Convergence of double Fourier series

Summary Let a2π-periodic function f(x, y) be continuous in some neighbourhood of the point (x, y) except possibly along finitely many lines l ₁ , l ₂ , ..., l_k terminating at (x, y). The problem of convergence of the Fourier series of f(x, y) at the point (x, y) is examined in some detail. It is established that under certain restrictions on the variation of f(x, y), and also on the lines l ₁ , l ₂ , ..., l_k, the fourier series converges to a value bounded above by the limit superior, and below by the limit inferior of f(x+u, y+v), u, v →0, this value depending on the manner in which the series is summed. The preparation of this paper was financed, in part, by a Canadian Mathematical Congress Summer Research Grant (1968). Entrata in Redazione il 3 febbraio 1969.     

Oligoacenes: theoretical prediction of open-shell singlet diradical ground states

A series of oligoacenes from benzene to decacene were studied computationally with DFT and CASSCF methods. In contrast to the common view that acenes are closed-shell systems or may have a triplet ground state, these results offer the first theoretical predictions for the singlet ground state and diradical character for oligoacenes. The nature of the ground states of these molecules arises from the disjoint nature of the NBMOs that are singly occupied in the diradical.     

Photochemistry of the SO2, 2-pentene system at 3660 Å

The photolysis of SO₂ in the presence of cis- and trans-2-pentene has been investigated at 3660 Å and 22°C. Quantum yield measurements of the SO₂ photosensitized conversion of one isomer into the other are consistent with a mechanism in which the only participating excited electronic state of SO₂ is the SO₂(³B₁) state. Quantum yield measurements were made for a variation in P_SO2/P_isomer reactant ratios of 4.01–283 and 57.5–351 for the cis and trans isomers, respectively. The data are consistent with a mechanism in which a (SO₂-olefin)³ collision intermediate is the precursor to the photosensitized isomeric products. The intermediate undergoes unimolecular decay to yield the cis and trans isomers with probabilities of 0.26 ± 0.05 and 0.69 ± 0.04, respectively. Estimates of the quenching rate constants at 22°C for removal of SO₂(³B₁) molecules by cis- and trans-2-pentene are (0.633 ± 0.125) × 10¹¹ l./mole/sec and (1.00 ± 0.27) × 10¹¹ l./mole/sec, respectively. An experimentally determined photostationary composition, [trans-2-pentene]/[cis-2-pentene] = 2.3 ± 0.1 was in fair agreement with that of 1.7 ± 0.7 as predicted from kinetic data derived in this study.     

Aspects of supersymmetric quantum mechanics

We review the properties of supersymmetric quantum mechanics for a class of models proposed by Witten. Using both Hamiltonian and path integral formulations, we give general conditions for which supersymmetry is broken (unbroken) by quantum fluctuations. The spectrum of states is discussed, and a virial theorem is derived for the energy. We also show that the euclidean path integral for supersymmetric quantum mechanics is equivalent to a classical stochastic process when the supersymmetry is unbroken (E₀ = 0). By solving a Fokker-Planck equation for the classical probability distribution, we find P_c(y) is identical to |Ψ₀(y)|² in the quantum theory.