CONSTRUCTING QUANTUM OBSERVABLES AND SELF-ADJOINT EXTENSIONS OF SYMMETRIC OPERATORS. III. SELF-ADJOINT BOUNDARY CONDITIONS

This paper completes the review of the theory of self-adjoint extensions of symmetric operators for physicists as a basis for constructing quantum-mechanical observables. It contains a comparative presentation of the well-known methods and a newly proposed method for constructing ordinary self-adjoint differential operators associated with self-adjoint differential expressions in terms of self-adjoint boundary conditions. The new method has the advantage that it does not require explicitly evaluating deficient subspaces and deficiency indices (these latter are determined in passing) and that boundary conditions are of explicit character irrespective of the singularity of a differential expression. General assertions and constructions are illustrated by examples of well-known quantum-mechanical operators like momentum and Hamiltonian.

Induced polarization, resistivity, and self-potential: a case history of contamination evaluation due to landfill leakage

This article compares the efficiency of induced polarization (IP) and resistivity in characterizing a contamination plume due to landfill leakage in a typical tropical environment. The resistivity survey revealed denser electrical current flow that induced lower resistivity values due to the high ionic content. The increased ionic concentration diminished the distance of the ionic charges close to the membrane, causing a decrease in the IP phenomena. In addition, the self-potential (SP) method was used to characterize the preferential flow direction of the area. The SP method proved to be effective at determining the flow direction; it is also fast and economical. In this study, the resistivity results were better correlated with the presence of contamination (lower resistivity) than the IP (lower chargeability) data.