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

Spectral characteristics for a spherically confined -1/r + br^2 potential

We consider the analytical properties of the eigenspectrum generated by a class of central potentials given by V(r) = -a/r + br^2, b>0. In particular, scaling, monotonicity, and energy bounds are discussed. The potential $V(r)$ is considered both in all space, and under the condition of spherical confinement inside an impenetrable spherical boundary of radius R. With the aid of the asymptotic iteration method, several exact analytic results are obtained which exhibit the parametric dependence of energy on a, b, and R, under certain constraints. More general spectral characteristics are identified by use of a combination of analytical properties and accurate numerical calculations of the energies, obtained by both the generalized pseudo-spectral method, and the asymptotic iteration method. The experimental significance of the results for both the free and confined potential V(r) cases are discussed.Comment: 16 pages, 4 figure

Energies and wave functions for a soft-core Coulomb potential

For the family of model soft Coulomb potentials represented by V(r) = -\frac{Z}{(r^q+\beta^q)^{\frac{1}{q}}}, with the parameters Z>0, \beta>0, q \ge 1, it is shown analytically that the potentials and eigenvalues, E_{\nu\ell}, are monotonic in each parameter. The potential envelope method is applied to obtain approximate analytic estimates in terms of the known exact spectra for pure power potentials. For the case q =1, the Asymptotic Iteration Method is used to find exact analytic results for the eigenvalues E_{\nu\ell} and corresponding wave functions, expressed in terms of Z and \beta. A proof is presented establishing the general concavity of the scaled electron density near the nucleus resulting from the truncated potentials for all q. Based on an analysis of extensive numerical calculations, it is conjectured that the crossing between the pair of states [(\nu,\ell),(\nu',\ell')], is given by the condition \nu'\geq (\nu+1) and \ell' \geq (\ell+3). The significance of these results for the interaction of an intense laser field with an atom is pointed out. Differences in the observed level-crossing effects between the soft potentials and the hydrogen atom confined inside an impenetrable sphere are discussed.Comment: 13 pages, 5 figures, title change, minor revision

Solutions for certain classes of Riccati differential equation

We derive some analytic closed-form solutions for a class of Riccati equation y'(x)-\lambda_0(x)y(x)\pm y^2(x)=\pm s_0(x), where \lambda_0(x), s_0(x) are C^{\infty}-functions. We show that if \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} and s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1}, n=1,2,..., then The Riccati equation has a solution given by y(x)=\mp s_{n-1}(x)/\lambda_{n-1}(x). Extension to the generalized Riccati equation y'(x)+P(x)y(x)+Q(x)y^2(x)=R(x) is also investigated.Comment: 10 page

Closed-form sums for some perturbation series involving associated Laguerre polynomials

Infinite series sum_{n=1}^infty {(alpha/2)_n / (n n!)}_1F_1(-n, gamma, x^2), where_1F_1(-n, gamma, x^2)={n!_(gamma)_n}L_n^(gamma-1)(x^2), appear in the first-order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator Hamiltonian H = -d^2/dx^2 + B x^2 + A/x^2 + lambda/x^alpha 0 0, A >= 0. It is proved that the series is convergent for all x > 0 and 2 gamma > alpha, where gamma = 1 + (1/2)sqrt(1+4A). Closed-form sums are presented for these series for the cases alpha = 2, 4, and 6. A general formula for finding the sum for alpha/2 = 2 + m, m = 0,1,2, ..., in terms of associated Laguerre polynomials, is also provided.Comment: 16 page

Short-term treatment with tolfenamic acid improves cognitive functions in Alzheimer\u27s disease mice

Tolfenamic acid lowers the levels of the amyloid precursor protein (APP) and amyloid beta (Aβ) when administered to C57BL/6 mice by lowering their transcriptional regulator specificity protein 1 (SP1). To determine whether changes upstream in the amyloidogenic pathway that forms Aβ plaques would improve cognitive outcomes, we administered tolfenamic acid for 34 days to hemizygous R1.40 transgenic mice. After the characterization of cognitive deficits in these mice, assessment of spatial learning and memory functions revealed that treatment with tolfenamic acid attenuated long-term memory and working memory deficits, determined using Morris water maze and the Y-maze. These improvements occurred within a shorter period of exposure than that seen with clinically approved drugs. Cognitive enhancement was accompanied by reduction in the levels of the SP1 protein (but not messenger RNA [mRNA]), followed by lowering both the mRNA and the protein levels of APP and subsequent Aβ levels. These findings provide evidence that tolfenamic acid can disrupt the pathologic processes associated with Alzheimer\u27s disease (AD) and are relevant to its scheduled biomarker study in AD patients

Synthesis of high-T_g hole-transporting polymers with different redox potentials and their performance in organic two-layer LEDs

Organic hole transport materials are used in organic LEDs, where they substantially improve device performance if placed as a hole transport layer (HTL) between the anode and the electroluminescent layer (EL). Soluble polymeric hole transport materials with high glass transition temperatures are of particular interest, because they allow for efficient device fabrication through spin casting of the HTL, and high glass transition temperatures have been found to improve thermal and long-term stability of the device. The redox potential of the hole transport material determines the facility of charge injection at the anode/HTL and the HTL/EL interfaces, thus affecting the overall device efficiency. We have synthesized a series of soluble hole-transporting polymers with glass transition temperatures in the range of 130 degrees C to 150 degrees C. The synthetic method allows facile substitution of the hole transport functionality with electron-withdrawing and electron-donating groups, which permits tuning of the redox potential of the polymer. These polymers have been used as HTL in tow-layer devices ITO/HTL/Alq/Mg. The maximum external quantum efficiency increase, if the redox potential is changed to facilitate reduction of the hole transport material at the HTL/EL interface. Electron-deficient derivatives show higher external quantum efficiencies. The device stability, however, follows the opposite trend

Solvable Systems of Linear Differential Equations

The asymptotic iteration method (AIM) is an iterative technique used to find exact and approximate solutions to second-order linear differential equations. In this work, we employed AIM to solve systems of two first-order linear differential equations. The termination criteria of AIM will be re-examined and the whole theory is re-worked in order to fit this new application. As a result of our investigation, an interesting connection between the solution of linear systems and the solution of Riccati equations is established. Further, new classes of exactly solvable systems of linear differential equations with variable coefficients are obtained. The method discussed allow to construct many solvable classes through a simple procedure.Comment: 13 page

Three-arm, randomized, phase 2 study of carboplatin and paclitaxel in combination with cetuximab, cixutumumab, or both for advanced non-small cell lung cancer (NSCLC) patients who will not receive bevacizumab-based therapy: An Eastern Cooperative Oncology Group (ECOG) study (E4508)

BACKGROUND: Preclinical evidence supports the clinical investigation of inhibitors to the insulin-like growth factor receptor (IGFR) and the epidermal growth factor receptor (EGFR) either alone or in combination as treatment for patients with non-small cell lung cancer (NSCLC). METHODS: Patients with chemotherapy-naïve, advanced NSCLC who had an Eastern Cooperative Oncology Group performance status of 0 or 1 were eligible. Patients were randomized to receive carboplatin intravenously at an area under the plasma drug concentration-time curve of 6.0 plus paclitaxel 200 mg/m(2) intravenously on day 1 every 3 weeks combined with either intravenous cetuximab weekly (arm A), intravenous cixutumumab every 2 weeks (arm B), or both (arm C). Patients who had nonprogessing disease after 12 weeks of therapy were permitted to continue on maintenance antibody therapy until they developed progressive disease. The primary endpoint was progression-free survival (PFS). The study design required 180 eligible patients and had 88% power to detect a 60% increase in median PFS for either comparison (arm A vs arm C or arm B vs arm C) using the log-rank test. RESULTS: From September 2009 to December 2010, 140 patients were accrued. The study was closed to accrual early because of an excessive number of grade 5 events reported on arms A and C. Thirteen patients died during treatment (6 patients on arm A, 2 patients on arm B, and 5 patients on arm C), including 9 within approximately 1 month of starting therapy. The estimated median PFS for arms A, B, and C were similar at 3.4 months, 4.2 months, and 4 months, respectively. CONCLUSIONS: On the basis of the apparent lack of efficacy and excessive premature deaths, the current results do not support the continued investigation of carboplatin, paclitaxel, and cixutumumab either alone or in combination with cetuximab for patients with advanced NSCLC