We consider three dimensional lossless periodic dielectric (photonic crystals) and acoustic media having a gap in the spectrum. If such a periodic medium is perturbed by a strong enough defect, defect eigenmodes arise, localized exponentially around the defect, with the corresponding eigenvalues in the gap. We use a modified Birman-Schwinger method to derive equations for these eigenmodes and c...

We develop a method for the exact determination of frequency responses for a class of infinite dimensional systems. In particular, we consider distributed systems in which a spatial independent variable belongs to a finite interval, and in which the inputs and outputs are spatially distributed over the same interval. We show that an explicit formula for the Hilbert–Schmidt norm of the operator-...

We study the Connes spectral distance of quantum states and analyse nonlocality a 4D generalized noncommutative phase space. By virtue Hilbert–Schmidt operatorial formulation, we obtain Dirac operator construct triple corresponding to Based on ball condition, some constraint relations about optimal elements, then calculate between two Fock states. Due noncommutativity, distances in spaces are s...

In this work, we introduce a theory of stochastic integration with respect to symmetric α-stable cylindrical Lévy processes. Since processes do not enjoy semi-martingale decomposition, our approach is based on decoupling inequality for the tangent sequence Radonified increments. This enables us characterise largest space predictable Hilbert-Schmidt operator-valued which are integrable an proces...

This article generalises the concept of realised covariation to Hilbert-space-valued stochastic processes. More precisely, based on high-frequency functional data, we construct an estimator trace-class operator-valued integrated volatility process arising in general mild solutions Hilbert space-valued evolution equations sense Da Prato and Zabczyk (2014). We prove a weak law large numbers for t...

1. Scheby-Buch.-Arch. f. Ophthal., p. 265, 1871. 2. Moll.-Centralbl. f. prakt Augenheilk, p. 2, 1896. 3. Weber.-Virchow's Arch., Vol. XXV, p. 114b, and Vol. XXVIII, p. 489, 1863. % 4. Schmidt-Rimpler.-Klin. Wochenschr., p. 92, 1884. Berlin. 5. Aubertin.-Rev. Gen. d'Ophtal., p. 280, 1933. 6. Lohmann.-Beticht iuber d. Vers. d. Ophthal. Gesellsch., p. 264, 1908. Heidelberg. 7. Stephenson.The Ophth...

We develop relative oscillation theory for one-dimensional Dirac operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type compariso...

In this paper we consider a class of weighted integral operators on L2(0,∞) and show that they are unitarily equivalent to little Hankel operators between weighted Bergman spaces of the right half plane. We use two parameters α, β ∈ (−1,∞) and involve two weights to define Bergman spaces of the domain and range of the little Hankel operators. We obtained conditions for the Hankel integral opera...

Although phase transition waves in atomic chains with double-well potential play a fundamental role in materials science, very little is known about their mathematical properties. In particular, the only available results about waves with large amplitudes concern chains with piecewisequadratic pair potential. In this paper we consider perturbations of a bi-quadratic potential and prove that the...

The Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form −d 2 /dx 2 + V (g; x), where the potential is an elliptic function depending on a coupling vector g ∈ R 4. Alternatively, this operator arises from the BC 1 specialization of the BC N elliptic nonrelativistic Calogero–Moser system (a.k.a. the Inozemtsev system). Under suitable restrict...