Self-adjoint extensions for confined electrons: From a particle in a spherical cavity to the hydrogen atom in a sphere and on a cone

In a recent study of the self-adjoint extensions of the Hamiltonian of a particle confined to a finite region of space, in which we generalized the Heisenberg uncertainty relation to a finite volume, we encountered bound states localized at the wall of the cavity. In this paper, we study this situation in detail both for a free particle and for a hydrogen atom centered in a spherical cavity. For appropriate values of the self-adjoint extension parameter, the bound states localized at the wall resonate with the standard hydrogen bound states. We also examine the accidental symmetry generated by the Runge–Lenz vector, which is explicitly broken in a spherical cavity with general Robin boundary conditions. However, for specific radii of the confining sphere, a remnant of the accidental symmetry persists. The same is true for an electron moving on the surface of a finite circular cone, bound to its tip by a 1/r1/r potential.

Osseointegration of hollow cylinder based spinal implants in normal and osteoporotic vertebrae: a sheep study

INTRODUCTION: Osteoporosis is not only responsible for an increased number of metaphyseal and spinal fractures but it also complicates their treatment. To prevent the initial loosening, we developed a new implant with an enlarged implant/bone interface based on the concept of perforated, hollow cylinders. We evaluated whether osseointegration of a hollow cylinder based implant takes place in normal or osteoporotic bone of sheep under functional loading conditions during anterior stabilization of the lumbar spine. MATERIALS AND METHODS: Osseointegration of the cylinders and status of the fused segments (ventral corpectomy, replacement with iliac strut, and fixation with testing implant) were investigated in six osteoporotic (age 6.9 +/- 0.8 years, mean body weight 61.1 +/- 5.2 kg) and seven control sheep (age 6.1 +/- 0.2 years, mean body weight 64.9 +/- 5.7 kg). Osteoporosis was introduced using a combination protocol of ovariectomy, high-dose prednisone, calcium and phosphor reduced diet and movement restriction. Osseointegration was quantified using fluorescence and conventional histology; fusion status was determined using biomechanical testing of the stabilized segment in a six-degree-of-freedom loading device as well as with radiological and histological staging. RESULTS: Intact bone trabeculae were found in 70% of all perforations without differences between the two groups (P = 0.26). Inside the cylinders, bone volume/total volume was significantly higher than in the control vertebra (50 +/- 16 vs. 28 +/- 13%) of the same animal (P<0.01), but significantly less (P<0.01) than in the near surrounding (60 +/- 21%). After biomechanical testing as described in Sect. "Materials and methods", seven spines (three healthy and four osteoporotic) were classified as completely fused and six (four healthy and two osteoporotic) as not fused after a 4-month observation time. All endplates were bridged with intact trabeculae in the histological slices. CONCLUSIONS: The high number of perforations, filled with intact trabeculae, indicates an adequate fixation; bridging trabeculae between adjacent endplates and tricortical iliac struts in all vertebrae indicates that the anchorage is adequate to promote fusion in this animal model, even in the osteoporotic sheep.