Oppenheimer Expansion Research Articles

New asymptotic results for generalized Oppenheim expansions

New asymptotic results for generalized Oppenheim expansions

Convergence results for Oppenheim expansions (II)

Convergence results for Oppenheim expansions (II)

On Exact Laws of Large Numbers for Oppenheim Expansions with Infinite Mean

In this work, we investigate the asymptotic behaviour of weighted partial sums of a particular class of random variables related to Oppenheim series expansions. More precisely, we verify convergence in probability as well as almost sure convergence to a strictly positive and finite constant without assuming any dependence structure or the existence of means. Results of this kind are known as exact weak and exact strong laws.

Convergence results for Oppenheim expansions

We prove convergence in probability for particular sequences defined in terms of the digits appearing in Oppenheim Series expansions and Oppenheim Continued Fractions expansions of real numbers. Our results are obtained by first proving a general theorem (Theorem 2.2) having both kinds of expansion as particular cases.

On singularity of distribution of random variables with independent symbols of Oppenheim expansions

The paper is devoted to the restricted Oppenheim expansion of real numbers ($ROE$), which includes already known Engel, Sylvester and Luroth expansions as partial cases. We find conditions under which for almost all (with respect to Lebesgue measure) real numbers from the unit interval their $ROE$-expansion contain arbitrary digit $i$ only finitely many times. Main results of the paper state the singularity (w.r.t. the Lebesgue measure) of the distribution of a random variable with i.i.d. increments of symbols of the restricted Oppenheim expansion. General non-i.i.d. case is also studied and sufficient conditions for the singularity of the corresponding probability distributions are found.

Exceptional sets of the Oppenheim expansions over the field of formal Laurent series

Let Fq be a finite field with q elements, Fq((z−1)) denote the field of all formal Laurent series with coefficients in Fq and I be the valuation ideal of Fq((z−1)). For any formal Laurent series x=∑n=ν∞cnz−n∈I, the series 1a1(x)+∑n=1∞r1(a1(x))⋯rn(an(x))s1(a1(x))⋯sn(an(x))1an+1(x) is the Oppenheim expansion of x. Suppose ϕ:N→R+ is a function satisfying ϕ(n)/n→∞ as n→∞. In this paper, we quantify the size, in the sense of Hausdorff dimension, of the setE(ϕ)={x∈I:limn→∞⁡∑j=0n−1Δj(x)ϕ(n)=1}, where Δ0(x)=deg⁡a1(x) and Δn(x)=deg⁡an+1(x)−2deg⁡an(x)−deg⁡rn(an(x))+deg⁡sn(an(x)) for all n≥1. As applications, we investigate the cases when ϕ(n) are the given polynomial or exponential functions. At the end of the article, we list some special cases (including Lüroth, Engel, Sylvester expansions of Laurent series and Cantor infinite products of Laurent series) to which we apply the conclusions above.

The generalized Oppenheim expansions for the direct product of non-Archimedean fields

The well-known Oppenheim expansion algorithm in the field ℚp is generalized to the ring ℚg, where g = p1 · ... · pN. The metric properties of the digits of this expansion and also the metric properties of the coefficients of some expansions of polyadic numbers are studied.

A Born–Oppenheimer Expansion in a Neighborhood of a Renner–Teller Intersection

We perform a rigorous mathematical analysis of the bending modes of a linear triatomic molecule that exhibits the Renner–Teller effect. Assuming the potentials are smooth, we prove that the wave functions and energy levels have asymptotic expansions in powers of \({\epsilon}\) , where \({\epsilon^4}\) is the ratio of an electron mass to the mass of a nucleus. To prove the validity of the expansion, we must prove various properties of the leading order equations and their solutions. The leading order eigenvalue problem is analyzed in terms of a parameter \({\tilde{b}}\) , which is equivalent to the parameter originally used by Renner. For \({0< \tilde{b}< 1}\) , we prove self-adjointness of the leading order Hamiltonian, that it has purely discrete spectrum, and that its eigenfunctions and their derivatives decay exponentially. Perturbation theory and finite difference calculations suggest that the ground bending vibrational state is involved in a level crossing near \({\tilde{b}=0.925}\) . We also discuss the degeneracy of the eigenvalues. Because of the crossing, the ground state is degenerate for \({0< \tilde{b}< 0.925}\) and non-degenerate for \({0.925< \tilde{b}< }\) .

Some Remarks on the Generalized St. Petersburg Games and Formal Laurent Series Expansions

We discuss the asymptotic behavior of the largest amount of winnings in the generalized St. Petersburg games and show that the results on the random variables associated with the amount of winnings can be applied to the random variables associated with the degree of the polynomial digits of the continued fraction and the Oppenheim expansions of formal Laurent series.

Theory of Vibrational Circular Dichroism and Infrared Absorption: Extension to Molecules with Low-Lying Excited Electronic States

A vibronic theory of unpolarized infrared absorption and vibrational circular dichroism (VCD) is presented that is valid in the limit of vibrational resonance with low-lying electronic states (LLESs). The theory is developed within the complete adiabatic vibronic coupling formalism that describes both the correlation of electron density to nuclear positions and the correlation of electron current density to nuclear velocities. It is found that additional terms contribute to the electric-dipole and magnetic-dipole transition moments that become zero in the limit where all excited electronic states in the molecule are much higher in energy relative to fundamental vibrational transitions. Two correction terms appear that are extensions of the lowest-order non-Born−Oppenheimer expansion term when vibronic detail is included in the energy denominators. One term is a resonance term with respect to the energy difference between the electronic and vibrational transitions, and the other is a nonresonance term. Und...

Metric Properties and Exceptional Sets of the Oppenheim Expansions over the Field of Laurent Series

We investigate metric properties of the polynomial digits occurring in a large class of Oppenheim expansions of Laurent series, including Luroth, Engel, and Sylvester expansions of Laurent series and Cantor infinite products of Laurent series. The obtained results cover those for special cases of Luroth and Engel expansions obtained by Grabner, A. Knopfmacher, and J. Knopfmacher. Our results applied in the cases of Sylvester expansions and Cantor infinite products are original. We also calculate the Hausdorff dimensions of different exceptional sets on which the above-mentioned metric properties fail to hold.

Born-Oppenheimer Corrections to the Effective Zero-Mode Hamiltonian in SYM Theory

We calculate the subleading terms in the Born--Oppenheimer expansion for the effective zero-mode Hamiltonian of N = 1, d=4 supersymmetric Yang--Mills theory with any gauge group. The Hamiltonian depends on 3r abelian gauge potentials A_i, lying in the Cartan subalgebra, and their superpartners (r being the rank of the group). The Hamiltonian belongs to the class of N = 2 supersymmetric QM Hamiltonia constructed earlier by Ivanov and I. Its bosonic part describes the motion over the 3r--dimensional manifold with a special metric. The corrections explode when the root forms \alpha_j(A_i) vanish and the Born--Oppenheimer approximation breaks down.

Adiabatic Perturbation Theory in Molecular Spectroscopy

The Born–Oppenheimer method or adiabatic perturbation theory has long provided a formal basis for separation of the electronic and nuclear motions in molecules. According to it solutions of the molecular Schrödinger equation can be obtained using expansions in power series in the parameter κ = (m/M)1/4 where m is the electronic mass and M is the average mass of the nuclei; the zeroth-order term in the eigenfunction expansion was shown to be a product of the electronic, vibrational, rotational, and translational eigenfunctions. However, this zeroth-order separation of motions was established only qualitatively because the Hamiltonian expansion was not obtained in explicit form. Therefore the adiabatic perturbation theory (to be referred to as APT) appeared to be inconvenient in applications, and practical investigations in molecular spectroscopy were based on simplified models and approximations rather than on the APT. The present review attempts to clarify and remove difficulties impeding wider applications of the APT and to give examples of its practical use. For this purpose a molecule-fixed moving coordinate system (to be referred to as mcs) is introduced by means of the Eckart conditions, and the Hamiltonian is transformed to the electronic and nuclear coordinates defined with the help of the mcs. For the transformed Hamiltonian the Born–Oppenheimer expansion is written explicitly and order-by-order calculations of the APT are carried out up to the fourth-order equation for normal and linear molecules. The results are used to discuss the stability of normal molecules and the Renner effect in linear molecules. The stability is considered for the cases of degenerate and near degenerate electronic levels. Lowering of vibrational frequencies caused by the near degeneracy is also studied, and the results are applied to infrared spectra of hexafluorides. The Renner equations for the bending vibrations of linear molecules are derived from the second-order equation of the APT. The four-atomic Renner equations are solved approximately using expansions in power series in the Renner parameters; the results are applied to vibronic spectra of C2H2 and C2D2 molecules. The influence of the Renner effect on the rotational spectra of triatomic molecules is considered with the help of the sixth-order energy correction of the APT. Higher order APT calculations are discussed briefly in the last section.

Possible New Type of Molecular Bound State

It is shown that the corrections to the Born–Oppenheimer approximation may lead to the appearance of a new type of bound state, the theory of which is mathematically analogous to that of Cooper pair states in superconductivity theory. Some of the properties of these states, and the conditions for their existence, are studied for the special case of diatomic molecules. In all cases of interest, one such state at most can appear below the continuum threshold belonging to the ground electronic state of the molecule, i.e., they have no rotational or vibrational structure. In the case most likely to arise, the binding energy, considered as a function of the Born–Oppenheimer expansion parameter κ, has an essential singularity at zero, so the state could never have been discovered by the power-series expansion method. Experimental observation of these states is expected to be difficult since their wavefunctions do not differ importantly from those for ordinary bound states of the same energy. However, the theory predicts that an accurate Born–Oppenheimer calculation of the number of bound states should sometimes be too low by one, and the isotope effect on the binding energy is different from that of ordinary bound states.