The Scattering Problem for a Discrete Sturm-Liouville Operator

Abstract

The previous work on the dosimetry of bone is briefly reviewed. A dosimetric theory for the response of detectors irradiated by fast neutrons is applied to the problem of bone dosimetry. In the theory the detector or cavity shape is characterised by distributions of chord lengths along which the neutron-produced charged particles travel and deposit energy. Cavities of different convex geometries can be treated. A simplified version of the theory uses a single mean chord length to characterise the cavity. The absorbed dose to individual marrow cavities in trabecular bone is calculated over a large range of marrow cavity size for monoenergetic neutrons ranging from 0.5 to 7.0 MeV and for 252Cf neutrons. The influence of cavity shape is explored by considering spheres and cylinders of different elongation. The difference in absorbed dose is not great. Also the simplified model using a single mean chord length gives results in close agreement with the results obtained with chord length distributions. The mean marrow dose to different human bones has been calculated in three ways. First by using measured chord length distributions for the marrow cavities in the bones, second by using a sphere with the same mean chord length as the measured distribution and third by applying the measured single mean chord length. The difference between the three approaches is small and the agreement is geod with results obtained by other workers who used the Monte Carlo technique. The dose to the endosteal cell layer has also been calculated by approximating the layer with an infinite slab. Dosimetry of bone irradiated by fast neutrons 957

Criterion for the Resolvent Set of Nonsymmetric Tridiagonal Operators

“…Proof and similarly for ßkN+i and ykx+i ■ Then it is known that the poles of the Padé approximants n" = Pn/Qn are essentially concentrated on a system of N closed intervals ( [4], [7]). The convergence of n" however depends on the behaviour of the so-called spurious poles.…”

“…This connection between the spectral theory and analysis is fruitful for various points of view. For example, the scattering problem for a Jacobi matrix can be treated on the basis of strong (or Szegö type) asymptotic results for orthogonal polynomials ( [4], [6], [10]). On the other hand, the perturbation theory gives new results for orthogonal polynomials ( [12]).…”

Orthogonal polynomials

“…In the last 20 years the fundamentals of matrix orthogonal polynomials have been developed mainly by A. Durán and his coauthors (see also the work [9] by A. I. Aptekarev and E. M. Nikishin). The theory shows many similarities with the scalar case, but there is an unexpected richness which is still to be explored.…”

Relations between the orthogonal matrix polynomials on [a, b], Dyukarev-Stieltjes parameters, and Schur complements

“…See Remark 3.2. Matrix continued fractions were studied in [1], [39], [42], [43], [18] and the references therein.…”