The Steep Nekhoroshev’s Theorem

Revising Nekhoroshev’s geometry of resonances, we provide a fully constructive and quantitative proof of Nekhoroshev’s theorem for steep Hamiltonian systems proving, in particular, that the exponential stability exponent can be taken to be \({1/(2n\alpha_1\cdots\alpha_{n-2}}\)) (\({\alpha_i}\)’s being Nekhoroshev’s steepness indices and \({n \ge 3}\) the number of degrees of freedom). On the base of a heuristic argument, we conjecture that the new stability exponent is optimal.     

A numerical study of the topology of normally hyperbolic invariant manifolds supporting Arnold diffusion in quasi-integrable systems

We investigate numerically the stable and unstable manifolds of the hyperbolic manifolds of the phase space related to the resonances of quasi-integrable systems in the regime of validity of the Nekhoroshev and KAM theorems. Using a model of weakly interacting resonances we explain the qualitative features of these manifolds characterized by peculiar ‘flower-like’ structures. We detect different transitions in the topology of these manifolds related to the local rational approximations of the frequencies. We find numerically a correlation among these transitions and the speed of Arnold diffusion.     

Effective Stability for Gevrey and Finitely Differentiable Prevalent Hamiltonians

For perturbations of integrable Hamiltonian systems, the Nekhoroshev theorem shows that all solutions are stable for an exponentially long interval of time, provided the integrable part satisfies a steepness condition and the system is analytic. This fundamental result has been extended in two distinct directions. The first one is due to Niederman, who showed that under the analyticity assumption, the result holds true for a prevalent class of integrable systems which is much wider than the steep systems. The second one is due to Marco-Sauzin but it is limited to quasi-convex integrable systems, for which they showed exponential stability if the system is assumed to be only Gevrey regular. If the system is finitely differentiable, we showed polynomial stability, still in the quasi-convex case. The goal of this work is to generalize all these results in a unified way, by proving exponential or polynomial stability for Gevrey or finitely differentiable perturbations of prevalent integrable Hamiltonian systems.     

The effect of an anti-hydrogen bond on Fermi resonance:A Raman spectroscopic study of the Fermi doublet ν1-ν12 of liquid pyridine

The effects of an anti-hydrogen bond on the ν1-ν12 Fermi resonance(FR) of pyridine are experimentally investigated by using Raman scattering spectroscopy.Three systems,pyridine/water,pyridine/formamide,and pyridine/carbon tetrachloride,provide varying degrees of strength for the diluent-pyridine anti-hydrogen bond complex.Water forms a stronger anti-hydrogen bond with pyridine than with formamide,and in the case of adding non-polar solvent carbon tetrachloride,which is neither a hydrogen bond donor nor an acceptor and incapable of forming a hydrogen bond with pyridine,the intermolecular distance of pyridine will increase and the interaction of pyridine molecules will reduce.The dilution studies are performed on the three systems.Comparing with the values of the Fermi coupling coefficient W of the ring breathing mode ν1 and triangle mode ν12 of pyridine at different volume concentrations,which are calculated according to the Bertran equations,in three systems,we find that the solution with the strongest anti-hydrogen bond,water,shows the fastest change in the ν1-ν12 Fermi coupling coefficient W with the volume concentration varying,followed by the formamide and carbon tetrachloride solutions.These results suggest that the stronger anti-hydrogen bond-forming effect will cause a greater reduction in the strength of the ν1-ν12 FR of pyridine.According to the mechanism of the formation of an anti-hydrogen bond in the complexes and the FR theory,a qualitative explanation for the anti-hydrogen bond effect in reducing the strength of the ν1-ν12 FR of pyridine is given.     

A consequence from the model description of a decomposing liquid mixture

In terms of representation [1] of a decomposing binary liquid mixture as a system of two coupled self-oscillators with partial frequencies ν _A and ν _B , the following conjecture is verified: given frequency ratios ν _A /ν _B and ν _A /ν _C for decomposing liquid mixtures A-B and A-C at the same temperature, it is possible to determine the ν _B /ν _C ratio, which is independent of the choice of the A component.     

On the mechanism of the broadening of the proton vibrational IR band in weak hydrogen bonds. Double adiabatic approximation

It is proposed that broadening of the ν_s (AH) IR band is due to the ν_s (AH) and ν_σ (AH…B) mode coupling and to the stochastic variation of the equilibrium distance R_e (A…B) modulating the proton vibration. The R_e disordering variation is caused by a coupling of the ν_σ mode with the low frequency oscillators (ν_Q) of the medium and of the complex as such. Besides, the (ν_σ, ν_Q) and (ν_s, ν_σ) couplings cause a ν_s frequency shift. Both the band broadening and the frequency shift increase with higher force constants responsible for (ν_σ, ν_Q) mode coupling. Furthermore the Q_i low frequency stochastic vibrations directly modulate the ν_σ (AH…B) vibration and the free AH groups stretching vibration (ν^o_s) causing some broadening at their bands which is however several times smaller than the ν_s band broadening in the complex. Several examples are reported to confirm the proposed model.     

二元溶液中基频拉曼散射系数随浓度变化对费米共振的影响

在二硫化碳(CS₂)和苯(C₆H₆)的二元溶液中,随着相对体积浓度而变化,C₆H₆的基频ν_１(992 cm-1) 的拉曼散射系数随浓度减小而增加,而CS₂的基频ν₁(656 cm-1)的拉曼散射系数则随浓度增加而增加。文章测量了这二元溶液相对不同体积浓度的拉曼光谱。结果表明,尽管CS₂和C₆H₆的基频ν₁的强度都随着浓度变化而发生较大变化,但C₆H₆的基频ν₁对其参与的ν₁+ν₆～ν₈费米共振几乎没有影响,而CS₂的基频ν₁强度的变化不仅对ν₁～2ν₂费米共振产生较大影响,还使基频ν₂发生变化。文章给出了实验结果,并用Bertran费米共振理论和群论给予了解释。     

Torsional Line Splittings in the (ν4+ν8)−ν4 Hot Transitions of C2H6 between 1400 and 1510 cm

A detailed investigation of the high-resolution infrared spectrum of ethane revealed the occurrence of features belonging to the hot perpendicular system (ν₄+ν₈)−ν₄ between 1400 and 1510 cm⁻¹. Transition lines of the subbranches with K″ΔK from −7 to 4, exhibiting torsional splittings of several tenths of a cm⁻¹, were observed and measured in this region. The observed line splittings are strongly influenced by the interaction between the ν₄+ν₈ and 2ν₄+ν₁₂ states and change with the values of K″ΔK, depending on the zero-order energy separation of the interacting levels. We found by numerical extrapolation that splittings still occur far from resonance, showing that the intrinsic torsional splittings of the combining states ν₄+ν₈ and ν₄ are quite different. We determined the intrinsic torsional splitting of ν₄+ν₈ to be less than 0.083 cm⁻¹, compared with 0.236 cm⁻¹ estimated for the ν₄ state. This result is in agreement with the expected effects of torsional Coriolis and head-tail coupling and is consistent with previous observations on vibrationally degenerate states of ethane-like molecules.     

Statistics of Poincaré recurrences for maps with integrable and ergodic components

Recurrence gives powerful tools to investigate the statistical properties of dynamical systems. We present in this paper some applications of the statistics of first return times to characterize the mixed behavior of dynamical systems in which chaotic and regular motion coexist. Our analysis is local: we take a neighborhood A of a point x and consider the conditional distribution of the points leaving A and for which the first return to A, suitably normalized, is bigger than t. When the measure of A shrinks to zero the distribution converges to the exponential e(-t) for almost any point x, if the system is mixing and the set A is a ball or a cylinder. We consider instead a system, a skew integrable map of the cylinder, which is not ergodic and has zero entropy. This map describes a shear flow and has a local mixing property. We rigorously prove that the statistics of first return is of polynomial type around the fixed points and we generalize around other points with numerical computations. The result could be extended to quasi-integrable area preserving maps such as the standard map for small coupling. We then analyze the distribution of return times in a region which is composed by two invariants subdomains: one with a mixing dynamics and the other with an integrable dynamics given by our shear flow. We show that the statistics of first return in this mixed region is asymptotically given by the exponential law, but this limit is attained by an intermediate regime where exponential and polynomial laws are linearly superposed and weighted by some factors which are proportional to the relative sizes of the chaotic and regular regions. The result on the statistics of first return times for mixed regions in the phase space can provide a basis to analyze such a property for area preserving maps in mixed regions even when a rigorous result is not available. To this end we present numerical investigations on the standard map which confirm the results of the model.     

The ν6, ν8 3ν4+ν12 infrared system of Si2H6 under high resolution: rotational and torsional analysis

A Fourier transform infrared spectrum of disilane has been measured at a Doppler limited resolution, and analysed in the region of the ν₆ and ν₈ fundamentals, from about 800 to 1020cm^?1. The torsional splittings are not resolved in the ν₆ band, showing that the splittings in the ν₆ = 1 state and in the ground state are almost identical. The torsional splittings in the reasonably unperturbed regions of the ν₈ fundamental are about 0.0146cm^?1, and a detailed rotation-torsion analysis shows that the intrinsic splittings in the ν₈ = 1 state are smaller than in the ground state by this amount. An intrinsic torsional splitting about 0.0150 cm^?1 is estimated in the vibrational ground state and in the ν₆ = 1 state, and almost vanishing in the ν₈ = 1 state (about 0.0004cm^?1), with a barrier height around 407cm^?1. This is in agreement with the expectation from theory. The ν₈ band, beyond a moderate x, y-Coriolis coupling with ν₆, is affected by several perturbations, also selective in the torsional components. The 3ν₄ + v¹² combination, with three quanta of the torsional mode excited and large torsional splittings, is the main perturber, causing both Fermi and Coriolis resonances in several regions of the spectrum. The vibrational origins of all four torsional components of 3ν₄ + v¹² were determined. Other perturbative effects are attributed to the systems 2ν₃ + ν₄, and ν₄ + 24₉(E + A). The spectrum was numerically analysed, and the relevant vibration-rotation-torsion parameters were determined.     

以新型分子距边矢量ν估计和预测烷烃核磁共振碳谱(13C NMR)化学位移和（CSS）

系统研究了核磁共振碳谱与化学位移和规律 ,以及分子拓扑指数在定量 [结 ]构 [波 ]谱关系 (QSSR)中的应用 .本文基于矢量路径长度矢量 p =(P1,P2 ,P3 ,… ,Pm)与分子中原子相互作用 ,提出了一种新型分子距边矢量并发现它与烷烃13 CNMR化学位移和有良好线性相关性 ,回归方程及其统计参数为 :CSS =bν +p3 =Σmj=0 bjνj+b10 p3 =b0 ν +b1ν1+b2 ν2 +b3 ν3 +b4 ν4 +b5ν5+b6ν6+b7ν7+b8ν8+b9ν9ν +b10 P3 =- 13.6 0 11+2 2 .2 133ν1+2 8.412 1ν2 +2 5 .9416ν3 +2 6 .6 70 9ν4 +14.4976ν5+5 .72 40ν6- 5 .3830ν7- 3.2 15 2ν8- 15 .0 2 13ν9- 2 5 .70 99ν10 +12 .2 786P3 (n =6 3,R =0 .9970 ,EV =99 .6 8% ,RMS =3.734 8,F =2 418.2 ;交互校验CV为 :R =0 .9893,EV =98.83% ,RMS =7.12 6 1,F =6 6 4.0 46 ) ;结果良好 .     

Intracavity spectroscopy of high-temperature water vapor in the region of 1.06 μm

The absorption spectrum of water vapor is studied in the region of 9375–9460 cm^?1 and at temperatures within 300–1200 K using an intracavity laser spectrometer based on a Nd laser having a threshold absorption sensitivity of 10^?8 cm^?1. More than 270 absorption lines are detected in the high-temperature spectrum of water vapor, 70% of which are assigned to ten vibrational bands: 3ν₂ + ν₃, 2ν₁ + ν₂, ν₁ + ν₂ + ν₃, ν₂ + 2ν₃, ν₁ + 3ν₂, 3ν₃ ? ν₂, 2ν₂ + 2ν₃ ? ν₂, ν₁ + 2ν₂ + ν₃ ? ν₂, 2ν₁ + ν₃ ? ν₂, and ν₂ + 3ν₃ ? 2ν₂. The vibrational-rotational energy levels are determined.     

High resolution infrared synchrotron study of CH2D81Br: ground state constants and analysis of the ν5, ν6 and ν9 fundamentals

The high resolution infrared absorption spectrum of CH₂D⁸¹Br has been recorded by Fourier transform spectroscopy in the range 550–1075?cm^?1, with an unapodized resolution of 0.0025?cm^?1, employing a synchrotron radiation source. This spectral region is characterized by the ν₆ (593.872?cm^?1), ν₅ (768.710?cm^?1) and ν₉ (930.295?cm^?1) fundamental bands. The ground state constants up to sextic centrifugal distortion terms have been obtained for the first time by ground-state combination differences from the three bands and subsequently employed for the evaluation of the excited state parameters. Watson's A-reduced Hamiltonian in the I^r representation has been used in the calculations. The ν ₆?=?1 level is essentially free from perturbation whereas the ν ₅?=?1 and ν ₉?=?1 states are mutually interacting through a-type Coriolis coupling. Accurate spectroscopic parameters of the three excited vibrational states and a high-order coupling constant which takes into account the interaction between ν₅ and ν₉ have been determined.     

Measurements and calculations of Ar-broadening parameters of water vapour transitions in a wide spectral region

The water vapour line-broadening (γ) and shift (δ) coefficients for 310 lines of 10 vibrational bands ν₁, ν₃, 2ν₂, ν₁+ ν₂, ν₂+ ν_3, 2ν₂ +ν₃, 2ν₁, ν₁+ ν₃, 2ν₃ and ν₁ +2ν₂ induced by argon pressure were measured with a Bruker IFS HR 125 spectrometer. The measurements were performed at room temperature, at the spectral resolution of 0.01 cm^–1 and over a wide pressure range of Ar. The calculations of the broadening coefficients γ and δ were performed in the framework of the semi-classical method. The intermolecular potential was taken as the sum of the atom–atom potential and the vibrationally and rotationally dependent isotropic induction+dispersion potential. The measured γ and δ were combined with literature data for the ν₂ and 3ν₁+ν₃, 2ν₁+2ν₂+ν₃ vibrational bands, and the optimal sets of potential parameters that gave the best agreement with the measured broadening coefficients for each vibrational band separately were found. Then, combined experimental data of 13 vibrational bands of H₂O perturbed by Ar were used to determine the analytical dependence of some potential parameters on vibrational quantum numbers.     

The Hot Bands of Methane between 5 and 10 μm

Experimental line intensities of 1727 transitions arising from nine hot bands in the pentad–dyad system of methane are fitted to first and second order using the effective dipole moment expansion in the polyad scheme. The observed bands are ν₃− ν₂, ν₃− ν₄, ν₁− ν₂, ν₁− ν₄, 2ν₄− ν₄, ν₂+ ν₄− ν₂, ν₂+ ν₄− ν₄, 2ν₂− ν₂, and 2ν₂− ν₄, and the intensities are obtained from long-path spectra recorded with the Fourier transform spectrometer located at Kitt Peak National Observatory. For the second order model, some of the 27 intensity parameters are not linearly independent, and so two methods (extrapolation and effective parameters) are proposed to model the intensities of the hot bands. In order to obtain stable values for three of these parameters, 1206 dyad (ν₄, ν₂) intensities are refitted simultaneously with the hot band lines. The simultaneous fits to first and second order lead to rms values respectively of 21.5% and 5.0% for the 1727 hot band lines and 6.5% and 3.0% for the 1206 dyad lines. The band intensities of all 10 pentad–dyad hot bands are predicted in units of cm⁻²atm⁻¹at 296 K to range from 0.931 (for 2ν₄− ν₄) to 7.67 × 10⁻⁵(for 2ν₄− ν₂). The total intensities are also estimated to first order for two other hot band systems (octad–pentad and tetradecad–octad) that give rise to weak transitions between 5 and 10 μm.     

Torsional tunneling splittings in the v4 = 1 excited torsional state of Si2H6: analysis of hot bands accompanying fundamental transitions in the high resolution infrared spectrum

The (ν₄?+?ν₆)???ν₄, (ν₄?+?ν₈)???ν₄ and (ν₄?+?ν₉)???ν₄ hot infrared systems of disilane (Si₂H₆) have been analysed at high resolution, and the values of the relative vibration–rotation–torsion parameters have been determined. The torsional splitting is about 0.500?cm^?1 in the ν₄ and ν₄?+?ν₆ states, and decreases strongly in the vibrationally degenerate upper states ν₄?+?ν₈ (about 0.0272?cm^?1 on average) and ν₄?+?ν₉ (about 0.3019?cm^?1), consistent with theoretical predictions. Comparison between the vibrational wavenumbers of cold transitions and hot transitions originating in the excited torsional state v₄?=?1 allows one to determine the change of the fundamental torsional frequency ν₄ caused by the excitation of small amplitude vibrations. A remarkable increase in ν₄ of about 8.599?cm^?1 is found in the v₉?=?1 state (E_1d SiH₃-rocking mode, asymmetric to inversion in the staggered geometry), and this corresponds to an increase in the torsional barrier height in this excited fundamental vibrational state by about 48.77?cm^?1. The mechanism responsible for the decrease of the torsional splittings in the degenerate vibrational states is briefly outlined by means of second-order perturbation theory, using torsion-hindered vibrational basis functions of E_1d and E_2d symmetries for the degenerate modes.     

Laser infrared spectroscopy of vinyl fluoride in the 1280–1400 cm−1 region

The infrared spectra of CH₂=CHF have been investigated in the ν₅ and ν₆ band regions between 1280 and 1400?cm^?1, at a resolution of about 0.002?cm^?1, using a tunable diode laser spectrometer. These vibrations of symmetry species A′ give rise to a/b-hybrid bands with different contributions from both the components. Spectral analysis resulted in the identification of 1565 (J≤46, K _a ≤11) and 1651 (J≤48, K _a ≤15) transitions of the ν₅ and ν₆ fundamentals, respectively. Both bands are perturbed by the nearby states ν₈?+?ν₉ and ν₉?+?ν₁₁ through different Coriolis resonances and an anharmonic interaction. Using Watson's A-reduction Hamiltonian in the I^r representation and perturbation operators almost all the transitions have been fitted simultaneously to a model including six resonances within the tetrad ν₅/ν₆/ν₈?+?ν₉/ν₉?+?ν₁₁. A set of spectroscopic constants for the ν₅ and ν₆ bands, as well as parameters for the dark states ν₈?+?ν₉ and ν₉?+?ν₁₁ and coupling constants, have been determined. From spectral simulations the dipole moment ratio |Δμ _a /Δμ _b | was estimated to be 0.6?±?0.1 and 2.0±0.3 for the ν₅ and ν₆ bands, respectively.     

High-resolution FTIR spectroscopy of HCFC-31 in the 950−1160 cm−1 region: rovibrational analysis and resonances in the ν4, ν9 and ν5+ν6 bands of CH235ClF

The FTIR spectrum of CH₂ClF (natural isotopic mixture) was investigated in the ν₄, ν₉ and ν₅+ν₆ band region between 950 and 1160 cm^?1 at the resolution of 0.004 cm^?1. The ν₄ and ν₅+ν₆ vibrations of A′ symmetry give rise to a/b hybrid bands with a predominant a-type component. The ν₉ vibration of A^″ symmetry, expected with a c-type band contour, shows an intense Coriolis-induced parallel component (ΔK_a = 0, ΔK_c = 0) derived from mixing with the v₄ = 1 vibrational state. The high-resolution spectra of ν₉ and ν₅+ν₆ have been analyzed for the first time, while the assignments of the ν₄ band, previously investigated, have been extended to higher J and K_a values in the b-type component. The spectral analysis resulted in the identification of 1508, 809 and 349 transitions for the ν₄, ν₉ and ν₅+ν₆ bands of CH₂³⁵ClF, respectively. Besides the strong first-order a- and b-type Coriolis resonances between ν₄ and ν₉, the ν₅+ν₆ vibration was found to interact through a c-type Coriolis with the ν₄ and 3ν₆. High-order anharmonic resonance (ΔK_a = ±2) between ν₄ and ν₅+ν₆ was also established. All the assigned data were simultaneously fitted using the Watson's A-reduction Hamiltonian in the I^r representation and the relevant perturbation operators. The model employed includes five types of resonances within the tetrad ν₄/ν₉/ν₅+ν₆/3ν₆. Α set of spectroscopic constants for ν₄, ν₉ and ν₅+ν₆ bands as well as parameters for the dark state 3ν₆ and seven coupling terms have been determined. The simulations performed in different spectral regions satisfactorily reproduce the experimental data.     

Intensity and transmission measurements in the ν3-fundamental of N2O at low temperatures

Spectral transmission of i.r. radiation through the nitrogen-broadened lines of the υ₃-fundamental of N₂O has been measured at 154°, 202° and 300°K. A value of S⁰_v = 1411±54 cm^-2atm^-1 at S.T.P. has been obtained for the combined strength of the ν₃ and ν₂¹+ν₃?ν₂¹ bands using the Wilson-Wells-Penner-Weber method. This value for S_v, the relative intensity calculations of Gray Young, the room-temperature data of Toth for nitrogen-broadened half-widths in the ν₁+ν₃ and 2ν₂⁰+ν₃ bands and the T^-0.75 variation of line width with temperature proposed by Varanasi and Sarangi are shown to yield excellent agreement between the measured and computed spectral transmittance throughout the band.