Dirac Operator Research Articles

Bismut–Cheeger Eta Form and Higher Spectral Flow

Abstract We prove the variation formula, the embedding formula, and the adiabatic limit formula for equivariant Bismut–Cheeger eta forms with a fiberwise compact Lie group action. We only need the condition that the kernels of corresponding fiberwise Dirac operators form vector bundles.

Decay Estimates for Unitary Representations with Applications to Continuous- and Discrete-Time Models

We present a new technique to obtain polynomial decay estimates for the matrix coefficients of unitary operators. Our approach, based on commutator methods, applies to nets of unitary operators, unitary representations of topological groups, and unitary operators given by the evolution group of a self-adjoint operator or by powers of a unitary operator. Our results are illustrated with a wide range of examples in quantum mechanics and dynamical systems, as for instance Schrödinger operators, Dirac operators, quantum waveguides, horocycle flows, adjacency matrices, Jacobi matrices, quantum walks or skew products.

Spectral properties of the Dirac operator coupled with $$\delta $$-shell interactions

Let $\Omega\subset\mathbb{R}^3$ be an open set, we study the spectral properties of the free Dirac operator $\mathcal{H}$ coupled with the singular potential $V_\kappa=(\epsilon I_4 +\mu\beta+\eta(\alpha\cdot N))\delta_{\partial\Omega}$. The open set $\Omega$ can be either a $\mathcal{C}^2$-bounded domain or a locally deformed half-space. In both cases, self-adjointness is proved and several spectral properties are given. In particular, we give a complete description of the essential spectrum of $\mathcal{H}+V_\kappa$ for the so-called critical combinations of coupling constants, when $\Omega$ is a locally deformed half-space. Finally, we introduce a new model of Dirac operators with $\delta$-interactions and deals with its spectral properties. More precisely, we study the coupling $\mathcal{H}_{\upsilon}=\mathcal{H}+i\upsilon\beta(\alpha\cdot N)\delta_{\partial\Omega}$. In particular, we show that $\mathcal{H}_{\pm2}$ is essentially self-adjoint and generates confinement.

$$j$$-Self-Adjointness Conditions for Jacobi Matrices and Schrödinger and Dirac Operators with Point Interactions

$$j$$-Self-Adjointness Conditions for Jacobi Matrices and Schrödinger and Dirac Operators with Point Interactions

Inverse Problems for Dirac Operators with Constant Delay: Uniqueness, Characterization, Uniform Stability

We initiate studying inverse spectral problems for Dirac-type functional-differential operators with constant delay. For simplicity, we restrict ourselves to the case when the delay parameter is not less than one half of the interval. For the considered case, however, we give answers to the full range of questions usually raised in the inverse spectral theory. Specifically, reconstruction of two complex $$L_{2}$$ -potentials is studied from either complete spectra or subspectra of two boundary value problems with one common boundary condition. We give conditions on the subspectra that are necessary and sufficient for the unique determination of the potentials. Moreover, necessary and sufficient conditions for the solvability of both inverse problems are obtained. For the inverse problem of recovering from the complete spectra, we establish also uniform stability in each ball of a finite radius. For this purpose, we use recent results on uniform stability of sine-type functions with asymptotically separated zeros.

Dirac Operators for the Dunkl Angular Momentum Algebra

We define a family of Dirac operators for the Dunkl angular momentum algebra depending on certain central elements of the group algebra of the Pin cover of the Weyl group inherent to the rational Cherednik algebra. We prove an analogue of Vogan's conjecture for this family of operators and use this to show that the Dirac cohomology, when non-zero, determines the central character of representations of the angular momentum algebra. Furthermore, interpreting this algebra in the framework of (deformed) Howe dualities, we show that the natural Dirac element we define yields, up to scalars, a square root of the angular part of the Calogero-Moser Hamiltonian.

On the spectrum of non-self-adjoint Dirac operators with quasi-periodic boundary conditions

In this paper, we consider non-self-adjoint Dirac operators on a finite interval with complex-valued potentials and quasi-periodic boundary conditions. Necessary and sufficient conditions for a set of complex numbers to be the spectrum of the indicated problem are established.

The UV prolate spectrum matches the zeros of zeta

SignificanceWe show that the eigenvalues of the self-adjoint extension (introduced by A.C. in 1998) of the prolate spheroidal operator reproduce the UV behavior of the squares of zeros of the Riemann zeta function, and we construct an isospectral family of Dirac operators whose spectra have the same UV behavior as those zeros.

The Orthogonal Branching Problem for Symplectic Monogenics

In this paper we study the $$\mathfrak {sp}(2m)$$ -invariant Dirac operator $$D_s$$ which acts on symplectic spinors, from an orthogonal point of view. By this we mean that we will focus on the subalgebra $$\mathfrak {so}(m) \subset \mathfrak {sp}(2m)$$ , as this will allow us to derive branching rules for the space of 1-homogeneous polynomial solutions for the operator $$D_s$$ (hence generalising the classical Fischer decomposition in harmonic analysis for a vector variable in $${\mathbb {R}}^m$$ ). To arrive at this result we use techniques from representation theory, including the transvector algebra $${\mathcal {Z}}(\mathfrak {sp}(4),\mathfrak {so}(4))$$ and tensor products of Verma modules.

On the calculation of the discrete spectra of one‐dimensional Dirac operators

In this work, we consider the one‐dimensional Dirac operator where is Pauli's matrix, is a ‐matrix representing a regular potential that includes the electrostatic and scalar interactions as well as the anomalous magnetic momentum, is a singular potential consisting of delta distributions ( ) are ‐matrices representing the strengths of Dirac deltas, and is a two‐spinor. We associate to the operator an unbounded in symmetric operator denoted by , where is the support of singular potential . The operator includes only the regular potential together with certain interaction conditions at each point . The paper presents a method for determining the discrete spectrum of the operator for arbitrary potential whose entries are given by ‐functions. The eigenvalues of the operator are the zeros of a dispersion equation , where the characteristic function is determined explicitly in terms of power series involving the spectral parameter . The construction of the characteristic function from a set of monodromy matrices and the interaction conditions is presented in the paper. Moreover, its power series representation leads to an efficient numerical method for calculating the eigenvalues of the Dirac operator from the zeros of certain approximate function which is obtained by truncating the series up to a finite number of terms. Several examples show the applicability and accuracy of the numerical method.

Generalized boundary triples, II. Some applications of generalized boundary triples and form domain invariant Nevanlinna functions

AbstractThe paper is a continuation of Part I and contains several further results on generalized boundary triples, the corresponding Weyl functions, and applications of this technique to ordinary and partial differential operators. We establish a connection between Post's theory of boundary pairs of closed nonnegative forms on the one hand and the theory of generalized boundary triples of nonnegative symmetric operators on the other hand. Applications to the Laplacian operator on bounded domains with smooth, Lipschitz, and even rough boundary, as well as to mixed boundary value problem for the Laplacian are given. Other applications concern with the momentum, Schrödinger, and Dirac operators with local point interactions. These operators demonstrate natural occurrence of ‐generalized boundary triples with domain invariant Weyl functions and essentially selfadjoint reference operators A0.

Heat kernel, spectral functions and anomalies in Weyl semimetals

Being motivated by applications to the physics of Weyl semimetals we study spectral geometry of Dirac operator with an abelian gauge field and an axial vector field. We impose chiral bag boundary conditions with variable chiral phase θ on the fermions. We establish main properties of the spectral functions which ensure applicability of the ζ function regularization and of the usual heat kernel formulae for chiral and parity anomalies. We develop computational methods, including a perturbation expansion for the heat kernel. We show that the terms in both anomalies which include electromagnetic potential are independent of θ.

A generalized MIT Bag operator on spin manifolds in the non-relativistic limit

We consider Dirac-like operators with piecewise constant mass terms on spin manifolds, and we study the behaviour of their spectra when the mass parameters become large. In several asymptotic regimes, effective operators appear: the extrinsic Dirac operator and a generalized MIT Bag Dirac operator. This extends some results previously known for the Euclidean spaces to the case of general spin manifolds.

Spectral statistics of Dirac ensembles

In this paper, we find spectral properties in the large N limit of Dirac operators that come from random finite noncommutative geometries. In particular, for a Gaussian potential, the limiting eigenvalue spectrum is shown to be universal, regardless of the geometry, and is given by the convolution of the semicircle law with itself. For simple non-Gaussian models, this convolution property is also evident. In order to prove these results, we show that a wide class of multi-trace multimatrix models have a genus expansion.

Pseudo-Hermitian Dirac operator on the torus for massless fermions under the action of external fields

The Dirac equation in (2+1) dimensions on the toroidal surface is studied for a massless fermion particle under the action of external fields. Using the covariant approach based on general relativity, the Dirac operator stemming from a metric related to the strain tensor is discussed within the pseudo-Hermitian operator theory. Furthermore, analytical solutions are obtained for two cases, namely, constant and position-dependent Fermi velocity.

Immersions and the unbounded Kasparov product: embedding spheres into Euclidean space

We construct an unbounded representative for the shriek class associated to the embeddings of spheres into Euclidean space. We equip this unbounded KK -cycle with a connection and compute the unbounded Kasparov product with the Dirac operator on \mathbb{R}^{n+1} . We find that the resulting spectral triple for the algebra C(\mathbb{S}^n) differs from the Dirac operator on the round sphere by a so-called index cycle, whose class in KK_0(\mathbb{C},\mathbb{C}) represents the multiplicative unit. At all points we check that our construction involving the unbounded Kasparov product is compatible with the bounded Kasparov product using Kucerovsky’s criterion and we thus capture the composition law for the shriek map for these immersions at the unbounded KK -theoretical level, while retaining the geometric information.

Harmonic Functions and the Mass of 3-Dimensional Asymptotically Flat Riemannian Manifolds

An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the positive mass theorem is achieved in dimension three. The proof has parallels with both the Schoen–Yau minimal hypersurface technique and Witten’s spinorial approach. In particular, the role of harmonic spinors and the Lichnerowicz formula in Witten’s argument is replaced by that of harmonic functions and a formula introduced by the fourth named author in recent work, while the level sets of harmonic functions take on a role similar to that of the Schoen–Yau minimal hypersurfaces.

Toward infinite‐dimensional Clifford analysis

Clifford analysis is a higher dimensional function theory and a refinement of harmonic analysis. We will extend Clifford analysis to the infinite‐dimensional setting. To do that, we have to introduce infinite‐dimensional Clifford algebras and the infinite‐dimensional Dirac operator. The paper's main result is the Wiener chaos decomposition in Clifford analysis. We apply the decomposition to white noise and Brownian motion.

The asymptotic approach to the continuum of lattice QCD spectral observables

We consider spectral quantities in lattice QCD and determine the asymptotic behaviour of their discretization errors. Wilson fermion with O(a)-improvement, (Möbius) Domain wall fermion (DWF), and overlap Dirac operators are considered in combination with the commonly used gauge actions. Wilson fermions and DWF with domain wall height M5=1+O(g02) have the same, approximate, form of the asymptotic cutoff effects: Ka2[g¯2(a−1)]0.760. A domain wall height M5=1.8, as often used, introduces large mass-dependent K′(m)a2[g¯2(a−1)]0.518 effects. Massless twisted mass fermions have the same form as Wilson fermions when the Sheikholeslami-Wohlert term [1] is included. For their mass-dependent cutoff effects we have information on the exponents Γˆi of g¯2(a−1) but not for the pre-factors. For staggered fermions there is only partial information on the exponents.We propose that tree-level O(a2) improvement, which is easy to do [2], should be used in the future – both for the fermion and the gauge action. It improves the asymptotic behaviour in all cases.

Spectral Properties of Relativistic Quantum Waveguides

We make a spectral analysis of the massive Dirac operator in a tubular neighborhood of an unbounded planar curve,subject to infinite mass boundary conditions. Under general assumptions on the curvature, we locate the essential spectrum and derive an effective Hamiltonian on the base curve which approximates the original operator in the thin-strip limit. We also investigate the existence of bound states in the non-relativistic limit and give a geometric quantitative condition for the bound states to exist.