Self-similar functions in $ L_2\lbrack0,1\rbrack$ and the Sturm-Liouville problem with a singular indefinite weight

Abstract

A new experimental method is described which allows the range of scattering angles to be extended to cover the whole of the backward hemisphere, i.e. up to 180 • , in measurements of elastic electron scattering. This is achieved by adding a localized magnetic field to the interaction region of a conventional electrostatic electron spectrometer. Measurements are presented of the differential cross section from 30-180 • for elastic scattering of electrons by argon at an energy of 10 eV, and of the resonance structures due to the 2 P 3/2,1/2 states of Ar − at the fixed scattering angles of 110 • , 150 • and 180 • .

Measure theoretic trigonometric functions

“…2.5]. Moreover, we have a pure point spectrum not only in the non-atomic case, see Vladimirov and Sheipak [29].…”

Degenerate self-similar measures, spectral asymptotics and small deviations of Gaussian processes

“…The construction of a self-similar measure on interval described in [33] (see also [19]) enables one to construct measures with positive Hausdorff dimension of support. Let us note that the primitive of such measure is always a continuous function, which is self-similar in the sense of [36,37]. On the other hand, a function f , self-similar in the mentioned sense, need not be continuous (the criteria of its continuity are established in [29,Section 3]).…”

“…This function is conjectured to be non-constant in all non-trivial cases, but this problem is still open. Only in the simplest case of 'Cantor ladder', this conjecture was proved in [35,36].…”

“…The results of [11,33] were generalized later in two directions: in [35][36][37], the more general (non-sign-definite) weight functions were considered whereas in [19] the Green functions of ordinary differential operators of arbitrary order were examined. The logarithmic asymptotics were obtained in [19] for corresponding processes as well.…”

Sturm-Liouville operators with measure-valued coefficients

“…This connection will be discussed in detail and exploited in a forthcoming paper [16]. Finally, the case where the weight coefficient is a measure is known as Krein string and has also attracted considerable interest recently [44]- [46] (see also the monograph [15]).…”