Unitary operator

Summary

In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating on a Hilbert space, but the same notion serves to define the concept of isomorphism between Hilbert spaces. A unitary element is a generalization of a unitary operator. In a unital algebra, an element U of the algebra is called a unitary element if UU = UU = I, where I is the identity element. Definition Definition 1. A unitary operator is a bounded linear operator U : H → H on a Hilbert space H that satisfies UU = UU = I, where U* is the adjoint of U, and I : H → H is the identity operator. The weaker condition UU = I defines an isometry. The other condition, UU = I, defines a coisometry. Thus a unitary operator is a bounded linear operator which is both an isometry and a coisometry, or, equivalently, a surjective isometry. An equivalen

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Generalized Norm-compatible Systems on Unitary Shimura Varieties

Mohamed Réda Boumasmoud

We define and study in terms of integral IwahoriâHecke algebras a new class of geometric operators acting on the Bruhat-Tits building of connected reductive groups over p-adic fields. These operators, which we call U-operators, generalize the geometric notion of "successors" for trees with a marked end. The first main contributions of the thesis are: (i) the integrality of the U-operator over the spherical Hecke algebra using the compatibility between Bernstein and Satake homomorphisms, (ii) in the unramified case, the U-operator attached to a cocharacter is a right root of the corresponding Hecke polynomial. In the second part of the thesis, we study some arithmetic aspects of special cycles on (products of) unitary Shimura varieties, these cycles are expected to yield new results towards the BlochâBeilinson conjectures. As a global application of (ii), we obtain: (iii) the horizontal norm relations for these GGP cycles for arbitrary n, at primes where the unitary group splits. The general local theory developed in the first part of the thesis, has the potential to result in a number of global applications along the lines of (iii) (involving other Shimura varieties and also vertical norm relations) and offers new insights into topics such as the BlasiusâRogawski conjecture as well.

EPFL2019

We present an open-source program irvsp, to compute irreducible representations of electronic states for all 230 space groups with an interface to the Vienna ab-initio Simulation Package. This code is fed with plane-wave-based wavefunctions (e.g. WAVECAR) and space group operators (listed in OUTCAR), which are generated by the VASP package. This program computes the traces of matrix presentations and determines the corresponding irreducible representations for all energy bands and all the k-points in the three-dimensional Brillouin zone. It also works with spin-orbit coupling (SOC), i.e., for double groups. It is in particular useful to analyze energy bands, their connectivities, and band topology, after the establishment of the theory of topological quantum chemistry. Accordingly, the associated library -irrep_bcs.a - is developed, which can be easily linked to by other ab-initio packages. In addition, the program has been extended to orthogonal tight-binding (TB) Hamiltonians, e.g. electronic or phononic TB Hamiltonians. A sister program is presented as well. Program summary Program title: irvsp CPC Library link to program files: http://doi.org/10.1763/y9ds5nnm2f.1 Licensing provisions: GNU Lesser General Public License Programming language: Fortran 90/77 Nature of problem: Determining irreducible representations for all energy bands and all the k-points in 230 space groups. It is in particular useful to analyze energy bands, their connectivities, and band topology. Solution method: By computing the traces of matrix presentations of space group operators for the eigen-wavefunctions at a certain k-point in a given space group, one can determine irreducible representations for them. (C) 2020 Elsevier B.V. All rights reserved.

ELSEVIER2021

Hidden and explicit quantum scale invariance

Sander Johannes Nicolaas Mooij, Thibault Voumard

There exist renormalization schemes that explicitly preserve the scale invariance of a theory at the quantum level. Imposing a scale-invariant renormalization breaks renormalizability and induces new nontrivial operators in the theory. In this work, we study the effects of such scale-invariant renormalization procedures. On the one hand, an explicitly quantum scale-invariant theory can emerge from the scale-invariant renormalization of a scale-invariant Lagrangian. On the other hand, we show how a quantum scale-invariant theory can equally emerge from a Lagrangian visibly breaking scale invariance renormalized with scale-dependent renormalization (such as the traditional (MS) over bar scheme). In this last case, scale invariance is hidden in the theory, in the sense that it only appears explicitly after renormalization.

2019