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Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials

Eckhardt J., Gesztesy F., Nichols R., Teschl G.

Opuscula Mathematica

2013

Vol. 33, no. 3

467-563

We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals (a, b) ⊆ R associated with rather general differential expressions of the type [formula] where the coefficients p, q, r, s are real-valued and Lebesgue measurable on (a, b), with p ≠ 0, r > 0 a.e. on (a, b), and p−1, q, r, [formula] , and ƒ is supposed to satisfy [formula]. In particular, this setup implies that τ permits a distributional potential coefficient, including potentials in [formula]. We study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator Tmax, or equivalently, all self-adjoint extensions of the minimal operator Tmin, all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of Tmin. In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira m-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum. We also deal with principal solutions and characterize the Friedrichs extension of Tmin. Finally, in the special case where τ is regular, we characterize the Krein-von Neumann extension of Tmin and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents (and hence semigroups).

Electrochemical and Computational Studies of Some 9,10-Anthraquinone Derivatives in Acetonitrile Solution

Shamsipur M., Mohammadi T., Alizadeh N., Sharghi H., Nichols R.

Polish Journal of Chemistry

2005

Vol. 79 / nr 8

1379-1388

The electrochemical behavior of some recently synthesized 9,10-anthraquinone derivatives on Au electrode in acetonitrile was investigated. In the absence of proton donors, the anthraquinones reduced in two successive one-electron steps. The first step is reversible or nearly reversible, while the second step is quasi-reversible. The influence of molecular structure on the reduction potential is addressed. The diffusion coefficients of the anthraquinone derivatives were determined from chronoamperometry and rotating disk electrode measurements. The heterogeneous electron transfer rate constants and charge transfer coefficients were evaluated from rotating disk voltammetry measurements. An ab-initio quantum mechanical method was carried out at the DFT-B3LYP level to compute the electrode's formal potentials for the four anthraquinones AQ1-AQ4 in acetonitrile solution. A nice linear relationship was observed between the theoretically predicted values and experimentally determined formal electrode potentials of the 9,10-anthraquinone derivatives.