Computing elements of certain form in ideals to prove properties of operators

Proving statements about linear operators expressed in terms of identities often leads to finding elements of certain form in noncommutative polynomial ideals. We illustrate this by examples coming from actual operator statements and discuss relevant algorithmic methods for finding such polynomials based on noncommutative Gr\"obner bases. In particular, we present algorithms for computing the intersection of a two-sided ideal with a one-sided ideal as well as for computing homogeneous polynomials in two-sided ideals and monomials in one-sided ideals. All methods presented in this work are implemented in the Mathematica package OperatorGB.Comment: 26 page

Formal proofs of operator identities by a single formal computation

A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.Comment: 22 page

Compatible rewriting of noncommutative polynomials for proving operator identities

The goal of this paper is to prove operator identities using equalities between noncommutative polynomials. In general, a polynomial expression is not valid in terms of operators, since it may not be compatible with domains and codomains of the corresponding operators. Recently, some of the authors introduced a framework based on labelled quivers to rigorously translate polynomial identities to operator identities. In the present paper, we extend and adapt the framework to the context of rewriting and polynomial reduction. We give a sufficient condition on the polynomials used for rewriting to ensure that standard polynomial reduction automatically respects domains and codomains of operators. Finally, we adapt the noncommutative Buchberger procedure to compute additional compatible polynomials for rewriting. In the package OperatorGB, we also provide an implementation of the concepts developed.Comment: 17 page

Algebraic proof methods for identities of matrices and operators: improvements of Hartwig's triple reverse order law

When improving results about generalized inverses, the aim often is to do this in the most general setting possible by eliminating superfluous assumptions and by simplifying some of the conditions in statements. In this paper, we use Hartwig's well-known triple reverse order law as an example for showing how this can be done using a recent framework for algebraic proofs and the software package OperatorGB. Our improvements of Hartwig's result are proven in rings with involution and we discuss computer-assisted proofs that show these results in other settings based on the framework and a single computation with noncommutative polynomials

Composing and Factoring Generalized Green's Operators and Ordinary Boundary Problems

We consider solution operators of linear ordinary boundary problems with "too many" boundary conditions, which are not always solvable. These generalized Green's operators are a certain kind of generalized inverses of differential operators. We answer the question when the product of two generalized Green's operators is again a generalized Green's operator for the product of the corresponding differential operators and which boundary problem it solves. Moreover, we show that---provided a factorization of the underlying differential operator---a generalized boundary problem can be factored into lower order problems corresponding to a factorization of the respective Green's operators. We illustrate our results by examples using the Maple package IntDiffOp, where the presented algorithms are implemented.Comment: 19 page

Experimental Measurement of the Berry Curvature from Anomalous Transport

Geometrical properties of energy bands underlie fascinating phenomena in a wide-range of systems, including solid-state materials, ultracold gases and photonics. Most famously, local geometrical characteristics like the Berry curvature can be related to global topological invariants such as those classifying quantum Hall states or topological insulators. Regardless of the band topology, however, any non-zero Berry curvature can have important consequences, such as in the semi-classical evolution of a wave packet. Here, we experimentally demonstrate for the first time that wave packet dynamics can be used to directly map out the Berry curvature. To this end, we use optical pulses in two coupled fibre loops to study the discrete time-evolution of a wave packet in a 1D geometrical "charge" pump, where the Berry curvature leads to an anomalous displacement of the wave packet under pumping. This is both the first direct observation of Berry curvature effects in an optical system, and, more generally, the proof-of-principle demonstration that semi-classical dynamics can serve as a high-resolution tool for mapping out geometrical properties

Evidence for the η_b(1S) Meson in Radiative Υ(2S) Decay

We have performed a search for the η_b(1S) meson in the radiative decay of the Υ(2S) resonance using a sample of 91.6 × 10^6 Υ(2S) events recorded with the BABAR detector at the PEP-II B factory at the SLAC National Accelerator Laboratory. We observe a peak in the photon energy spectrum at E_γ = 609.3^(+4.6)_(-4.5)(stat)±1.9(syst) MeV, corresponding to an η_b(1S) mass of 9394.2^(+4.8)_(-4.9)(stat) ± 2.0(syst) MeV/c^2. The branching fraction for the decay Υ(2S) → γη_b(1S) is determined to be [3.9 ± 1.1(stat)^(+1.1)_(-0.9)(syst)] × 10^(-4). We find the ratio of branching fractions B[Υ(2S) → γη_b(1S)]/B[Υ(3S) → γη_b(1S)]= 0.82 ± 0.24(stat)^(+0.20)_(-0.19)(syst)

Solitary waves in the Nonlinear Dirac Equation

In the present work, we consider the existence, stability, and dynamics of solitary waves in the nonlinear Dirac equation. We start by introducing the Soler model of self-interacting spinors, and discuss its localized waveforms in one, two, and three spatial dimensions and the equations they satisfy. We present the associated explicit solutions in one dimension and numerically obtain their analogues in higher dimensions. The stability is subsequently discussed from a theoretical perspective and then complemented with numerical computations. Finally, the dynamics of the solutions is explored and compared to its non-relativistic analogue, which is the nonlinear Schr{\"o}dinger equation. A few special topics are also explored, including the discrete variant of the nonlinear Dirac equation and its solitary wave properties, as well as the PT-symmetric variant of the model