Asymptotic Analysis

Let H_β^±:=−Δ±βδ(|x|=R), be the quantum mechanical Hamiltonian describing a delta interaction living on a sphere of radius R in three dimensions, with strength β>0. Let D_β^±:=(−Δ+1)⁻¹−(H_β^±+1)⁻¹ and D_∞ be the strong limit w.r.t. β of the operators D_β⁺. For suitable sequence (β_n) we prove uniform convergence of D_βn⁻ towards D_∞ with optimal rate, namely β_n⁻¹. Convergence of the above mentioned operators against each other within Schatten–von Neumann ideals S_p, is established as well and the rate of convergence (depending on p) is determined. At the final step we discuss aspects of differences between negative and positive perturbations at large coupling.

We study the weighted averages of resonance clusters for the hydrogen atom with a Stark electric field in the weak field limit. We prove a semiclassical Szegö-type theorem for resonance clusters showing that the limiting distribution of the resonance shifts concentrates on the classical energy surface corresponding to a rescaled eigenvalue of the hydrogen atom Hamiltonian. This result extends Szegö-type results on eigenvalue clusters to resonance clusters. There are two new features in this work: first, the study of resonance clusters requires the use of non-self-adjoint operators, and second, the Stark perturbation is unbounded so control of the perturbation is achieved using localization properties of coherent states corresponding to hydrogen atom eigenvalues.

We consider a financial market with liquidity cost as in [Çetin, Jarrow and Protter, Finance and Stochastics 8 (2004), 311–341], where the supply function S^ε(s,ν) depends on a parameter ε≥0 with S⁰(s,ν)=s corresponding to the perfect liquid situation. Using the PDE characterization of Çetin, Soner and Touzi [Finance and Stochastics 14(3) (2010), 317–341], of the super-hedging cost of an option written on such a stock, we provide a Taylor expansion of the super-hedging cost in powers of ε. In particular, we explicitly compute the first term in the expansion for a European Call option and give bounds for the order of the expansion for a European Digital Option.

We consider a nearly-elastic model system with one degree of freedom. In each collision with the “wall”, the system can either lose or gain a small amount of energy due to stochastic perturbation. The weak limit of the corresponding slow motion, which is a stochastic process on a graph, is calculated. A large deviation type asymptotics and the metastability of the system are also considered.

In this paper, we study the theory of orthogonal trigonometric polynomials (OTPs). We obtain asymptotics of OTPs with positive and analytic weight functions by Riemann–Hilbert approach and find that they have relations with orthogonal polynomials on the unit circle (OPUC). By the relations and the theory of OPUC, we also get four-terms recurrent formulae, Christoffel–Darboux formula and some algebraic and asymptotic properties of zeros for orthogonal trigonometric polynomials.

We prove polynomial and exponential decay at infinity of eigen-vectors of partial differential operators related to radiation problems for time-harmonic generalized Maxwell systems in an exterior domain Ω⊂R^N, N≥1, with non-smooth inhomogeneous, anisotropic coefficients converging near infinity with a rate r^−τ, τ>1, towards the identity. As a canonical application we show that the corresponding eigen-values do not accumulate and that by means of Eidus' limiting absorption principle a Fredholm alternative holds true.

We obtain, by exploiting Jost functions, asymptotic expansions of Green functions of perturbed Hill operators with respect to the spectral parameter around the minima of the continuous spectra. From the Green function expansions, we derive asymptotic expansions for large time of heat kernels for perturbed Hill operators.