have been calculated³. The wave functions of the valence electrons were found by the tight-binding method⁴ while plane waves were used as wave functions for the excited electrons. From a theoretical point of view it is more reasonable to use orthogonalized plane waves to describe the electron states in the conduction band. Both the core and valence states are to be orthogonalized.The present work reports calculation of the shape expected for X-ray photo-electron spectra of diamond and silicon valence electrons in the approximation of orthogonalized plane wave. Investigation was also made of the influence caused by the orthogonality terms on the ratio of photoionization cross-sections of s- and p-electrons. The electronic structure of diamond and silicon was calculated also by the tight-binding method with the use of the parameters from ref. 3. X-ray photo-electron spectra of valence electrons were calculated over 5230 points in 1/48 part of the Brillouin zone. For simplicity it was assumed that the polarization vector of the electromagnetic wave A is directed along the crystal Z-axis. As was done in ref. 3, the X-ray photoemission intensity was averaged over angular variables. As an example we shall give the formula used for calculation of the X-ray photoelectron spectrum for diamond I(ω, E) ～ ∝E σ_n,k [($\text{1}\text{3}$ET_2s² + $\text{1}\text{3}$T_2p²U_2p?2s² - $\text{2}\text{3}$ ∝ET_2sT_2pU_2p?2s)|C_sⁿ(K|² + $\text{1}\text{5}$ET_2p² (|C_xⁿ(K|² + $\text{3}\text{5}$ET_2p² + $\text{1}\text{3}$T_1sU_1s-2p + T_2sU_2s-2p)² + $\text{2}\text{3}$ √ET_2p(T_1sU_1s-2p + T_2sU_2s-2p) √C_zⁿ(k)|²] where T_nl(E) = fx_O^∞ r²R_nl(tr)_jl(∝Erdr and U_ij = (E_i-E_j fx8r³R_iR_j(r)R_j(r)dr R_i(r) is the radial part of the atomic wave function, C_iⁿ(k), is found from the Schrödinger equation by the tight-binding method. A similar formula is valid for silicon but the number of integrals T_nl and U_ij will be larger owing to the fact that there are more electron states in the silicon atomic core. In eqn. (1) the terms | C_xⁿ|²,|C_yⁿ|² and |C_zⁿ|² have different factors because the A vector is directed along the crystal Z-axis. These factors will be the same when the A vector is directed along (111). Therefore the contribution of p-electrons to the photoelectron spectrum will be proportional to the partial density of p-states.The formula (1) is simplified in the approximation of a plane wave I(ω, E ～ E^{$\text{3}\text{2}$} Σ_n,k[T_s²$\text{1}\text{3}$|C_sⁿ(k)|² + T_p² ($\text{1}\text{5}$|C_xⁿ(k)|² + C_yⁿ(k)|² + C_zⁿ(k)|²)] (2) Figures 1 and 2 show the results obtained from eqns. (1), (2). Here both the spectra calculated in the plane wave approximation and those found experimentally are given. The ratio between maximum III (p-states prevail) and maximum I (s-states dominate) is given in Table 1.As can be seen from Table 1 and Figs. 1 and 2, the spectra calculated in the

4. Relationship between maximum III and maximum I and between photoionization cross-sections of s- and p- electrons in diamond and silicon

     

11.

Nonexponential decay law     

Asher Peres 《Annals of Physics》1980,129(1):33-46

The decay of a nonstationary state usually starts as a quadratic function of time and ends as an inverse power law (possibly with oscillations). Between these two extremes, the familiar exponential decay law may be approximately valid. The main purpose of this paper is to find the conditions which must be satisfied by the Hamiltonian and by the initial state, for the exponential law to have a significant domain of validity. It is shown that the evolution of a nonstationary state is governed by a nonnegative function W(E), having the dimensions of an energy. Among its properties are: the energy uncertainty is given by $\text{(ΔH)}{}^2\text{= ?W(E)dE}$, and the inverse lifetime by Γ = 2πW(E₀), where E₀ is the expectation value of H. The detailed shape of W(E) defines two characteristic times between which the exponential decay law is a good approximation: roughly speaking, the smoother W(E), the larger the domain of validity of the exponential law. For instance, if W(E) is very smooth ($\text{|}\text{dW}\text{dE}\text{| ? 1}$) except for a sharp threshold at E = E_thr, the transition from quadratic to exponential decay occurs for $\text{t ?}\text{1}\text{(E}{}_0\text{? E}{}_{\mathrm{<mtext>thr</mtext>}}\text{)}$, and the transition from exponential to inverse power law when $\text{Γt ?}\text{log}\text{[}\text{(E}{}_0\text{? E}{}_{\mathrm{<mtext>thr</mtext>}}\text{)}\text{Γ}\text{]}$.     

12.

Energy gaps in the fractional quantum hall effect     

《Solid State Communications》1986,60(6):501-504

Based on a density-of-states N(E) ∝ (E − μ₀)² Poisson's equation is solved perpendicular to the current direction in the two-dimensional Hall-plate. By comparing the result with the structure obtained for an Ising-like chain with occupation numbers $\text{σ = 0,}\text{1}\text{2}$, 1 to account for vanishing energy gaps for filling factors $\text{v =}\text{m}\text{n}$ with n even, the energy gaps for n odd are found essentially to be proportional to $\text{B}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{n}{}^{\mathrm{-2}}$ in accordance with recent results.     

13.

Computable upper bounds for the adiabatic approximation errors     

BaoMin Yu  HuaiXin Cao  ZhiHua Guo  WenHua Wang 《中国科学:物理学 力学 天文学(英文版)》2014,57(11):2031-2038

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.     

14.

Fission-fragment energy correlation measurements for (n_th, f) of ²³²U and ²³³U     

M. Asghar  F. Caïtucoli  B. Leroux  P. Perrin  G. Barreau 《Nuclear Physics A》1981,368(2):328-336

Fission-fragment mass and kinetic energy distributions and their correlations have been measured for ²³²U and ²³³U. The results on these uranium isotopes and ²³⁵U are compared. The mass peak/valley ratio of 785 ± 68 for ²³²U is the highest of the three isotopes. The 〈E_K〉(μ_H) distributions show significant differences. The dip ΔE_K at symmetry is $\text{16.2 ± 1.0}\text{MeV}\text{(}{}^{232}\text{U}\text{)}$, $\text{17.0 ± 1.0}\text{MeV}\text{(}{}^{233}\text{U}\text{)}\text{and}\text{20.6 ± 1.1}\text{MeV}\text{(}{}^{235}\text{U}\text{)}$. In the yields for high-kinetic-energy-selected events, the mass 134 dominates for ²³³U and ²³⁵U, but for ²³²U it is μ_H≈144, which dominates. This complete reversal of profiles can be understood in terms of fragment shells.     

15.

Full O(αew) electroweak corrections to e+e−→HHZ     

《Physics letters. [Part B]》2004,578(3-4):349-358

We calculate the full $\text{O}\text{(α}{}_{\mathrm{<mtext>ew</mtext>}}\text{)}$ electroweak corrections to the Higgs pair production process e⁺e⁻→HHZ at an electron–positron linear collider in the standard model, and analyze the dependence of the Born cross section and the corrected cross section on the Higgs boson mass m_H and the c.m. energy $\text{s}$. To see the origin of some of the large corrections clearly, we calculate the QED and genuine weak corrections separately. The numerical results show that the corrections significantly suppress or enhance the Born cross section, depending on the values of m_H and $\text{s}$. For the c.m. energy $\text{s}\text{=500}$ GeV, which is the most favorable colliding energy for HHZ production with intermediate Higgs boson mass, the relative correction decreases from −5.3% to −11.5% as m_H increases from 100 to 150 GeV. For the range of the c.m. energy where the cross section is relatively large, the genuine weak relative correction is small, less than 5%.     

16.

Long-wavelength excitations in a Bose gas at zero temperature     

Victor K Wong  Harvey Gould 《Annals of Physics》1974,83(2):252-302

The long-wavelength excitations in a simple model of a dilute Bose gas at zero temperature are investigated from a purely microscopic viewpoint. The role of the interaction and the effects of the condensate are emphasized in a dielectric formulation, in which the response functions are expressed in terms of regular functions that do not involve an isolated single-interaction line nor an isolated single-particle line. Local number conservation is incorporated into the formulation by the generalized Ward identities, which are used to express the regular functions involving the density in terms of regular functions involving the longitudinal current. A perturbation expansion is then developed for the regular functions, producing to a given order in the perturbation expansion an elementary excitation spectrum without a gap and simultaneously response functions that obey local number conservation and related sum rules.Explicit results to the first order beyond the Bogoliubov approximation in a simple one-parameter model are obtained for the elementary excitation spectrum ω_k, the dynamic structure function $\text{S}$(k, ω), the associated structure function $\text{S}$_m(k), and the one-particle spectral function $\text{A}$(k, ω), as functions of the wavevector k and frequency ω. These results display the sharing of the gapless spectrum ω_k by the various response functions and are used to confirm that the sum rules of interest are satisfied. It is shown that ω_k and some of the $\text{S}$_m(k) are not analytic functions of k in the long wavelength limit. The dynamic structure function $\text{S}$(k, ω) can be conveniently separated into three parts: a one-phonon term which exhausts the f sum rule, a backflow term, and a background term. The backflow contribution to the static structure function $\text{S}$₀(k) leads to the breakdown of the one-phonon Feynman relation at order k³. Both $\text{S}$(k, ω) and $\text{A}$(k, ω) display broad backgrounds because of two-phonon excitations. Simple arguments are given to indicate that some of the qualitative features found for various physical quantities in the first-order model calculation might also be found in superfluid helium.     

17.

Shallow acceptor bound exciton and free exciton states in high-purity zinc telluride     

H. Venghaus  P.J. Dean 《Solid State Communications》1979,31(11):897-900

We investigated excitons bound to shallow acceptors in high-purity ZnTe and measured excitation spectra of two-hole luminescence lines at 1.6 K using a tunable dye-laser. The electron-hole coupling in the bound exciton (BE) states appears to be very different for the various acceptors even for almost identical exciton localisation energies. Three different types of BE are reported. For the Li-acceptor BE we observe three sub-components separated by 0.22 and 0.17 meV and interpreted as J = $\text{1}\text{2}$, $\text{3}\text{2}$, $\text{5}\text{2}$ states. The Ag-acceptor BE exhibits a strong ground state and a weak excited state at 1.3 meV higher energy. For the as yet unidentified k-acceptor we observe a single BE level, degenerate with the Ag-acceptor BE ground state. Dips in the excitation spectra due to absorption into free exciton 1S, 2S, and 3S states yield an exciton Rydberg R₀ = 12.8±0.3 meV and a free exciton binding energy FE(1S) = 13.2±0.3 meV.     

18.

States on the current algebra     

J. Alcantara  D.A. Dubin 《Reports on Mathematical Physics》1984,19(1):13-26

We introduce the field algebra Σ$\text{D}$(M;ⁿ?ⁿ$\text{g}$) associated with the current algebra $\text{D}$_r(M;$\text{g}$) for the Lie algebra $\text{g}$ over physical space M. The Heisenberg magnet model is generalized to this continuum. It is shown that the Hamiltonian can be given meaning as implementing a derivation of the field algebra in certain representations.We introduce new representations of the current algebra. For example, if G = SU(2), a representation in L²(R³)?³ is [σ(?)F]^j = ε^jkl?_kψ_l for (?_k) = ? in $\text{D}$_r(M;$\text{g}$)(ψ_l = F. This has cyclic subrepresentations with prime parts.     

19.

Holtzmark electric field distribution in a two-dimensional electron fluid     

C. Deutsch 《Physics letters. A》1983,94(1):40-41

The Holtzmark distribution for the thermal electric field at a neutral point in a two-dimensional and uncorrelated electron fluid is given by $\text{β(1 + β}{}^2\text{)}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$, with $\text{β =}\text{E}\text{E}{}_0$, and E₀ = πnZ_pe in terms of the density n and perturber Z_pe.     

20.

Effective Hamiltonian for polyatomic linear molecules     

Koichi M.T. Yamada  F.W. Birss  M.R. Aliev 《Journal of Molecular Spectroscopy》1985,112(2):347-356

The leading terms of an effective Hamiltonian for a linear molecule in a given vibrational state are presented up to κ¹⁰T_v order of magnitude, whereby higher-order l-dependent terms such as $\text{H}\text{?}{}_{12.0}$, $\text{H}\text{?}{}_{8.0}$, and $\text{H}\text{?}{}_{8.2}$ have been neglected because in spectroscopic application they are of minor importance. This Hamiltonian therefore includes all those l-type interactions which could contribute to the fitting procedure, within a vibrational state where one or more bending vibrations are excited.     

	Diamond	Silicon
IIII/II
Density of states of valence electrons	2.64	2.29
XPS (in PW-approximation)	0.27	1.65
XPS (in OPW-approximation)	0.41	1.12
XPS (experimental)	0.44	1.40
σs/σp
PW-approximation	25.0	1.7
OPW-approximation	13.6	2.4
Free atom	12.0	3.4

Nonexponential decay law

The decay of a nonstationary state usually starts as a quadratic function of time and ends as an inverse power law (possibly with oscillations). Between these two extremes, the familiar exponential decay law may be approximately valid. The main purpose of this paper is to find the conditions which must be satisfied by the Hamiltonian and by the initial state, for the exponential law to have a significant domain of validity. It is shown that the evolution of a nonstationary state is governed by a nonnegative function W(E), having the dimensions of an energy. Among its properties are: the energy uncertainty is given by $\text{(ΔH)}{}^2\text{= ?W(E)dE}$, and the inverse lifetime by Γ = 2πW(E₀), where E₀ is the expectation value of H. The detailed shape of W(E) defines two characteristic times between which the exponential decay law is a good approximation: roughly speaking, the smoother W(E), the larger the domain of validity of the exponential law. For instance, if W(E) is very smooth ($\text{|}\text{dW}\text{dE}\text{| ? 1}$) except for a sharp threshold at E = E_thr, the transition from quadratic to exponential decay occurs for $\text{t ?}\text{1}\text{(E}{}_0\text{? E}{}_{\mathrm{<mtext>thr</mtext>}}\text{)}$, and the transition from exponential to inverse power law when $\text{Γt ?}\text{log}\text{[}\text{(E}{}_0\text{? E}{}_{\mathrm{<mtext>thr</mtext>}}\text{)}\text{Γ}\text{]}$.     

Energy gaps in the fractional quantum hall effect

Based on a density-of-states N(E) ∝ (E − μ₀)² Poisson's equation is solved perpendicular to the current direction in the two-dimensional Hall-plate. By comparing the result with the structure obtained for an Ising-like chain with occupation numbers $\text{σ = 0,}\text{1}\text{2}$, 1 to account for vanishing energy gaps for filling factors $\text{v =}\text{m}\text{n}$ with n even, the energy gaps for n odd are found essentially to be proportional to $\text{B}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{n}{}^{\mathrm{-2}}$ in accordance with recent results.     

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.     

Fission-fragment energy correlation measurements for (nth, f) of 232U and 233U

Fission-fragment mass and kinetic energy distributions and their correlations have been measured for ²³²U and ²³³U. The results on these uranium isotopes and ²³⁵U are compared. The mass peak/valley ratio of 785 ± 68 for ²³²U is the highest of the three isotopes. The 〈E_K〉(μ_H) distributions show significant differences. The dip ΔE_K at symmetry is $\text{16.2 ± 1.0}\text{MeV}\text{(}{}^{232}\text{U}\text{)}$, $\text{17.0 ± 1.0}\text{MeV}\text{(}{}^{233}\text{U}\text{)}\text{and}\text{20.6 ± 1.1}\text{MeV}\text{(}{}^{235}\text{U}\text{)}$. In the yields for high-kinetic-energy-selected events, the mass 134 dominates for ²³³U and ²³⁵U, but for ²³²U it is μ_H≈144, which dominates. This complete reversal of profiles can be understood in terms of fragment shells.     

Full O(αew) electroweak corrections to e+e−→HHZ

We calculate the full $\text{O}\text{(α}{}_{\mathrm{<mtext>ew</mtext>}}\text{)}$ electroweak corrections to the Higgs pair production process e⁺e⁻→HHZ at an electron–positron linear collider in the standard model, and analyze the dependence of the Born cross section and the corrected cross section on the Higgs boson mass m_H and the c.m. energy $\text{s}$. To see the origin of some of the large corrections clearly, we calculate the QED and genuine weak corrections separately. The numerical results show that the corrections significantly suppress or enhance the Born cross section, depending on the values of m_H and $\text{s}$. For the c.m. energy $\text{s}\text{=500}$ GeV, which is the most favorable colliding energy for HHZ production with intermediate Higgs boson mass, the relative correction decreases from −5.3% to −11.5% as m_H increases from 100 to 150 GeV. For the range of the c.m. energy where the cross section is relatively large, the genuine weak relative correction is small, less than 5%.     

Long-wavelength excitations in a Bose gas at zero temperature

The long-wavelength excitations in a simple model of a dilute Bose gas at zero temperature are investigated from a purely microscopic viewpoint. The role of the interaction and the effects of the condensate are emphasized in a dielectric formulation, in which the response functions are expressed in terms of regular functions that do not involve an isolated single-interaction line nor an isolated single-particle line. Local number conservation is incorporated into the formulation by the generalized Ward identities, which are used to express the regular functions involving the density in terms of regular functions involving the longitudinal current. A perturbation expansion is then developed for the regular functions, producing to a given order in the perturbation expansion an elementary excitation spectrum without a gap and simultaneously response functions that obey local number conservation and related sum rules.Explicit results to the first order beyond the Bogoliubov approximation in a simple one-parameter model are obtained for the elementary excitation spectrum ω_k, the dynamic structure function $\text{S}$(k, ω), the associated structure function $\text{S}$_m(k), and the one-particle spectral function $\text{A}$(k, ω), as functions of the wavevector k and frequency ω. These results display the sharing of the gapless spectrum ω_k by the various response functions and are used to confirm that the sum rules of interest are satisfied. It is shown that ω_k and some of the $\text{S}$_m(k) are not analytic functions of k in the long wavelength limit. The dynamic structure function $\text{S}$(k, ω) can be conveniently separated into three parts: a one-phonon term which exhausts the f sum rule, a backflow term, and a background term. The backflow contribution to the static structure function $\text{S}$₀(k) leads to the breakdown of the one-phonon Feynman relation at order k³. Both $\text{S}$(k, ω) and $\text{A}$(k, ω) display broad backgrounds because of two-phonon excitations. Simple arguments are given to indicate that some of the qualitative features found for various physical quantities in the first-order model calculation might also be found in superfluid helium.     

Shallow acceptor bound exciton and free exciton states in high-purity zinc telluride

We investigated excitons bound to shallow acceptors in high-purity ZnTe and measured excitation spectra of two-hole luminescence lines at 1.6 K using a tunable dye-laser. The electron-hole coupling in the bound exciton (BE) states appears to be very different for the various acceptors even for almost identical exciton localisation energies. Three different types of BE are reported. For the Li-acceptor BE we observe three sub-components separated by 0.22 and 0.17 meV and interpreted as J = $\text{1}\text{2}$, $\text{3}\text{2}$, $\text{5}\text{2}$ states. The Ag-acceptor BE exhibits a strong ground state and a weak excited state at 1.3 meV higher energy. For the as yet unidentified k-acceptor we observe a single BE level, degenerate with the Ag-acceptor BE ground state. Dips in the excitation spectra due to absorption into free exciton 1S, 2S, and 3S states yield an exciton Rydberg R₀ = 12.8±0.3 meV and a free exciton binding energy FE(1S) = 13.2±0.3 meV.     

States on the current algebra

We introduce the field algebra Σ$\text{D}$(M;ⁿ?ⁿ$\text{g}$) associated with the current algebra $\text{D}$_r(M;$\text{g}$) for the Lie algebra $\text{g}$ over physical space M. The Heisenberg magnet model is generalized to this continuum. It is shown that the Hamiltonian can be given meaning as implementing a derivation of the field algebra in certain representations.We introduce new representations of the current algebra. For example, if G = SU(2), a representation in L²(R³)?³ is [σ(?)F]^j = ε^jkl?_kψ_l for (?_k) = ? in $\text{D}$_r(M;$\text{g}$)(ψ_l = F. This has cyclic subrepresentations with prime parts.     

Holtzmark electric field distribution in a two-dimensional electron fluid

The Holtzmark distribution for the thermal electric field at a neutral point in a two-dimensional and uncorrelated electron fluid is given by $\text{β(1 + β}{}^2\text{)}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$, with $\text{β =}\text{E}\text{E}{}_0$, and E₀ = πnZ_pe in terms of the density n and perturber Z_pe.     

Effective Hamiltonian for polyatomic linear molecules

The leading terms of an effective Hamiltonian for a linear molecule in a given vibrational state are presented up to κ¹⁰T_v order of magnitude, whereby higher-order l-dependent terms such as $\text{H}\text{?}{}_{12.0}$, $\text{H}\text{?}{}_{8.0}$, and $\text{H}\text{?}{}_{8.2}$ have been neglected because in spectroscopic application they are of minor importance. This Hamiltonian therefore includes all those l-type interactions which could contribute to the fitting procedure, within a vibrational state where one or more bending vibrations are excited.