The high energy semiclassical asymptotics of loci of roots of fundamental solutions for polynomial potentials

In the case of polynomial potentials all solutions to 1-D Schroedinger equation are entire functions totally determined by loci of their roots and their behaviour at infinity. In this paper a description of the first of the two properties is given for fundamental solutions for the high complex energy limit when the energy is quantized or not. In particular due to the fact that the limit considered is semiclassical it is shown that loci of roots of fundamental solutions are collected of selected Stokes lines (called exceptional) specific for the solution considered and are distributed along these lines in a specific way. A stable asymptotic limit of loci of zeros of fundamental solutions on their exceptional Stokes lines has island forms and there are infintely many of such roots islands on exceptional Stokes lines escaping to infinity and a finite number of them on exceptional Stokes lines which connect pairs of turning points. The results obtained for asymptotic roots distributions of fundamental solutions in the semiclassical high (complex) energy limit are of a general nature for polynomial potentials.Comment: 41 pages, 14 figure

International Journal of Farming and Allied Sciences Effects of Seed ageing on Seedling emergence of Borage (Borago officinalis L.) Medicinal Plant

ABSTRACT: Borage (Borago officinalis L.), is an annual herbaceous plant and native to Europe, North Africa, and Asia Minor . Reduced seedling growth is a consequence of seed deterioration.. It is not clear which of these component (s) is affected by seed deterioration. To study this subject, germination characteristics and seedling growth from deteriorated Borage (Borago officinalis L.) seeds were studied; seeds were incubated at two different temperatures of 30 ºC and 42 ºC for varying times. The results indicated that seed deterioration results in decreased percentage and rate of germination and decreased percentage of normal seedlings. Seedling growth and the fraction of seed reserve mobilization indicated a significant decrease with the advance of deterioration. However, the effect of seed deterioration on the conversion efficiency of mobilized reserves to seedling tissues was not significant. Thus efforts to improve seedling germination and growth in plant breeding programs should focus on improvement of fraction of mobilized seed reserves