We develop a specification test for the transition density of a discretely-sampled continuous-time diffusion process, based on a comparison of a nonparametric estimate of the transition density or distribution function to their corresponding parametric counterparts assumed by the null hypothesis. We provide a direct comparison of the two densities for an arbitrary specification of the null parametric model using three different discrepancy measures between the null and alternative transition density and distribution functions. We establish the asymptotic null distributions of proposed test statistics and compute their power functions. The finite sample properties are critically investigated via simulation studies and are compared with other tests. Our approaches are illustrated by applications to implied volatility data and Treasury bill data.

We analyze two-dimensional Schrdinger operators with the potential |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} where |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} and |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} which exhibit an abrupt change of spectral properties at a critical value of the coupling constant . We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for below the critical value the spectrum is purely discrete and we establish a LiebThirring-type bound on its moments. In the critical case where the essential spectrum covers the positive halfline while the negative spectrum can only be discrete, we demonstrate numerically the existence of a ground-state eigenvalue.

Graph consists of a set of vertices V={Xj: j I}, a set of finite edges L={Ljn:(Xj, Xn) IL I I} and a set of infinite edges L={Lj: Xj IC} attached to them. We regard it as a configuration space of a quantum system with the Hilbert space

This paper is devoted to the study of orthogonality and disjointness preserving linear maps between Fourier and FourierStieltjes algebras of locally compact groups. We show that a linear bijection : A (G 1) A (G 2)(resp. : B (G 1) B (G 2)) between two Fourier algebras (resp. FourierStieltjes algebras) of locally compact groups will induce a topological group isomorphism between G 1 and G 2, provided that preserves both disjointness and some kind of orthogonality. This improves earlier results of JJ Font and MS Monfared, where amenability of the groups or continuity of the linear maps are assumed. We also study the structure of bounded and unbounded disjointness preserving linear functionals of Fourier algebras. In the development, general results about disjointness and orthogonality preserving linear maps between C-algebras, W-algebras and their preduals are obtained.

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