Academic literature on the topic 'Microlocal spectrum condition'

Some years ago, Radzikowski has found a characterization of Hadamard states for scalar quantum fields on a four-dimensional globally hyperbolic spacetime in terms of a specific form of the wavefront set of their two-point functions (termed "wavefront set spectrum condition"), thereby initiating a major progress in the understanding of Hadamard states and the further development of quantum field theory in curved spacetime. In the present work, we extend this important result on the equivalence of the wavefront set spectrum condition with the Hadamard condition from scalar fields to vector fields (sections in a vector bundle) which are subject to a wave-equation and are quantized so as to fulfill the covariant canonical commutation relations, or which obey a Dirac equation and are quantized according to the covariant anti-commutation relations, in any globally hyperbolic spacetime having dimension three or higher. In proving this result, a gap which is present in the published proof for the scalar field case will be removed. Moreover we determine the short-distance saling limits of Hadamard states for vector-bundle valued fields, finding them to coincide with the corresponding flat-space, massless vacuum states.

We develop a new scheme of proofs for spectral theory of the [Formula: see text]-body Schrödinger operators, reproducing and extending a series of sharp results under minimum conditions. Our main results include Rellich’s theorem, limiting absorption principle bounds, microlocal resolvent bounds, Hölder continuity of the resolvent and a microlocal Sommerfeld uniqueness result. We present a new proof of Rellich’s theorem which is unified with exponential decay estimates studied previously only for [Formula: see text]-eigenfunctions. Each pair-potential is a sum of a long-range term with first-order derivatives, a short-range term without derivatives and a singular term of operator- or form-bounded type, and the setup includes hard-core interaction. Our proofs consist of a systematic use of commutators with ‘zeroth order’ operators. In particular, they do not rely on Mourre’s differential inequality technique.

AbstractFefferman (Acta Math 24:9–36, 1970, Theorem 2$$'$$ ′ ) has proved the weak (1,1) boundedness for a class of oscillating singular integrals that includes the oscillating spectral multipliers of the Euclidean Laplacian $$\Delta ,$$ Δ , namely, operators of the form $$\begin{aligned} T_{\theta }(-\Delta ):= (1-\Delta )^{-\frac{n\theta }{4}}e^{i (1-\Delta )^{\frac{\theta }{2}}},\,0\le \theta <1. \end{aligned}$$ T θ ( - Δ ) : = ( 1 - Δ ) - n θ 4 e i ( 1 - Δ ) θ 2 , 0 ≤ θ < 1 . The aim of this work is to extend Fefferman’s result to oscillating singular integrals on any arbitrary compact Lie group. We also consider applications to oscillating spectral multipliers of the Laplace–Beltrami operator. The proof of our main theorem illustrates the delicate relationship between the condition on the kernel of the operator, its Fourier transform (defined in terms of the representation theory of the group) and the microlocal/geometric properties of the group.

The assessment of the spatial and functional features of rare species populations without considering the ontogenetic groups, which are being rarely distinguished at international literature, does not give a complete understanding of the current status of populations and prospects of their development under various management conditions. This paper is aimed to determine the status of a threatened orchid species, Cephalanthera rubra, at the eastern border of its range (Republic of Tatarstan, European Russia). For this purpose, a complex of various population parameters was used. Ontogenetic groups of C. rubra individuals have been reliably identified on the basis of morphometric traits of reproductive and vegetative organs. The obtained results showed that the fruit set is low, ranging at average from 24% for young reproductive individuals (g1) to 39% for mid-mature reproductive individuals (g2); it reflects prospects for seed reproduction of this species at the eastern edge of its range. The abundance dynamics of populations has a fluctuation type; it is related to climatic factors of the growing season. So, we found its significant positive correlations with air humidity (from r = 0.6 to r = 0.7) and precipitation (from r = 0.5 to r = 0.6), and a negative correlation with temperature (from r = -0.5 to r = -0.6). In the Republic of Tatarstan, the base spectrum of C. rubra populations is of the centred type, 1:10:51:38 (j:im:v:g); it corresponds to the general ontogenetic spectrum of rhizomatous orchids. The spatial-ontogenetic structure of populations, and especially its dynamics, reflects the intraspecific relationships of various ontogenetic groups involved in maintaining the stability of C. rubra population in space and time. Under optimal conditions, the spatial structure of all individuals and reproductive groups is characterised by a spatial randomness, which probably reduces intraspecific competition. In contrary, pre-reproductive groups form aggregations with 0.5–0.9-m radius in microloci, favourable for seed germination. A characteristic feature of the spatial structure is the formation of aggregations of reproductive and pre-reproductive individuals with a 0.7–1.0-m radius with a 0.2–0.4-m zone of the random spatial positioning of individuals, which aims to reduce intraspecific competition between them. The probability of meeting an individual of another ontogenetic group increases towards the periphery of the formed aggregation. In C. rubra populations, the abundance and density of individuals, and the fruit set decrease in pessimal conditions of landslides, soil erosion, and habitat shading. Under these conditions, pre-reproductive individuals do not form aggregations, nor aggregations with reproductive individuals. In general, the spatial structure of a population depends on the life-form type of the species, the mechanism of spatial growth of underground organs; it is considered a diagnostic sign of the population status.

Le sujet de la thèse consiste à appliquer des techniques d'analyse microlocale à la construction d'états de Hadamard pour des champs de Dirac, pour des espaces-temps généraux sous des hypothèses faibles sur leur comportement à l'infini. Dans une deuxième partie il s'agira, pour des espaces temps asymptotiquement statiques, de construire les états de vide entrants et sortants et de montrer leur propriété de Hadamard
This thesis is about applying microlocal techniques to the construction of Hadamard states for Dirac fields on curved spacetimes with weak conditions on their asymptotical behaviors. In a second time we will focus on the construction of in and out vacuum states and prove their Hadamard property