A second eigenvalue bound for the Dirichlet Schroedinger operator

Let $\lambda_i(\Omega,V)$ be the $i$th eigenvalue of the Schr\"odinger operator with Dirichlet boundary conditions on a bounded domain $\Omega \subset \R^n$ and with the positive potential $V$. Following the spirit of the Payne-P\'olya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential $V_\star$, we prove that $\lambda_2(\Omega,V) \le \lambda_2(S_1,V_\star)$. Here $S_1$ denotes the ball, centered at the origin, that satisfies the condition $\lambda_1(\Omega,V) = \lambda_1(S_1,V_\star)$. Further we prove under the same convexity assumptions on a spherically symmetric potential $V$, that $\lambda_2(B_R, V) / \lambda_1(B_R, V)$ decreases when the radius $R$ of the ball $B_R$ increases. We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density

Thermoanalytical studies of carbamazepine: hydration/dehydration, thermal decomposition, and solid phase transitions

Carbamazepine (CBZ), a widely used anticonvulsant drug, can crystallize and exhibits four polymorphic forms and one dihydrate. Anhydrous CBZ can spontaneously absorb water and convert to the hydrate form whose different crystallinity leads to lower biological activity. The present study was concerned to the possibility of recovering the hydrated form by heating. The thermal behavior of spontaneously hydrated carbamazepine was investigated by TG/DTG-DTA and DSC in dynamic atmospheres of air and nitrogen, which revealed that the spontaneous hydration of this pharmaceutical resulted in a Form III hydrate with 1.5 water molecules. After dehydration, this anhydrous Form III converted to Form I, which melted and decomposed in a single event, releasing isocyanic acid, as shown by evolved gas analysis using TG-FTIR. Differential scanning calorimetry analyses revealed that Form III melted and crystallized as Form I, and that subsequent cooling cycles only generated Form I by crystallization. Solid state decomposition kinetic studies showed that there was no change in the substance after the elimination of water by heating to 120 °C. Activation energies of 98 ± 2 and 93 ± 2 kJ mol-1 were found for the hydrated and dried samples, respectively, and similar profiles of activation energy as a function of conversion factor were observed for these samples