Rectangular Well as Perturbation

We discuss a finite rectangular well as a perturbation for the infinite one with a depth $\lambda^2$ of the former as a perturbation parameter. In particular we consider a behaviour of energy levels in the well as functions of complex $\lambda$. It is found that all the levels of the same parity are defined on infinitely sheeted Riemann surfaces which topological structures are described in details. These structures differ considerably from those found in models investigated earlier. It is shown that perturbation series for all the levels converge what is in contrast with the known results of Bender and Wu. The last property is shown to hold also for the finite rectangular well with Dirac delta barier as a perturbation considered earlier by Ushveridze.Comment: 19 pages, 5 Postscript figures, uses psfig.st

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

Fundamental solution method applied to time evolution of two energy level systems: exact and adiabatic limit results

A method of fundamental solutions has been used to investigate transitions in two energy level systems with no level crossing in a real time. Compact formulas for transition probabilities have been found in their exact form as well as in their adiabatic limit. No interference effects resulting from many level complex crossings as announced by Joye, Mileti and Pfister (Phys. Rev. {\bf A44} 4280 (1991)) have been detected in either case. It is argued that these results of this work are incorrect. However, some effects of Berry's phases are confirmed.Comment: LaTeX2e, 23 pages, 8 EPS figures. Style correcte

Multiple benefits of manure: the key to maintenance of soil fertility and restoration of depleted sandy soils on African smallholder farms

Manure is a key nutrient resource on smallholder farms in the tropics, especially on poorly buffered sandy soils, due to its multiple benefits for soil fertility. Farmers preferentially apply manure to fields closest to homesteads (homefields), which are more fertile than fields further away (outfields). A three-year experiment was established on homefields and outfields on sandy and clayey soils to assess the effects of mineral nitrogen (N) fertilizer application in combination with manure or mineral phosphorus (P) on maize yields and soil chemical properties. Significant maize responses to application of N and manure were observed on all fields except the depleted sandy outfield. Large amounts of manure (17 t ha¿1 year¿1) were required to significantly increase soil organic carbon (SOC), pH, available P, and base saturation, and restore productivity of the depleted sandy outfield. Sole N as ammonium nitrate (100 kg N ha¿1) or in combination with single superphosphate led to acidification of the sandy soils, with a decrease of up to 0.8 pH units after three seasons. In a greenhouse experiment, N and calcium (Ca) were identified as deficient in the sandy homefield, while N, P, Ca, and zinc (Zn) were deficient or low on the sandy outfield. The deficiencies of Ca and Zn were alleviated by the addition of manure. This study highlights the essential role of manure in sustaining and replenishing soil fertility on smallholder farms through its multiple effects, although it should be used in combination with N mineral fertilizers due to its low capacity to supply N

The Quantum Galilei Group

The quantum Galilei group $G_{\varkappa}$ is defined. The bicrossproduct structure of $G_{\varkappa}$ and the corresponding Lie algebra is revealed. The projective representations for the two-dimensional quantum Galilei group are constructed.Comment: AMSTe

Irreducible Representations of an Algebra underlying Hidden Symmetries of a class of Quasi Exactly Solvable Systems of Equations

The set of linear, differential operators preserving the vector space of couples of polynomials of degrees n and n-2 in one real variable leads to an abstract associative graded algebra A(2). The irreducible, finite dimensional representations of this algebra are classified into five infinite discrete sets and one exceptional case. Their matrix elements are given explicitely. The results are related to the theory of quasi exactly solvable equations.Comment: 38 pages, late

Fractal properties of quantum spacetime

We show that in general a spacetime having a quantum group symmetry has also a scale dependent fractal dimension which deviates from its classical value at short scales, a phenomenon that resembles what observed in some approaches to quantum gravity. In particular we analyze the cases of a quantum sphere and of \k-Minkowski, the latter being relevant in the context of quantum gravity.Comment: 4 pages, 2 figures; some minor corrections; reference adde

Quadratic integrals of motion for the systems of identical particles

The dynamical systems of identical particles admitting quadratic integrals of motion are classified. The relevant integrals are explicitly constructed and their relation to separation of variables in H-J equation is clarified.Comment: 6 pages, no figure