Threshold conditions and bound states for locally periodic delta potentials

We present a systematic study of the conditions for the generation of threshold energy eigen states and also the energy spectrum generated by two types of locally periodic delta potentials each having the same strength λV and separation distance parameter a: (a) sum of N attractive potentials and (b) sum of pairs of attractive and repulsive potentials. Using the dimensionless parameter g = λV a in case (a) the values of g = g _n , n = 1, 2, …, N at which threshold energy bound state gets generated are shown to be the roots of Nth order polynomial D ₁(N, g) in g. We present an algebraic recursive procedure to evaluate the polynomial D ₁(N, g) for any given N. This method obviates the need for the tedious mathematical analysis described in our earlier work to generate D ₁(N, g). A similar study is presented for case (b). Using the properties of D ₁(N, g) we establish that in case (a) the critical minimum value of g which guarantees the generation of the maximum possible number of bound states is g = 4. The corresponding result for case (b) is g = 2. A typical set of numerical results showing the pattern of variation of g _n as a function of n and several interesting features of the energy spectrum for different values of g and N are also described.     

Dynamics of a three level atom interacting with a detuned SU (1,1) quantum system

We study the stability of attractive atomic Bose-Einstein condensate and the macroscopic quantum many-body tunneling (MQT) in the anharmonic trap. We utilize correlated two-body basis function which keeps all possible two-body correlations. The anharmonic parameter (λ) is slowly tuned from harmonic to anharmonic. For each choice of λ the many-body equation is solved adiabatically. The use of the van der Waals interaction gives realistic picture which substantially differs from the mean-field results. For weak anharmonicity, we observe that the attractive condensate gains stability with larger number of bosons compared to that in the pure harmonic trap. The transition from resonances to bound states with weak anharmonicity also differs significantly from the earlier study of [N. Moiseyev, L.D. Carr, B.A. Malomed, Y.B. Band, J. Phys. B 37, L193 (2004)]. We also study the tunneling of the metastable condensate very close to the critical number N _cr of collapse and observe that near collapse the MQT is the dominant decay mechanism compared to the two-body and three-body loss rate. We also observe the power law behavior in MQT near the critical point. The results for pure harmonic trap are in agreement with mean-field results. However, we fail to retrieve the power law behavior in anharmonic trap although MQT is still the dominant decay mechanism.     

The construction of the eigenstates for nonlinear Schrödinger model with supermatrices and attractive coupling

A quantum nonlinear Schrödinger model with supermatrices and attractive coupling is studied by using the quantum inverse scattering method. The eigenstates of the Hamiltonian and the infinite number of the conserved quantities of the system are constructed. In particular, theN-particle bound states with the mixture of bosons and fermions are found. The energy of theN-particle eigenstate are Σ _i=1 ^N andNp ² ?N(N ²?1)c ²/12 for the scattering state and the bound state respectively.     

Bounds on the number of eigenvalues of the Schrödinger operator

Inequalities on eigenvalues of the Schrödinger operator are re-examined in the case of spherically symmetric potentials. In particular, we obtain:

1. A connection between the moments of order (n ? 1)/2 of the eigenvalues of a one-dimensional problem and the total number of bound statesN _n, inn space dimensions;
2. optimal bounds on the total number of bound states below a given energy in one dimension;
3. alower bound onN ₂;
4. a self-contained proof of the inequality for α ≧ 0,n ≧ 3, leading to the optimalC ₀₄,C ₃;
5. solutions of non-linear variation equations which lead, forn ≧ 7, to counter examples to the conjecture thatC _0n is given either by the one-bound state case or by the classic limit; at the same time a conjecture on the nodal structure of the wave functions is disproved.

     

On the left-hand cut in potential scattering

Partial wave $\text{N}\text{D}$ equations in potential scattering are solved for the exponential, Hulthén and Morse potentials. The driving terms are taken to be either the contributions of a finite number of Born terms or the total contributions of only the nearest singularities (first n poles). For repulsive potentials one observes ghosts, anomalous bound states or resonances if the order of approximation is small with respect to the potential strength. The origin and meaning of these unphysical phenomena are explained. For attractive potentials such anomalies occur only at very large potential strengths if at all. Input-equivalent Bargmann potentials are employed to determine the quality and nature of the approximate $\text{N}\text{D}$ solutions. Rough criteria for the validity of approximations within the $\text{N}\text{D}$ approach are formulated.     

The localization of local and gap modes by the sign count method

In the theory of local states it is usual to bring the linear problem ¦glI _N ?D¦=0 into a non-linear one ¦R(λ)¦=0, where the ordern of the condensed stiffness matrixR(λ) is much smaller than the orderN of the matrixD. Powerful methods exist for determining the number of the eigenvalues ofD in some interval without solving the equation ¦glI _N ?D¦=0 explicitly. It is shown, that analogous methods can be applied to the condensed matrix R(λ), when computing the eigen-valuesλ outside the frequency spectrum of the ideal lattice.     

Band structure and optical transitions in semiconducting double-wall carbon nanotubes

The electronic structure of semiconducting double-wall carbon nanotubes (CNTs) is calculated using the linearized augmented cylindrical wave method. The consideration is performed in the framework of the local density functional theory and the muffin-tin (MT) approximation for the one-electron Hamiltonian. The electronic spectrum of a double-wall CNT is determined by the free motion of electrons in the interatomic space of the two cylindrical layers, scattering by the MT spheres, and tunneling through the classically impenetrable region. Calculated results for double-wall CNTs of the (n, 0)@(n′, 0) zigzag type indicate that the shift of the band-gap width depends on whether n and n′ are divided by 3 with a remainder of 1 or 2. It is found that, regardless of the type of the inner tube, the energy gap E _g of the outer tube decreases by 0.15–0.22 eV if the tube belongs to the sequence n = 2 (mod 3). For the outer tubes of the sequence n = 1 (mod 3), the shifts of the band gap ΔE _g are always negative ?0.15 ≤ ΔE _g ≤ ?0.05 eV. In both cases, the shifts ΔE _g weakly oscillate rather than decrease in going to tubes of a larger diameter d. For the inner tubes, the changes in the band gap ΔE _g are more sensitive to the diameter. At 10 ≤ n ≤ 16, the shifts ΔE _g are positive and the maximum value of ΔE _g equals 0.39 and 0.32 for the sequences n = 2 (mod 3) and n = 1 (mod 3), respectively. In going to the inner tubes of a larger diameter, ΔE _g rapidly drops and then oscillates in the range from ?0.05 to 0.06 eV. The calculated results indicate that the shifts of the optical band gaps in core and shell tubes upon the formation of double-wall CNTs are significant, which must hinder the identification of double-wall CNTs by optical methods. On the other hand, the obtained results open up possibilities for a more detailed classification of double-wall nanotubes.     

Chiral symmetry probed with pionic atoms

Study of pionic atom has been revisited by a recent discovery of deeply bound pionic states in the context of possible sensitivity to the partial chiral restoration in the nuclear medium. Precisely measured binding energies and widths of the deeply bound π^? Is states in ^115,119,123Sn in ^116,120,124Sn(d,³He) reactions at recoil-free conditions are used to deduce the medium-modified isovector scattering amplitude as b₁ = 0.115 ± 0.007m _π ^?1. The observed enhancement of b ₁ over the free πN value infers 34% reduction of $\left\langle {\bar qq} \right\rangle $ at the normal nuclear density, ρ = 0.17 fm^?3.     

Nitrogen trap bound states in In1−xGaxP

Photoluminescence and absorption data are reported for nitrogen-doped In_1?xGa_xP (0.64 ? x ? 1.0) demonstrating the existence of a shallow N-trap bound state, N_Γ, in the direct-gap composition region. The N_Γ state, which is derived from the Γ conduction band minimum, exists along with a deeper bound state, N_X, derived from the X minima. These states, which are not observed in N-doped GaP, exhibit non-crossing behavior in the composition region of strong coupling between the N_Γ and N_X states. This observation is in accord with recent experimental and theoretical work indicating that the N-trap bound states in the related alloy GaAs_1-xP_x are produced by a combination of long and short range potentials which should also exist in In_1-xGa_xP.     

Towards a finite N = 2 susy GUT

We consider finite, N = 2 supersymmetric GUTs based on gauge groups SU(n) and SO(n). As an example, we discuss a semirealistic model based on SO(12). We argue that in finite, N = 2 supersymmetric GUTs, gauge symmetry breaking should occur dynamically. We present a heuristic picture in which this is induced by soft, finiteness preserving SUSY breaking terms. The bound states formed cause a very rapid evolution of the SO(12) coupling constant and break SO(12) into SU(4)×SU(3)_C×U(1).     

Investigation of statistically perturbed angular correlations of154Gd and156Gd in different liquid sources

The statistical perturbations of γγ-angular correlations involving the first 2⁺ rotational state of¹⁵⁴Gd and¹⁵⁶Gd have been time differentially investigated. We used liquid sources of 3⁺ ions of Gadolinium in lN perchloric acid, 0.5 N and 1N hydrochloric acid, 1.3N and 2.6 N sulfuric acid. Influences of the various chemical surroundings on the ratio λ₄/λ₂ of the attenuation parameters have been found. A simultaneous measurement of the angular correlation of the 874 keV–123 keV cascade and the 2,098 keV–89 keV cascade of¹⁵⁴Gd and¹⁵⁶Gd, respectively, has been performed in perchloric solution. From the ratio of the attenuation parameters λ₂, obtained by this experiment, we have derived the ratio of the magneticg-factors of the first 2⁺ levels of these isotopes asg ¹⁵⁴/g ¹⁵⁶=1.11±0.08.     

Microwave spectrum,dipole moment,and structure of 3-aminopropanol

The microwave spectra of 3-aminopropanol and three of its deuterium substituted isotopic species have been investigated in the 26.5 to 40 GHz frequency region. The rotational spectrum of only one conformer has been assigned in which presumably a hydrogen bond of the OH---N type exists. The rotational spectra of a number of excited vibrational states have been observed and assignments made for some of these excited states. The average intensity ratio for the rotational transitions between the ground and excited vibrational states indicates that the first excited state is about 120 cm^?1 above the ground state.and the next higher state is roughly 200 cm^?1 above the ground vibrational state. The dipole moment was determined from the Stark effect measurements to be 3.13 ± 0.04 D with its principal axes components as |μ_a| = 2.88 ± 0.03 D, |μ_b| = 1.23 ± 0.04 D and |μ_c| = 0.06 ± 0.01 D. The possibility of another conformer where the hydrogen bond could be of NH---O type was explored, but the spectra of such a conformer could not be identified.     

The N-mode squeezed state with enhanced squeezing

It is known that exp [iλ (Q1P1i/2)] is a unitary single-mode squeezing operator,where Q1,P1 are the coordinate and momentum operators,respectively.In this paper we employ Dirac’s coordinate representation to prove that the exponential operator S n ≡ exp [iλ sum((QiPi+1+Qi+1Pi))) from i=1 to n ],(Qn+1=Q1,Pn+1=P1),is an n-mode squeezing operator which enhances the standard squeezing.By virtue of the technique of integration within an ordered product of operators we derive S n ’s normally ordered expansion and obtain new n-mode squeezed vacuum states,its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.     

High-resolution Fourier spectrometry of 14N2 and 15N2 infrared emission spectrum: Extensive analysis of the W3Δu - B3Πg system

The infrared spectrum of the nitrogen molecule, excited in a microwave discharge, has been recorded in high resolution by Fourier spectrometry in the range 2500–15 000 cm^?1. The precision of the measurements is estimated to be about 0.003 cm^?1. We have analyzed 19 bands of the W³Δ_u-B³Π_g system of ¹⁴N₂, with 0 ≤ v′ ≤ 7, and three bands of ¹⁵N₂ lying between 2500 and 5900 cm^?1. The molecular constants of the ³Δ_u and B³Π_g states have been determined by direct approach using an iterative nonlinear least-squares procedure. It has proved convenient to describe the levels of W³Δ_u state in a case a basis although in fact they approximate those of Hund's case b. Derived values of equilibrium constants of W³Δ_u are, in cm^?1: T_e = 8875.3347 (with origin taken in A, ³Σ_u⁺v = 0 level); ω_e = 1506.4859; ω_eχ_e = 12.5469; B_e = 1.4702537; α_B = 0.0170389; D_e = 0.55958 × 10^?5. RKR potential energy curves for the two states are constructed, and the Franck-Condon factors calculated for the W-B system.     

High-resolution Fourier spectrometry of 14N2 infrared emission spectrum: Extensive analysis of the B3Πg-A3Σu+ system

The B³Π_g-A³Σ_u⁺ system of N₂ excited in a microwave discharge was recorded between 3 000 and 18 000 cm^?1 with a high-resolution Fourier spectrometer. There are no observed irregularities in the levels of the A³Σ_u⁺ and B³Π_g states at least for the values of v and J considered here, except the predissociation in the B³Π_g state for v = 12 and J higher than 33. This predissociation will be checked with more complete data in another article. Thirty three bands of the first positive system with 0 ≤ v′ ≤ 12 and 0 ≤ v″ ≤ 8 are analyzed. The molecular parameters of the B³Π_g and A³Σ_u⁺ are obtained by a complete fitting procedure. Derived values of equilibrium constants are deduced; RKR potential energy curves for the two states are constructed, and the Franck-Condon factors calculated for the A-B system.     

Scattering of slow neutrons by arsenic and selenium isotopes

Coherent neutron scattering lengths and free cross sections were measured for arsenic, for ordinary selenium and its isolated isotopes. By means of the Cristiansen filter technique the following scattering lengths for the bound atoms were obtained (in fm):b(As) =6.58(1) [b ₊=6.04(5) andb _?=7.47(8)],b(Se)=7.970(9),b(⁷⁶Se)=12.2(1),b(⁷⁷Se) =08.25(8),b(⁷⁸Se)=8.24(9),b(⁸⁰Se)=7.48(3) andb(⁸²Se)=6.34(8). Transmission measurements with neutrons of 1.26 eV and 5.19 eV resulted in an energy independent free scattering cross section for arsenic ofσ _{s, t}=5.40(3)b (in the eV-region). For ordinary selenium energy dependent free cross sections ofσ _{s, t}(1.26 eV)=7.9(1)b andσ _{s, t} (5.19 eV)=7.55(3)b were found. The results were compared with the resonance parameters of the nuclei and with different sets of potential scattering radii. Thus it could be concluded that there is no evidence for a bound state of the neutron-nucleus compound⁷⁵As+n but a clear evidence for strong bound levels of⁷⁶Se+n and⁷⁷Se+n.     

Semiclassical approach for multiparticle production in scalar theories

We propose a semiclassical approach to calculate multiparticle cross sections in scalar theories, which have been strongly argued to have the exponential form exp(λ⁻¹ F(λn, ϵ)) in the regime λ → 0, λn, ϵ = fixed, where λ is the scalar coupling, n is the number of produced particles, and ϵ is the kinetic energy per final particle. The formalism is based on singular solutions to the field equation, which satisfy certain boundary and extremizing conditions. At low multiplicities and small kinetic energies per final particle we reproduce in the framework of this formalism the main perturbative results. We also obtain a lower bound on the tree-level cross section in the ultra-relativistic regime.     

Propagation of excitation in long 1D chains: Transition from regular quantum dynamics to stochastic dynamics

The quantum dynamics problem for a 1D chain consisting of 2N + 1 sites (N ? 1) with the interaction of nearest neighbors and an impurity site at the middle differing in energy and in coupling constant from the sites of the remaining chain is solved analytically. The initial excitation of the impurity is accompanied by the propagation of excitation over the chain sites and with the emergence of Loschmidt echo (partial restoration of the impurity site population) in the recurrence cycles with a period proportional to N. The echo consists of the main (most intense) component modulated by damped oscillations. The intensity of oscillations increases with increasing cycle number and matrix element C of the interaction of the impurity site n = 0 with sites n = ±1 (0 < C ≤ 1; for the remaining neighboring sites, the matrix element is equal to unity). Mixing of the components of echo from neighboring cycles induces a transition from the regular to stochastic evolution. In the regular evolution region, the wave packet propagates over the chain at a nearly constant group velocity, embracing a number of sites varying periodically with time. In the stochastic regime, the excitation is distributed over a number of sites close to 2N, with the populations varying irregularly with time. The model explains qualitatively the experimental data on ballistic propagation of the vibrational energy in linear chains of CH₂ fragments and predicts the possibility of a nondissipative energy transfer between reaction centers associated with such chains.