Marco Maioli

Double wells: Nevanlinna analyticity, distributional Borel sum and asymptotics

AbstractWe consider the energy levels of a Stark family, in the parameterj, of quartic double wells with perturbation parameterg:

The Distributional Borel Summability and the Large Coupling Φ4 Lattice Fields

H(g,j) = p^2 + x^2 (1 - gx)^2 - j\left( {gx - \frac{1}{2}} \right).

Stochastic and empirical models of the absolute asymmetric synthesis by the Soai-autocatalysis

For non-evenj (and for the symmetric casej=0) we prove analyticity in the full Nevanlinna disk ℜg−2 >R−1 of theg2-plane, as predicted by Crutchfield. By means of an approximation we give a heuristic estimate of the asymptotic smallg behaviour, showing the relation between the avoided crossings and the failure of the usual perturbation series.

Stark ladders of resonances: Wannier ladders and perturbation theory

We prove the Pade (Stieltjes) summability of the perturbation series of any energy level En,1(β), , of the cubic anharmonic oscillator, , as suggested by the numerical studies of Bender and Weniger. At the same time, we give a simple proof of the positivity of every level of the -symmetric Hamiltonian H1(β) for positive β (Bessis–Zinn Justin conjecture). The n zeros, of a state ψn,1(β), stable at β = 0, are confined for β on the cut complex plane, and are related to the level En,1(β) by the Bohr–Sommerfeld quantization rule (semiclassical phase-integral condition). We also prove the absence of non-perturbative eigenvalues and the simplicity of the spectrum of our Hamiltonians.

Stark resonances in disordered systems

We complete and correct some proofs of an earlier paper on distributional Borel summability and we add an application which can be useful in the discussion of semiclassical problems.

The top resonances of the cubic oscillator

We prove that the Stark effect perturbation theory of a class of bound states uniquely determines the position and the width of the resonances by Distributional Borel Sum. In particular the small field asymptotics of the width is uniquely related to the large order asymptotics of the perturbation coefficients. Similar results apply to all the “resonances” of the anharmonic and double well oscillators.

Wannier ladders and perturbation theory

Absolute asymmetric synthesis (AAS) is the preparation of pure (or excess of one) enantiomer of a chiral compound from achiral precursor(s) by a chemical reaction, without enantiopure chiral additive and/or without applied asymmetric physical field. Only one well-characterized example of AAS is known today: the Soai-autocatalysis. In an attempt at clarification of the mechanism of this particular reaction we have undertaken empirical and stochastic analysis of several parallel AAS experiments. Our results show that the initial steps of the reaction might be controlled by simple normal distribution (“coin tossing”) formalism. Advanced stages of the reaction, however, appear to be of a more complicated nature. Symmetric beta distribution formalism could not be brought into correspondence with the experimental observations. A bimodal beta distribution algorithm provided suitable agreement with the experimental data. The parameters of this bimodal beta function were determined by a Pólya-urn experiment (simulated by computer). Interestingly, parameters of the resulting bimodal beta function give a golden section ratio. These results show, that in this highly interesting autocatalysis two or even perhaps three catalytic cycles are cooperating. An attempt at constructing a “designed” Soai-type reaction system has also been made.

Double wells: perturbation series summable to the eigenvalues and directly computable approximations

LetHB be any fixed one-dimensional Bloch Hamiltonian with only the firstm gaps open andHF=HB+Fx be the corresponding Stark Hamiltonian. For any positiveF small enoughHF has onlym ladders of sharp resonances given by the analytic translation method, the decoupled band approximation and the regular perturbation theroy. This way, the Wannier conjecture becomes a definite regular perturbation theory for the Stark ladders as eigenvalues of the translated Hamiltonian.