Abstract The half-line Dirac operators with $L^{2}$ L 2 -potentials can be characterized by their spectral data. It is known that the spectral correspondence is a homeomorphism: close potentials give rise to close spectral data and vice versa. We prove the first explicit two-sided uniform estimate related to this continuity in the general $L^{2}$ L 2 -case. The proof is based on an exact solution of the inverse spectral problem for Dirac operators with $\delta $ δ -interactions on a half-lattice in terms of the Schur’s algorithm for analytic functions.

We compute the one-loop QED [Formula: see text]-function coefficient directly from heat kernel data of the twisted Spin c Dirac operator on [Formula: see text]. Using [Formula: see text]-function regularization, the logarithmic scale dependence is encoded in the [Formula: see text] coefficient of the spectral expansion. The [Formula: see text] term in [Formula: see text] yields exactly [Formula: see text], independent of [Formula: see text], [Formula: see text], or background, verifying spectral RG flow without flat-space propagators. The result is independent of the radii of [Formula: see text] and [Formula: see text] and of the choice of gauge background, providing a parameter-free consistency check that spectral data on compact manifolds encode renormalization group information. Beyond a mere verification of the coupling flow, this result serves as a non-trivial consistency check of the Spectral Action Principle in a curved background. It demonstrates that universal quantum corrections can be extracted purely from geometric spectral invariants, distinguishing this geometric spectral derivation from momentum-space propagator methods.

The Stueckelberg wave equation is transformed into a quantum telegraph equation and a set of stationary states is obtained as unitary solutions. As it has been shown previously that this PDE relates to the Dirac operator, and on the other hand it is a linearized Hamilton–Jacobi–Bellman PDE, from which the Schrödinger equation can be deduced in a nonrelativistic limit, it is clear that it is the key equation in relativistic quantum mechanics. We give a Bayesian interpretation for the measurement problem. The stationary solution is understood as a maximum entropy prior distribution and measurement is understood as a Bayesian update. We discuss the interpretation of the single electron experiments in the light of finite speed propagation of the transition probability field and how it relates to the interpretation of quantum mechanics more broadly.

Almansi Decomposition Theorems of Polynomial Weighted Dirac Operators and Their Applications

Clifford analysis focuses in the so-called monogenic functions, which are recognized as natural generalizations of the holomorphic functions of the complex plane. Due to the non-commutativity of the product in Clifford algebras, the inframonogenic functions arise as a non-commutative version of the harmonic ones. The construction of Dirac operators with arbitrary orthonormal bases of makes possible the emergence of a new subclass of biharmonic functions that generalize to inframonogenic functions. In this work, a Cauchy integral formula and a jump problem for this type of functions will be discussed, as well as the connection with the Lamé-Navier system. At the end, well-posed boundary problems and Fischer decompositions for the polynomial space will be shown.

We construct a traceless Ricci metric of Taub-NUT type whose total mass can be negative. For the Taub-NUT metric and its negative NUT charge counterpart, we prove that the spaces of twistor and parallel spinors coincide and are complex 2-dimensional. We use them to construct L 2 harmonic spinors and Rarita-Schwinger fields. For the traceless Ricci Taub-NUT type metric, we study the Dirac and Rarita-Schwinger equations by separating them into angular and radial equations, and obtain explicit solutions in certain special cases.

Abstract In this paper, the equation formed by a combination of the equations AKNS ⁡ ( + 1 ) \operatorname{AKNS}(+1) and AKNS ⁡ ( - 1 ) {\operatorname{AKNS}(-1)} was integrated using the inver problem method for the self-adjoint periodic Dirac operator in the class of periodic infinite-gap functions. In addition, an infinite system of Dubrovin differential equations that represents evolution of spectral data of the Dirac operator is derived and it is proved that the Cauchy problem for the system of Dubrovin differential equations has a unique solution in the class of three times continuously differentiable periodic infinite-gap functions.

Dirac operators with interactions on composite curves

For closed connected Riemannian spin manifolds, an upper estimate of the smallest eigenvalue of the Dirac operator in terms of the hyperspherical radius is proved. When combined with known lower Dirac eigenvalue estimates, this has a number of geometric consequences. Some are known and include Llarull’s scalar curvature rigidity of the standard metric on the sphere, Geroch’s conjecture on the impossibility of positive scalar curvature on tori, and a mean curvature estimate for spin fill-ins with nonnegative scalar curvature due to Gromov, including its rigidity statement recently proved by Cecchini, Hirsch and Zeidler. New applications provide a comparison of the hyperspherical radius with the Yamabe constant and improved estimates of the hyperspherical radius for Kähler manifolds, Kähler–Einstein manifolds, quaternionic Kähler manifolds, and manifolds with a harmonic 1-form of constant length.

Designing stable cluster synchronization patterns is a fundamental challenge in nonlinear dynamics of networks with great relevance to understanding neuronal and brain dynamics. So far, cluster synchronization has been studied exclusively in a node-based dynamical approach, according to which oscillators are associated only with the nodes of the network. Here we propose a topological synchronization dynamics model based on the use of the topological Dirac operator, which allows us to design cluster synchronization patterns for topological oscillators associated with both nodes and edges of a network. In particular, by modulating the ground state of the free energy associated with the dynamical model, we construct topological cluster synchronization patterns. These are aligned with the eigenstates of the topological Dirac equationthat provide a very useful decomposition of the dynamical state of node and edge signals associated with the network. We use linear stability analysis to predict the stability of the topological cluster synchronization patterns and provide numerical evidence of the ability to design several stable topological cluster synchronization states on real connectome data, random graphs, and on stochastic block models.

This paper aims to show that, in the limit of strong magnetic fields, the optimal domains for eigenvalues of magnetic Laplacians tend to exhibit symmetry. We establish several asymptotic bounds on magnetic eigenvalues to support this conclusion. Our main result implies that if, for a bounded simply-connected planar domain, the n-th eigenvalue of the magnetic Dirichlet Laplacian with uniform magnetic field is smaller than the corresponding eigenvalue for a disk of the same area, then the Fraenkel asymmetry of that domain tends to zero in the strong magnetic field limit. Comparable results are also derived for the magnetic Dirichlet Laplacian on rectangles, as well as the magnetic Dirac operator with infinite mass boundary conditions on smooth domains. As part of our analysis, we additionally provide a new estimate for the torsion function on rectangles.

On complete non-compact manifolds with bounded sectional curvature, we consider a class of self-adjoint Dirac-type operators called Dirac–Schrödinger operators. Assuming two Dirac–Schrödinger operators coincide at infinity, by previous work, one can define their relative eta invariant. A typical example of Dirac–Schrödinger operators is the (twisted) spin Dirac operators on spin manifolds which admit a Riemannian metric of uniformly positive scalar curvature. In this case, using the relative eta invariant, we get a geometric formula for the spectral flow on non-compact manifolds, which induces a new proof of Gromov–Lawson’s result about compact area enlargeable manifolds in odd dimensions. When two such spin Dirac operators are the boundary restriction of an operator on a manifold with non-compact boundary, under certain conditions, we obtain an index formula involving the relative eta invariant. This generalizes the Atiyah–Patodi–Singer index theorem to non-compact boundary situation. As a result, we can use the relative eta invariant to study the space of uniformly positive scalar curvature metrics on some non-compact connected sums.

Spectral theory for non-self-adjoint Dirac operators with periodic potentials and inverse scattering transform for the focusing nonlinear Schrödinger equation with periodic boundary conditions

We study a two-dimensional massive Dirac operator with a singular potential supported on a periodic graph, and examine the self-adjointness and the Fredholmness of the associated unbounded operator.

Dirac Operators on Conformal Manifolds

ABSTRACT This paper explores ‐hyperholomorphic functions in Clifford analysis, which satisfy the equation , where is the Dirac operator in , and . Utilizing endomorphisms on Clifford algebra and the method of normalized function systems with respect to the Dirac operator, we derive a Borel–Pompeiu formula for the operator , construct a transmutation operator for , and establish an Almansi‐type decomposition for solutions of the operator .

The spectral torsion for the one form rescaled Dirac operator

We consider spinors on the total space of a Kaluza–Klein model with fuzzy sphere fiber and geometrically realised Dirac operator on the product. We show that a single massless spinor on the product appears on spacetime as multiplets of spinors with a particular signature of differing masses and SU(2) Yang–Mills charges. For a finite-dimensional fiber, these become finite multiplets rather than infinite towers. For example, for the reduced fuzzy sphere isomorphic to M2(C), a massless spinor appears as two SU(2) doublets and an SU(2) quadruplet in mass ratios 1:5/3:7/3. Although such signatures do not match known fermions in the Standard Model, the paper provides a new mechanism which could be further explored for other noncommutative fiber algebras.

We study how far APS boundary conditions for a Lorentzian Dirac operator may be perturbed without destroying Fredholmness of the Dirac operator. This is done by developing criteria under which the perturbation of a compact pair of projections is a Fredholm pair.

Homogenization of the Dirac operator with position-dependent mass