Dirac Operator Research Articles

Nonrelativistic Limit of Generalized MIT Bag Models and Spectral Inequalities

For a family of self-adjoint Dirac operators -ic(α·∇)+c22\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-i c (\\alpha \\cdot \ abla ) + \\frac{c^2}{2}$$\\end{document} subject to generalized MIT bag boundary conditions on domains in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}, it is shown that the nonrelativistic limit in the norm resolvent sense is the Dirichlet Laplacian. This allows to transfer spectral geometry results for Dirichlet Laplacians to Dirac operators for large c.

Pseudo-supersymmetric approach to the Dirac operator in the Schwarzschild spacetime

Abstract We have discussed the Dirac equation in Schwarzschild spacetime using pseudo-supersymmetric quantum mechanics and have obtained the partner Hamiltonian of the initial Hamiltonian operator. We demonstrate that the partner metric tensors, corresponding to these Hamiltonians, can be derived using the intertwining relations inherent in pseudo-supersymmetric approaches. We have seen that the Schwarzschild metric may be expanded into Schwarzschild–Tangherlini metric using the aspects of pseudo-supersymmetry. Furthermore, we have derived approximate solutions for the radial component within the framework of N = 2 supersymmetry with the corresponding radial potential graphs presented. Subsequently, our discussion extends to the quasinormal modes, focusing particularly on the asymptotic limits as r approaches ± ∞ .

Convergence of generalized MIT bag models to Dirac operators with zigzag boundary conditions

Convergence of generalized MIT bag models to Dirac operators with zigzag boundary conditions

The cubic Dirac operator on compact quotients of the oscillator group

We study Kostant’s cubic Dirac operator D^{1/3} on locally symmetric Lorentzian manifolds of the form \Gamma\backslash\operatorname{Osc}_{1} , where \operatorname{Osc}_{1} is the four-dimensional oscillator group and \Gamma\subset\operatorname{Osc}_{1} is a cocompact lattice. These quotients are the only four-dimensional, compact Lorentzian G -homoge­neous spaces for a solvable but non-abelian Lie group G . We determine the spectrum of D^{1/3} . We also give an explicit decomposition of the regular representation of \operatorname{Osc}_{1} on L^{2} -sections of the spinor bundle into irreducible subrepresentations and we determine the eigenspaces of D^{1/3} .

Spectral Convergence of the Dirac Operator on Typical Hyperbolic Surfaces of High Genus

AbstractIn this article, we study the Dirac spectrum of typical hyperbolic surfaces of finite area, equipped with a nontrivial spin structure (so that the Dirac spectrum is discrete). For random Weil–Petersson surfaces of large genus g with $$o(\sqrt{g})$$ o ( g ) cusps, we prove convergence of the spectral density to the spectral density of the hyperbolic plane, with quantitative error estimates. This result implies upper bounds on spectral counting functions and multiplicities, as well as a uniform Weyl law, true for typical hyperbolic surfaces equipped with any nontrivial spin structure.

Inverse problems for the eigenparameter Dirac operator with complex weight

Abstract Inverse spectral problems are considered for the discontinuous Dirac operator with complex-value weight and the spectral parameter boundary conditions. We investigate some properties of spectral characteristics and show that the potential can be uniquely determined by the Weyl-type function or by two spectra on the whole interval.

A note on higher order Dirac operators in Clifford analysis

Abstract In the framework of Clifford analysis, we study higher order Dirac operators constructed with k-vectors. We find a necessary and sufficient condition to determine whether a function cancels them.

The centre of the Dunkl total angular momentum algebra

For a finite dimensional representation V of a finite reflection group W, we consider the rational Cherednik algebra Ht,c(V,W) associated with (V,W) at the parameters t≠0 and c. The Dunkl total angular momentum algebra Ot,c(V,W) arises as the centraliser algebra of the Lie superalgebra osp(1|2) containing a Dunkl deformation of the Dirac operator, inside the tensor product of Ht,c(V,W) and the Clifford algebra generated by V.We show that when dim⁡V≥3 and for every value of the parameter c, the centre of Ot,c(V,W) is isomorphic to a univariate polynomial ring. Notably, the generator of the centre changes depending on whether or not (−1)V is an element of the group W. Using this description of the centre, and using the projection of the pseudo scalar from the Clifford algebra into Ot,c(V,W), we establish results analogous to “Vogan's conjecture” for a family of operators depending on suitable elements of the double cover W˜.

Non-local relativistic δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}-shell interactions

In this paper, new self-adjoint realizations of the Dirac operator in dimension two and three are introduced. It is shown that they may be associated with the formal expression D0+|FδΣ⟩⟨GδΣ|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {D}_0+|F\\delta _\\Sigma \\rangle \\langle G\\delta _\\Sigma |$$\\end{document}, where D0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {D}_0$$\\end{document} is the free Dirac operator, F and G are matrix valued coefficients, and δΣ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta _\\Sigma $$\\end{document} stands for the single layer distribution supported on a hypersurface Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document}, and that they can be understood as limits of the Dirac operators with scaled non-local potentials. Furthermore, their spectral properties are analysed.

Integration of the mKdV equation with time-dependent coefficients, with an additional term and with an integral source in the class of rapidly decreasing functions

The work is devoted to the integration of the modified Korteweg–de Vries equation with time-dependent coefficients, an additional term and an integral source in the class of rapidly decreasing functions using the inverse scattering problem method. In this paper, we consider the case when the Dirac operator included in the Lax pairs is not self-adjoint, therefore the eigenvalues of the Dirac operator can be multiples. The evolution of scattering data is obtained for the non-self-adjoint Dirac operator, the potential of which is a solution of the modified Korteweg–de Vries equation with time-dependent coefficients, with an additional term and with an integral source of a class of rapidly decreasing functions. An example is given to illustrate the application of the results obtained.

Floquet–Bloch Functions on Non-simply Connected Manifolds, the Aharonov–Bohm Fluxes, and Conformal Invariants of Immersed Surfaces

We define spectral (Bloch) varieties of multidimensional differential operators on non-simply connected manifolds. In their terms we give a description of the analytic dependence of the spectra of magnetic Laplacians on non-simply connected manifolds on the values of the Aharonov–Bohm fluxes, construct analogs of spectral curves for two-dimensional Dirac operators on Riemann surfaces, and thereby find new conformal invariants of immersions of surfaces into three- and four-dimensional Euclidean spaces.

On Recovering Dirac Operators with Two Delays

On Recovering Dirac Operators with Two Delays

Perturbed Dirac Operators and Boundary Value Problems

In this paper, the time-independent Klein-Gordon equation in R3 is treated with a decomposition of the operator Δ−γ2I by the Clifford algebra Cl(V3,3). Some properties of integral operators associated the kind of equations and some Riemann-Hilbert boundary value problems for perturbed Dirac operators are investigated.

Characterization of continuous homomorphisms on entire slice monogenic functions

Abstract This paper is inspired by a class of infinite order differential operators arising in quantum mechanics. They turned out to be an important tool in the investigation of evolution of superoscillations with respect to quantum fields equations. Infinite order differential operators act naturally on spaces of holomorphic functions or on hyperfunctions. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e. functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra or also slice monogenic functions. This function theory has a very reach associated spectral theory and both the function theory and the operator theory in this setting are subjected to intensive investigations. Here we introduce the concept of proximate order and establish some fundamental properties of entire slice monogenic functions that are crucial for the characterization of infinite order differential operators acting on entire slice monogenic functions.

Harmonic Spinors in the Ricci Flow

This paper provides a new definition of the Ricci flow on closed manifolds admitting harmonic spinors. It is shown that Perelman’s Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms of the energy of Seiberg–Witten monopoles. Consequently, Ricci flow is the gradient flow of these energies. The proof relies on a weighted version of the monopole equations, introduced here. Further, a sharp parabolic Hitchin–Thorpe inequality for simply-connected, spin 4-manifolds is proven. From this, it follows that the normalized Ricci flow on any exotic K3 surface must become singular.

Spectral metric and Einstein functionals for the Hodge–Dirac operator

We examine the metric and Einstein bilinear functionals of differential forms introduced by Dąbrowski et al. (2023), for the Hodge–Dirac operator d+\delta on an oriented, closed, even-dimensional Riemannian manifold. We show that they are equal (up to a numerical factor) to these functionals for the canonical Dirac operator on a spin manifold. Furthermore, we demonstrate that the spectral triple for the Hodge–Dirac operator is spectrally closed, which implies that it is torsion-free.

Bohr–Sommerfeld quantization condition for self-adjoint Dirac operators

We study the eigenvalue problem for a self-adjoint 1D Dirac operator. It is known that, near an energy level where the square of the potential makes a simple well, the eigenvalues are approximated by a Bohr–Sommerfeld type quantization rule. A remarkable difference from the Schrödinger case appears in the Maslov correction term. This fact was recently found by Hirota [Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov–Shabat operator, J. Math. Phys. 58 (2017) 102–108] under the analyticity condition of the potential using a complex WKB method. In this paper, we approach this problem with a microlocal technique focusing on the asymptotic behavior of the eigenfunction along the characteristic set to generalize the result to [Formula: see text] potentials.

Scalar and mean curvature comparison via the Dirac operator

Scalar and mean curvature comparison via the Dirac operator

Monopoles in Dirac spin liquids and their symmetries from instanton calculus

The Dirac spin liquid (DSL) is a two-dimensional (2D) fractionalized Mott insulator featuring massless Dirac spinon excitations coupled to a compact U(1) gauge field, which allows for flux-tunneling instanton events described by magnetic monopoles in (2+1)D Euclidean spacetime. The state-operator correspondence of conformal field theory has been used recently to define associated monopole operators and determine their quantum numbers, which encode the microscopic symmetries of conventional ordered phases proximate to the DSL. In this work, we utilize semiclassical instanton methods not relying on conformal invariance to construct monopole operators directly in (2+1)D spacetime as instanton-induced ’t Hooft vertices, i.e., fermion-number-violating effective interactions originating from zero modes of the Euclidean Dirac operator in an instanton background. In the presence of a flavor-adjoint fermion mass, resummation of the instanton gas is shown to select the correct monopole to be proliferated, in accordance with predictions of the state-operator correspondence. We also show that our instanton-based approach is able to determine monopole quantum numbers on bipartite lattices.

Half inverse problem and interior inverse problem for the Dirac operators with discontinuity

Half inverse problem and interior inverse problem for the Dirac operators with discontinuity