A Second Eigenvalue Bound for the Dirichlet Schrödinger Operator

Let λ _i (Ω,V) be the i ^th eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain $\Omega \subset \mathbb{R}^nLet λ _i (Ω,V) be the i ^th eigenvalue of the Schr?dinger operator with Dirichlet boundary conditions on a bounded domain and with the positive potential V. Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential V _*, we prove that λ₂(Ω,V) ≤ λ₂(S ₁,V _*). Here S ₁ denotes the ball, centered at the origin, that satisfies the condition λ₁(Ω,V)=λ₁(S ₁,V _*).Further we prove under the same convexity assumptions on a spherically symmetric potential V, that λ₂(B _R , V) / λ₁(B _R , V) decreases when the radius R of the ball B _R increases.We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density.R.B. was supported by FONDECYT project # 102-0844.H.L. gratefully acknowledges financial support from DIPUC of the Pontifí cia Universidad Católica de Chile and from CONICYT.     

The infrared spectrum of isoxazole in the range 600–1400 cm−l,including a high-resolution study of the v 7(A′) band at 1370.9 cm−1 and the v 16(A″) band at 764.9 cm−l,together with ab initio studies of the full spectrum

Abstract

The high resolution infrared gas phase spectrum of isoxazole in the range 600–1400 cm^?1 has been recorded and more precise centres obtained for a number of bands; analyses of the v ₇(A′) band at 1370.9cm^?1 and the v ₁₆(A″) band at 764.9cm^?1 have been performed. Use of the Watson Hamiltonian, A-reduction, III^r-representation and simultaneous analysis of the present IR and previous microwave data, has led to rotation constants and quartic centrifugal distortion constants Δ _J , Δ _JK , ΔK, δ _J and δ _K for the ground state and for the v ₁₆ vibrationally excited states. The equilibrium structures and the derived harmonic frequencies have been calculated by ab initio methods using triple zeta+polarization (TZVP) and cc-pVTZ basis sets, with MP2, MP4 and CCSD(T) methods. At each methodology, for closest numerical agreement between the calculated and observed rotation constants, the optimum basis set seems to be TZVP rather than cc-pVTZ basis sets. However, the order of the highest A″ and lowest A′ symmetry vibrations is only resolved by the cc-pVTZ basis set with the MP4 methodology, which does generate the experimental sequence (v ₁₄>v ₁₃) The CCSD(T) methodology does not lead to significant difference over either MP2 or MP4 with the TZVP basis set.     

On the stability of scalar-vacuum space-times

We study the stability of static, spherically symmetric solutions to the Einstein equations with a scalar field as the source. We describe a general methodology of studying small radial perturbations of scalar-vacuum configurations with arbitrary potentials V(ϕ), and in particular space-times with throats (including wormholes), which are possible if the scalar is phantom. At such a throat, the effective potential for perturbations V _eff has a positive pole (a potential wall) that prevents a complete perturbation analysis. We show that, generically, (i) V _eff has precisely the form required for regularization by the known S-deformation method, and (ii) a solution with the regularized potential leads to regular scalar field and metric perturbations of the initial configuration. The well-known conformal mappings make these results also applicable to scalar-tensor and f(R) theories of gravity. As a particular example, we prove the instability of all static solutions with both normal and phantom scalars and V(ϕ)≡0 under spherical perturbations. We thus confirm the previous results on the unstable nature of anti-Fisher wormholes and Fisher’s singular solution and prove the instability of other branches of these solutions including the anti-Fisher “cold black holes.”     

The infrared spectrum of isothiazole in the range 600–1500 cm−1, including a high-resolution study of the v 11(A′) band at 818 cm−1 and the v 16(A″) band at 727 cm−1, together with ab initio studies of the full spectrum

Abstract

The gas-phase high-resolution infrared spectrum of isothiazole in the range 600–1500 cm^?1 has been recorded, and revised band centres obtained for a number of vibrations. An analysis of the v ₁₁(A′) band at 818 cm^?1 and the v ₁₆(A″) band at 727 cm^?1 has been performed, using the Watson Hamiltonian, A-reduction, III^r representation. These were combined with previous microwave spectral data to provide combined analyses for rotational constants and quartic centrifugal distortion constants Δ _J , Δ _JK , Δ _K , δ _j and δ _K . These extend the knowledge derived from previous microwave and IR spectral studies. The equilibrium structures and the derived harmonic frequencies were calculated by ab initio methods, using a variety of basis sets with MP2, MP4 and CCSD(T) methods, and a comparison of these with experimental data is discussed. At a pragmatic level, the closest correlation of the rotational constants with experiment is not obtained with the most sophisticated methodology. Similarly, the vibration frequencies and intensities also vary strongly with the procedure. In particular, we found that the cc-pVTZ+MP2 results probably provide the best numerical comparison with experimental IR data for this molecule.     

Contribution of the inhomogeneous waves in angular-spectrum representations

Angular-spectrum representations of wave fields separate naturally into two parts, V_h(R,t) containing only homogeneous plane waves and V_i(R,t) containing only inhomogeneous plane waves. Some properties of V_i(R,t) are presented here. We conclude that, for some problems, V_i(R,t) has several unphysical properties, and that under certain specified conditions, V_i(R,t) can not be neglected compared to V_h(R,t), even far from the sources of the field.     

Intermittent discharging of nickel–metal hydride secondary battery

A nickel–metal hydride secondary battery is discharged intermittently through a switch of 30 s on/30 s off mode via an external load R _r. The discharge current i _g is measured during switch-on and an open-circuit voltage V _oc during switch-off. In the course of discharge run, i _g drops at some time by depletion of an active material within the positive electrode. The pulse shapes of i _g and V _oc in each on/off duration are featureless before the i _g drop, satisfying the relation V _oc=i _g(R _i+R _r) (R _i being the internal resistance of a battery). After the i _g drop, they become time dependent; yet, the above relation still holds. It is found that V _oc changes continuously through the onset times of switch-on and -off.     

高次激发相关能在构建HCN-He亚光谱精度势能面中的角色:通过实验高分辨率光谱进行严格检测

Interpreting high-resolution rovibrational spectra of weakly bound complexes commonly requires spectroscopic accuracy (<1 cm^-1) potential energy surfaces (PES). Constructing high-accuracy ab initio PES relies on the high-level electronic structure approaches and the accurate physical models to represent the potentials. The coupled cluster approaches including single and double excitations with a perturbational estimate of triple excitations (CCSD(T)) have been termed the "gold standard" of electronic structure theory, and widely used in generating intermolecular interaction energies for most van der Waals complexes. However, for HCN-He complex, the observed millimeter-wave spectroscopy with high-excited resonance states has not been assigned and interpreted even on the ab initio PES computed at CCSD(T) level of theory with the complete basis set (CBS) limit. In this work, an effective three-dimensional ab initio PES for HCN-He, which explicitly incorporates dependence on the Q₁ (C-H) normal-mode coordinate of the HCN monomer has been calculated at the CCSD(T)/CBS level. The post-CCSD(T) interaction energy has been examined and included in our PES. Analytic two-dimensional PESs are obtained by least-squares fitting vibrationally averaged interaction energies for v₁(C-H)=0, and 1 to the Morse/Long-Range potential function form with root-mean-square deviations (RMSD) smaller than 0.011 cm^-1. The role and significance of the post-CCSD(T) interaction energy contribution are clearly illustrated by comparison with the predicted rovibrational energy levels. With or without post-CCSD(T) corrections, the value of dissociation limit (D₀) is 8.919 or 9.403 cm^-1, respectively. The predicted millimeter-wave transitions and intensities from the PES with post-CCSD(T) excitation corrections are in good agreement with the available experimental data with RMS discrepancy of 0.072 cm^-1. Moreover, the infrared spectrum for HCN-He complex is predicted for the first time. These results will serve as a good starting point and provide reliable guidance for future infrared studies of HCN doped in (He)_n clusters.     

Multiple photon excitation of the v2 + v6 combination band of SF6 in a bulk and in a pulsed free jet

Multiple photon excitation of the v₂ + v₆ combination band of SF₆ in a bulk at T ≈ 295 K and cooled in a pulsed free jet up to T_V ≈ 160 K and T_R ≈ 40 K by a pulsed TEA CO₂ laser has been investigated. Obtained results are compared with the data on the v₃ vibration excitation. At exciting energy fluences ø = 0.1?2.5 J cm^-2 the levels in the region of the discrete vibrational states (v=3?5) are found mainly to be excited. Multiphoton absorption spectra at room temperature have a sharp resonant structure. The fraction of interacting molecules is considerably (3–7 times) less compared than that for the case of v₃ vibration excitation. Multiphoton absorption of the v₂ + v₆ and v₃ vibrations of SF₆ is shown to be proportional to the dipole moments of the corresponding transitions.     

FTIR spectroscopy of the 2v3 overtone band for different BrCN isotopomers: an improved evaluation of the anharmonic force field of cyanogen bromide

The 2v₃ overtone band of six different isotopomers of cyanogen bromide has been recorded in the range from 4200 to 4400 cm^?1 with a resolution of 0.02 cm^?1 using a Fourier transform infrared (FTIR) spectrometer. This allowed us to achieve complete knowledge of the energies for all levels corresponding to double vibrational excitation. An improved evaluation of the quartic force field of cyanogen bromide has been performed using the new data obtained together with those already known from previous works. Theoretical values derived from a recent ab initio calculation have been used to constrain the potential terms which cannot be determined with sufficient precision by the least-squares analysis of the available experimental data.     

Zur Anisotropie träger Massen in tensoriellen Gravitationstheorien

According to the general theory of relativity an anisotropy of the space -like componentsT^k_i of the matter-tensor T_μ^v is generating an anisotropy of the three-dimensional space-metric in any non-inertial reference system, i.l., in any non-freemoving laboratory. Hence, the three-dimensional space-metric is not conformal euclidian. Then, the effective inertial masses m are anisotropic tensors ∽g_ik in such non-inertial reference systems, and the three-dimensional momentum and the threedimensional velocity are no longer parallel vectors (FOKKER 1917, 1965). — A laboratory at rest on the earth is a non-inertial reference system relative to the gravitational field of the earth. The matter tensor T_ik of the earth has an anisotropic term which results from the rotation of the earth. According to EINSTEINS equations this term generates an anisotropy correction ?γ_ik to the three-dimensional metric approximately given by γ_ik = ? 4f/3c⁴ M/R ?_i?_k (with M = earth mass; R = earth radius) (In this formula ?_i is the velocity of rotation at the equator). The anisotropy of the three-dimensional metric gives rise to an anisotropy of the inertial masses m: The inertial masses become greater for motions in the east-west-direction than for motions in the north-south-direction, the difference amounting to Δm = 4/3 f/c⁴ M/R ?²m ≈ 2 ? 10^?21 m Therefore, the moments of inertia of revolving bodies are distorted. According to COCCONI and SALPETER (1958/60) and HUGHES (1960/64) the anisotropy of the inertial masses gives rise to a quadrupol-like correction term in the HAMILTON ian with the consequence of a splitting of degenerate energies. This splitting would appear as a broadened line, with an increased width, in the case of nuclear magnetic resonance lines of atomes with the nuclear spin S = 3/2. The expected line broadening resulting from mass anisotropy is given by the term shifts ΔT = 8/9 f/c⁴ M/R ?²T?P?₂ (cos ?) P₂ (cos β) (P₂ = LEGENDRE -function) and amounts to Δv ≈ 2 ? 5 ∣P₂(cos β)∣ HZ, For the nucleus Li⁷ in the state P_3/2. (In this formula β is the angle of the magnetic field with the direction of the earth is rotation.) — Therefore, the corresponding resonance frequency would have a minimal width Δv. This may be a strange result with difficulties for the general relativity theory of gravitation. The general relativistic mass anisotropy is an effect of the second post-NEWTON ian approximation: Δm ? fM/Rc² V²/c² m The effect is by the factor 10^?6 smaller than the smallest relativistic effect examined till now (POUND -REBKA -experiment). However, the experimental accuracy is by the factor 10⁷ better, too.     

Instanton effects on the interaction potential between light quarks and the isomultiplet mass splitting of baryons

On the basis of current assumptions of instanton theory we derive strictly the explicit dependence on the masses and spins of the instanton induced potential between a pair of light quarks in baryons, namelyV ₁₂=γ+β(m _* ^1+α )(m _* ^2+α )η(1-σ ₁ησ ₁), wherem _i ^* andσ _i (i=1.2) are respectively the mass and Pauli spin of theith quark. On the additional basis of the MIT bag model, we obtain γ=c/R ³ and β=b/R ³>0, and α>0 is independent of the radiusR of the baryon. The magnitudes of the parametersb and α are also estimated. The MIT bag model is improved by taking into account this potential. Isomultiplet mass splitting formulas are derived in good agreement with experiment.     

AsH(X3Σ－)及AsH2(X2B1)自由基的分子结构与解析势能函数

运用Gaussian 03程序包中的单双迭代三重激发耦合簇理论和相关一致五重基优化了AsH₂的基态结构,并在优化结构的基础上计算了它的离解能和振动频率. 结果表明：AsH₂基态的平衡构型具有C_2v对称性,键长R_As-H=01508 nm,键角∠HAsH=912231°,离解能D_e(HAs-H)=28795 eV,振动频率ν 关键词： 2')" href="#">AsH₂ Murrell-Sorbie函数 多体项展式理论 解析势能函数     

Analysis of the high resolution spectrum of the fundamental bandsv2 andv5 of12CH3F and their interactions

Thev ₂(A₁) andv ⁵(E) fundamental vibration-rotation bands of¹²CH₃F have been recorded under high resolution (0.015 to 0.020 cm^–1) in the spectral range of 1460 cm^–1. About 1100 transitions have been assigned. The Coriolis interaction between v₂=1 and v₅=1, and the l(2,-1) interaction in v₅=1 have been rigorously treated. Sixteen molecular constants have been determined from a least squares analysis. They reproduce the observed data with an overall standard deviation of 0.0037 cm^–1.     

Indirect Analytic Representation of Foucault's Pendulum

A complex representation of the equations of motion of the Foucault's pendulum is considered and the inverse problem is solved to derive an indirect analytic representation. Both real L _R ⁽ⁱ⁾ and imaginary L _I ⁽ⁱ⁾ parts of the derived complex valued Lagrangian are found to reproduce the equations of motion via the Euler-Lagrange equations. The expressions for L _R ⁽ⁱ⁾ and L _I ⁽ⁱ⁾ are not connected by a gauge term thereby forming a set of inequivalent Lagrangians. In an appropriate limit L _I ⁽ⁱ⁾ is found to reproduce the Lagrangian obtained by implementing the usual Coriolis theorem while in some other limit L _R ⁽ⁱ⁾ and L _I ⁽ⁱ⁾ give the indirect and the direct analytic representations for a set of two uncoupled harmonic oscillators.     

The reference potential in the ATA

Precise limitations on the choice of the reference potential for the ATA are investigated, and it is shown that V⁽ⁱ⁾_r?〈V_i〉=O(xy).     

Paramagnetic resonance of some three-nuclear clusters of iron

ESR of exchange-coupled iron triads in carboxilates [Fe₃O(RCOO)₆(H₂O)₃·(RCOO)_n where R = ClCH₂, FCH₂, Cl₂CH or Cl₃C is studied at 4.2 K (microwave frequency v = 9.5 kMc/sec). ESR spectra and observed values of g-factors are discussed according to the model of a symmetrical cluster. Fine structure δ splittings of the triad ground state are determined on bases of the theory and ESR data: δ ≈ 0.03-0.16 cm^?1.     

The pair distributions and the osmotic coefficient in anomalous electrolytes up to concentrations c≈ 1 mol/1

The osmotic coefficient of anomalous electrolytes up to concentrations c ≈ 1 mol/l is explained by the pair distributions $\text{n(r) =}\text{exp}\text{[-β(}\text{V}{}_{\mathrm{c}}\text{(r) + V(}{}^{\mathrm{hs}}\text{)(r) + V}{}_1\text{(r))]}$. Here $\text{V}$_c(r) is a screened Coulomb potential, V(^hs)(r) a hard sphere potential and V₁(r) = ?A/r⁶ a short range attractive potential. For the contact distances R₊₊, R_?? and R_+? of the hard sphere potentials between ions with the same sign of their charges (++,??) and ions of opposite charges (+?) the relations R₊₊ = R_?? = R and R_+? = q₁R with 0 < q₁ < 1 are assumed. In contrast to a previous paper the parameter q₁ takes a fixed value q₁ ≈ 0,8. The constant A is determined by the fraction q₂ defined by A/R⁶ = q₂(Z²e²/DR) where the positive integer Z is the charge number of the ions and D the dielectric constant of the solvent. The numerical calculation of the osmotic coefficient of 1-1-valent hydrous electrolytes in the range of temperature 273 K ? T ? 293 K shows that the anomalous electrolytes are described by fractions q₂ in the range 0,25 ? q₂ ? 0,5 if the contact distances R are in the range $\text{3}\text{A}\text{?}\text{? R ? 7}\text{A}\text{?}$.     

Determination of the positions of impurity centers in a dislocation core in a NaCl crystal from magnetoplasticity spectra

The spectra of the mean free paths l(ν) of edge dislocations have been studied in NaCl crystals exposed in the electron paramagnetic resonance scheme to the crossed magnetic fields: the Earth’s field (50 μT) and the pump field (2.5 μT, 5–440 kHz). The spectra have been measured for a series of angles θ = 0°–5° of rotation of the sample around its edge [100] with respect to the Earth’s field. The fine structure of the spectra contains a series of peaks whose resonance frequencies are described by the empirical expression v _i ^± = Asin(θ ± Δθ _i ) ≈ A(θ ± Δθ _i ). The parameters Δθ _i are independent of the angle θ within the experimental errors. Within the model of “frozen” magnetic moments associated with impurity center Ca⁺-Cl⁰, the angles Δθ _i characterize the deviation of the axis of the center from the 〈100〉 direction in the core of a dislocation. These angles can be expressed in terms of the spectra obtained: Δθ _i = (? _i ⁺ ? v _i ^? )/2A. The computer simulation of the edge dislocation core provides the set of the angles Δθ _i close to the measured values. The spin-lattice relaxation time of the center on dislocation has been estimated from the low-frequency edge of the spectrum l(ν) as τ _{s ? l} ～ 10^?4 s.     

Ab initio study of the Ar–CS2(V1B2) intermolecular potential surface: effect of van der Waals interaction on the emission of CS2 molecule

In this work, the excited state intermolecular potential energy surface of the Ar–CS₂(V¹B₂) van der Waals complex was evaluated for the first time. The calculation of more than 4000 single-point interaction energies for the complex using an equation-of-motion coupled-cluster model with single and double substitutions level of theory with extended basis set involving bond functions has been performed. After fitting the interaction energies to analytical functions, the emission spectra of the Ar–CS₂(V¹B₂) complex related to the different stationary points on the potential energy surface were calculated. It was seen that the intensity and the position of the emission spectra are dependent on the orientation of the Ar atom around the bent excited CS₂ and the distance between two components. The information about the structural parameters of the complex related to the global minimum was obtained under the pseudodiatomic approximation with assistance of ab initio potential. The presented investigation could be useful for further theoretical and experimental studies of Ar–CS₂(V¹B₂) complex.     

Conservation of Energy,Entropy Identity,and Local Stability for the Spatially Homogeneous Boltzmann Equation

For nonsoft potential collision kernels with angular cutoff, we prove that under the initial condition f ₀(v)(1+|v|²+|logf ₀(v)|)L ¹(R ³), the classical formal entropy identity holds for all nonnegative solutions of the spatially homogeneous Boltzmann equation in the class L ([0, ); L ¹ ₂(R ³))C ¹([0, ); L ¹(R ³)) [where L ¹ _s (R ³)={ff(v)(1+|v|²) ^s/2L ¹(R ³)}], and in this class, the nonincrease of energy always implies the conservation of energy and therefore the solutions obtained all conserve energy. Moreover, for hard potentials and the hard-sphere model, a local stability result for conservative solutions (i.e., satisfying the conservation of mass, momentum, and energy) is obtained. As an application of the local stability, a sufficient and necessary condition on the initial data f ₀ such that the conservative solutions f belong to L ¹ _loc([0, ); L ¹ ₂₊ (R ³)) is also given.