Criterion for polynomial solutions to a class of linear differential equation of second order

We consider the differential equations y''=\lambda_0(x)y'+s_0(x)y, where \lambda_0(x), s_0(x) are C^{\infty}-functions. We prove (i) if the differential equation, has a polynomial solution of degree n >0, then \delta_n=\lambda_n s_{n-1}-\lambda_{n-1}s_n=0, where \lambda_{n}= \lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1}\hbox{and}\quad s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1},\quad n=1,2,.... Conversely (ii) if \lambda_n\lambda_{n-1}\ne 0 and \delta_n=0, then the differential equation has a polynomial solution of degree at most n. We show that the classical differential equations of Laguerre, Hermite, Legendre, Jacobi, Chebyshev (first and second kind), Gegenbauer, and the Hypergeometric type, etc, obey this criterion. Further, we find the polynomial solutions for the generalized Hermite, Laguerre, Legendre and Chebyshev differential equations.Comment: 12 page

Physical applications of second-order linear differential equations that admit polynomial solutions

Conditions are given for the second-order linear differential equation P3 y" + P2 y'- P1 y = 0 to have polynomial solutions, where Pn is a polynomial of degree n. Several application of these results to Schroedinger's equation are discussed. Conditions under which the confluent, biconfluent, and the general Heun equation yield polynomial solutions are explicitly given. Some new classes of exactly solvable differential equation are also discussed. The results of this work are expressed in such way as to allow direct use, without preliminary analysis.Comment: 13 pages, no figure

Strain Gradients in Epitaxial Ferroelectrics

X-ray analysis of ferroelectric thin layers of Ba1/2Sr1/2TiO3 with different thickness reveals the presence of internal strain gradients across the film thickness and allows us to propose a functional form for the internal strain profile. We use this to calculate the direct influence of strain gradient, through flexoelectric coupling, on the degradation of the ferroelectric properties of thin films with decreasing thickness, in excellent agreement with the observed behaviour. This work highlights the link between strain relaxation and strain gradients in epitaxial films, and shows the pressing need to avoid strain gradients in order to obtain thin ferroelectrics with bulk-like properties.Comment: 4 pages, 3 embedded figures (1 color), revTex

Eigenvalue bounds for polynomial central potentials in d dimensions

If a single particle obeys non-relativistic QM in R^d and has the Hamiltonian H = - Delta + f(r), where f(r)=sum_{i = 1}^{k}a_ir^{q_i}, 2\leq q_i < q_{i+1}, a_i \geq 0$, then the eigenvalues E = E_{n\ell}^{(d)}(\lambda) are given approximately by the semi-classical expression E = \min_{r > 0}[\frac{1}{r^2} + \sum_{i = 1}^{k}a_i(P_ir)^{q_i}]. It is proved that this formula yields a lower bound if P_i = P_{n\ell}^{(d)}(q_1), an upper bound if$P_i = P_{n\ell}^{(d)}(q_k) and a general approximation formula if P_i = P_{n\ell}^{(d)}(q_i). For the quantum anharmonic oscillator f(r)=r^2+\lambda r^{2m},m=2,3,... in d dimension, for example, E = E_{n\ell}^{(d)}(\lambda) is determined by the algebraic expression \lambda={1\over \beta}({2\alpha(m-1)\over mE-\delta})^m({4\alpha \over (mE-\delta)}-{E\over (m-1)}) where \delta={\sqrt{E^2m^2-4\alpha(m^2-1)}} and \alpha, \beta are constants. An improved lower bound to the lowest eigenvalue in each angular-momentum subspace is also provided. A comparison with the recent results of Bhattacharya et al (Phys. Lett. A, 244 (1998) 9) and Dasgupta et al (J. Phys. A: Math. Theor., 40 (2007) 773) is discussed.Comment: 13 pages, no figure

Green's function for a Schroedinger operator and some related summation formulas

Summation formulas are obtained for products of associated Lagurre polynomials by means of the Green's function K for the Hamiltonian H = -{d^2\over dx^2} + x^2 + Ax^{-2}, A > 0. K is constructed by an application of a Mercer type theorem that arises in connection with integral equations. The new approach introduced in this paper may be useful for the construction of wider classes of generating function.Comment: 14 page

Preconditioned Bi-Conjugate Gradient Method for Radiative Transfer in Spherical Media

A robust numerical method called the Preconditioned Bi-Conjugate Gradient (Pre-BiCG)method is proposed for the solution of radiative transfer equation in spherical geometry.A variant of this method called Stabilized Preconditioned Bi-Conjugate Gradient (Pre-BiCG-STAB) is also presented. These are iterative methods based on the construction of a set of bi-orthogonal vectors. The application of Pre-BiCG method in some benchmark tests show that the method is quite versatile, and can handle hard problems that may arise in astrophysical radiative transfer theory.Comment: 19 pages, 12 figure

On composite systems of dilute and dense couplings

Composite systems, where couplings are of two types, a combination of strong dilute and weak dense couplings of Ising spins, are examined through the replica method. The dilute and dense parts are considered to have independent canonical disordered or uniform bond distributions; mixing the models by variation of a parameter $\gamma$ alongside inverse temperature $\beta$ we analyse the respective thermodynamic solutions. We describe the variation in high temperature transitions as mixing occurs; in the vicinity of these transitions we exactly analyse the competing effects of the dense and sparse models. By using the replica symmetric ansatz and population dynamics we described the low temperature behaviour of mixed systems.Comment: 35 pages, 9 figures, submitted to JPhys

Construction of exact solutions to eigenvalue problems by the asymptotic iteration method

We apply the asymptotic iteration method (AIM) [J. Phys. A: Math. Gen. 36, 11807 (2003)] to solve new classes of second-order homogeneous linear differential equation. In particular, solutions are found for a general class of eigenvalue problems which includes Schroedinger problems with Coulomb, harmonic oscillator, or Poeschl-Teller potentials, as well as the special eigenproblems studied recently by Bender et al [J. Phys. A: Math. Gen. 34 9835 (2001)] and generalized in the present paper to higher dimensions.Comment: 10 page

Diffusion Tensor Imaging of Dolphin Brains Reveals Direct Auditory Pathway to Temporal Lobe

The brains of odontocetes (toothed whales) look grossly different from their terrestrial relatives. Because of their adaptation to the aquatic environment and their reliance on echolocation, the odontocetes’ auditory system is both unique and crucial to their survival. Yet, scant data exist about the functional organization of the cetacean auditory system. A predominant hypothesis is that the primary auditory cortex lies in the suprasylvian gyrus along the vertex of the hemispheres, with this position induced by expansion of ‘associative0 regions in lateral and caudal directions. However, the precise location of the auditory cortex and its connections are still unknown. Here, we used a novel diffusion tensor imaging (DTI) sequence in archival post-mortem brains of a common dolphin (Delphinus delphis) and a pantropical dolphin (Stenella attenuata) to map their sensory and motor systems. Using thalamic parcellation based on traditionally defined regions for the primary visual (V1) and auditory cortex (A1), we found distinct regions of the thalamus connected to V1 and A1. But in addition to suprasylvian-A1, we report here, for the first time, the auditory cortex also exists in the temporal lobe, in a region near cetacean-A2 and possibly analogous to the primary auditory cortex in related terrestrial mammals (Artiodactyla). Using probabilistic tract tracing, we found a direct pathway from the inferior colliculus to the medial geniculate nucleus to the temporal lobe near the sylvian fissure. Our results demonstrate the feasibility of postmortem DTI in archival specimens to answer basic questions in comparative neurobiology in a way that has not previously been possible and shows a link between the cetacean auditory system and those of terrestrial mammals. Given that fresh cetacean specimens are relatively rare, the ability to measure connectivity in archival specimens opens up a plethora of possibilities for investigating neuroanatomy in cetaceans and other species