Dirac operator on the sphere with attached wires

An explicitly solvable model for tunnelling of relativistic spinless particles through a sphere is suggested. The model operator is constructed by an operator extensions theory method from the orthogonal sum of the Dirac operators on a semi- axis and on the sphere. The transmission coefficient is obtained. The dependence of the transmission coefficient on the particle energy has a resonant character. One observes pairs of the Breit-Wigner and the Fano resonances. It correlates with the corresponding results for a non-relativistic particle.

Ambarzumyan's Theorem for the Dirac Operator on Equilateral Tree Graphs

The classical Ambarzumyan’s theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator-d^(2)/dx^(2)+q with an integrable real-valued potential q on[0,π] are {n^(2):n≥0},then q=0 for almost all x∈[0,π].In this work,the classical Ambarzumyan’s theorem is extended to the Dirac operator on equilateral tree graphs.We prove that if the spectrum of the Dirac operator on graphs coincides with the unperturbed case,then the potential is identically zero.

Inverse Problems for the Dirac Operator on a Star Graph

Following the previous work,we shall study some inverse problems for the Dirac operator on an equilateral star graph.It is proven that the so-called Weyl function uniquely determines the potentials.Furthermore,we pay attention to the inverse problem of recovering the potentials from the spectral data,which consists of the eigenvalues and weight matrices,and present a constructive algorithm.The basic tool in this paper is the method of spectral mappings developed by Yurko.

Dirac method for nonlinear and non-homogenous boundary value problems of plates

The boundary value problem plays a crucial role in the analytical investigation of continuum dynamics. In this paper, an analytical method based on the Dirac operator to solve the nonlinear and non-homogeneous boundary value problem of rectangular plates is proposed. The key concept behind this method is to transform the nonlinear or non-homogeneous part on the boundary into a lateral force within the governing function by the Dirac operator, which linearizes and homogenizes the original boundary, allowing one to employ the modal superposition method for obtaining solutions to reconstructive governing equations. Once projected into the modal space, the harmonic balance method(HBM) is utilized to solve coupled ordinary differential equations(ODEs)of truncated systems with nonlinearity. To validate the convergence and accuracy of the proposed Dirac method, the results of typical examples, involving nonlinearly restricted boundaries, moment excitation, and displacement excitation, are compared with those of the differential quadrature element method(DQEM). The results demonstrate that when dealing with nonlinear boundaries, the Dirac method exhibits more excellent accuracy and convergence compared with the DQEM. However, when facing displacement excitation, there exist some discrepancies between the proposed approach and simulations;nevertheless, the proposed method still accurately predicts resonant frequencies while being uniquely capable of handling nonuniform displacement excitations. Overall, this methodology offers a convenient way for addressing nonlinear and non-homogenous plate boundaries.

Dirac Operators on Quadratic Lie Superalgebras

Assume thatτis a finite dimensional complex Lie superalgebra with a non-degenerate super-symmetric invariant bilinear form,p is a finite dimensional complex super vector space with a nondegenerate super-symmetric bilinear form,and v:τ→osp(p)is a homomorphism of Lie superalgebras.In this paper,we give a necessary and sufficient condition forτ■p to be a quadratic Lie superalgebra.Then,we define the cubic Dirac operator D(g,τ)on g and give a formula of(D(g,τ))^(2).Finally,we get the Vogan’s conjecture for quadratic Lie superalgebras by D(g,τ).

Ambarzumyan Theorems for Dirac Operators

We consider the inverse eigenvalue problems for stationary Dirac systems with differentiable selfadjoint matrix potential.The theorem of Ambarzumyan for a Sturm-Liouville problem is extended to Dirac operators,which are subject to separation boundary conditions or periodic(semi-periodic)boundary conditions,and some analogs of Ambarzumyan's theorem are obtained.The proof is based on the existence and extremal properties of the smallest eigenvalue of corresponding vectorial Sturm-Liouville operators,which are the second power of Dirac operators.

LYAPUNOV EXPONENT AND ROTATION NUMBER FOR STOCHASTIC DIRAC OPERATORS

In this paper, we consider the stochastic Dirac operatoron a polish space (Ω,β, P). The relation between the Lyapunov index, rotation number andthe spectrum of Lis discussed. The existence of the Lyapunov index and rotation number isshown. By using the W-T functions and W-function we prove the theorems for Las in Kotani[1], [2] for Schrodinger operatorB, and in Johnson [5] for Dirac operators on compact space.

Semi-Commutative Differential Operators Associated with the Dirac Opetator and Darboux Transformation

In the present paper, the semi-commutative differential oparators associated with the 1-dimensional Dirac operator are constructed. Using this results, the hierarchy of the mKdV (-) polynomials are expressed in terms of the KdV polynomials. These formulas give a new interpretation of the classical Darboux transformation and the Miura transformation. Moreover, the recursion operator associated with the hierarchy of the mKdV (-) polynomials is constructed by the algebraic method.

QUANTIZATION COMMUTES WITH REDUCTION,A SURVEY

We review the themes relating to the proposition that“quantization commutes with reduction”([Q,R]=0),from symplectic manifolds to Cauchy-Riemann manifolds.

ON SELF-ADJOINTNESS OF SINGULAR DIRAC OPERATORS

In this paper, applying the method of , we give the complete description of self adjointness of singular Dirac operators, their deficiency indices are supposed to be (2.2) and (1.1), respectively.

Spectral flow, Llarull's rigidity theorem in odd dimensions and its generalization

For a compact spin Riemannian manifold(M,g^(TM))of dimension n such that the associated scalar curvature kT M verifies that k^(TM)≥n(n-1),Llarull’s rigidity theorem says that any area-decreasing smooth map f from M to the unit sphere Sn of nonzero degree is an isometry.In this paper,we present a new proof of Llarull’s rigidity theorem in odd dimensions via a spectral flow argument.This approach also works for a generalization of Llarull’s theorem when the sphere Sn is replaced by an arbitrary smooth strictly convex closed hypersurface in Rn+1.The results answer two questions by Gromov(2023).

Dirac Cohomology and Character Lifting

The endoscopic transfer factor is expressed as difference of characters for the even and odd parts of the spin modules,or Dirac index of the trivial representation.The lifting of tempered characters in terms of index of Dirac cohomology is calculated explicitly.

Continuity in Weak Topology: First Order Linear Systems of ODE

In this paper we study important quantities defined from solutions of first order linear systems of ordinary differential equations. It will be proved that many quantities, such as solutions, eigenvalues of one-dimensional Dirac operators, Lyapunov exponents and rotation numbers, depend on the coefficients in a very strong way. That is, they are not only continuous in coefficients with respect to the usual L^p topologies, but also with respect to the weak topologies of the Lp spaces. The continuity results of this paper are a basis to study these quantities in a quantitative way.

Integral Representation Formulas Related to the Lamé–Navier System

The paper provides integral representations for solutions to a certain first order partial differential equation natural arising in the factorization of the Lamé–Navier system with the help of Clifford analysis techniques.These representations look like in spirit to the Borel–Pompeiu and Cauchy integral formulas both in three and higher dimensional setting.