Uniqueness of the potential function for the vectorial Sturm-Liouville equation on a finite interval

[[abstract]]In this paper, the vectorial Sturm-Liouville operator L Q =−d 2 dx 2 +Q(x) is considered, where Q(x) is an integrable m×m matrix-valued function defined on the interval [0,π] . The authors prove that m 2 +1 characteristic functions can determine the potential function of a vectorial Sturm-Liouville operator uniquely. In particular, if Q(x) is real symmetric, then m(m+1) 2 +1 characteristic functions can determine the potential function uniquely. Moreover, if only the spectral data of self-adjoint problems are considered, then m 2 +1 spectral data can determine Q(x) uniquely.[[notice]]補正完畢[[incitationindex]]SCI[[cooperationtype]]國外[[booktype]]電子

Transmutations and spectral parameter power series in eigenvalue problems

We give an overview of recent developments in Sturm-Liouville theory concerning operators of transmutation (transformation) and spectral parameter power series (SPPS). The possibility to write down the dispersion (characteristic) equations corresponding to a variety of spectral problems related to Sturm-Liouville equations in an analytic form is an attractive feature of the SPPS method. It is based on a computation of certain systems of recursive integrals. Considered as families of functions these systems are complete in the $L_{2}$-space and result to be the images of the nonnegative integer powers of the independent variable under the action of a corresponding transmutation operator. This recently revealed property of the Delsarte transmutations opens the way to apply the transmutation operator even when its integral kernel is unknown and gives the possibility to obtain further interesting properties concerning the Darboux transformed Schr\"{o}dinger operators. We introduce the systems of recursive integrals and the SPPS approach, explain some of its applications to spectral problems with numerical illustrations, give the definition and basic properties of transmutation operators, introduce a parametrized family of transmutation operators, study their mapping properties and construct the transmutation operators for Darboux transformed Schr\"{o}dinger operators.Comment: 30 pages, 4 figures. arXiv admin note: text overlap with arXiv:1111.444

One-Dimensional and Multi-Dimensional Integral Transforms of Buschman–Erdélyi Type with Legendre Functions in Kernels

This paper consists of two parts. In the first part we give a brief survey of results on Buschman–Erdélyi operators, which are transmutations for the Bessel singular operator. Main properties and applications of Buschman–Erdélyi operators are outlined. In the second part of the paper we consider multi-dimensional integral transforms of Buschman–Erdélyi type with Legendre functions in kernels. Complete proofs are given in this part, main tools are based on Mellin transform properties and usage of Fox H-functions

Responses of marine benthic microalgae to elevated CO<inf>2</inf>

Increasing anthropogenic CO2 emissions to the atmosphere are causing a rise in pCO2 concentrations in the ocean surface and lowering pH. To predict the effects of these changes, we need to improve our understanding of the responses of marine primary producers since these drive biogeochemical cycles and profoundly affect the structure and function of benthic habitats. The effects of increasing CO2 levels on the colonisation of artificial substrata by microalgal assemblages (periphyton) were examined across a CO2 gradient off the volcanic island of Vulcano (NE Sicily). We show that periphyton communities altered significantly as CO2 concentrations increased. CO2 enrichment caused significant increases in chlorophyll a concentrations and in diatom abundance although we did not detect any changes in cyanobacteria. SEM analysis revealed major shifts in diatom assemblage composition as CO2 levels increased. The responses of benthic microalgae to rising anthropogenic CO2 emissions are likely to have significant ecological ramifications for coastal systems. © 2011 Springer-Verlag

Nonlocal supersymmetric deformations of periodic potentials

Irreducible second-order Darboux transformations are applied to the periodic Schrodinger's operators. It is shown that for the pairs of factorization energies inside of the same forbidden band they can create new non-singular potentials with periodicity defects and bound states embedded into the spectral gaps. The method is applied to the Lame and periodic piece-wise transparent potentials. An interesting phenomenon of translational Darboux invariance reveals nonlocal aspects of the supersymmetric deformations.Comment: 15 pages, latex, 9 postscript figure