On the Spectral Flow of Families of Dirac Operators with Constant Symbol

We consider families of generalized Dirac operators D_t with constant principal symbol and constant essential spectrum such that the endpoints are gauge equivalent, i.e., D₁ = W*D₀W. The spectral flow un any gap in the essential spectrum we express as the Fredholm index of 1 + (W - 1)P where P is the spectral projection on the interval d, ∞) with respect to D₀ and d is in the gap. We reduce the computation of this index to the Atiyah-Singer index theorem for elliptic pseudodifferential operators. We find an invariant of the Riemannian geometry for odd dimensional spin manifolds estimating the length of gaps in the spectrum of the Dirac operator.     

On Two-Dimensional Finite-Gap Potential Schroedinger and Dirac Operators with Singular Spectral Curves

We describe a wide class of two-dimensional potential Schroedinger and Dirac operators which are finite-gap at the zero energy level and whose spectral curves at this level are singular, in particular may have n-multiple points with n3.     

Local Boundary Value Problems for Dirac Type Operators

To each Dirac type operator there is assigned a formally exact elliptic complex of length 2. We study the Neumann problem for this complex in a bounded domain with smooth boundary in Rⁿ, as it was formulated by Spencer in 1957.     

Graded Differential Equations and Dirac Type Operators

Singularities of solutions of graded differential equations as filtrations are studied. Symbols and the transfer operator associated with the h-adic filtration are computed. The condition that the transfer operator is a differential operator over the characteristic manifold of codimension 1 for this filtration is given. The Dirac operator is introduced by using the De Broglie principle: singularities of wave-like solutions move as material points. Use of the transfer operator associated with the h-adic filtration implies an invariant definition of the Pauli operator. A natural parallel translation arises as quantum analog of the Levi-Civita connection.     

On the Asymptotics of the Spectral Density of Radial Dirac Operators with Divergent Potential

The radial Dirac operator with a potential tending to infinity at infinity and satisfying a mild regularity condition is known to have a purely absolutely continuous spectrum covering the whole real line. Although having two singular end-points in the limit-point case, the operator has a simple spectrum and a generalised Fourier expansion in terms of a single solution. In the present paper, a simple formula for the corresponding spectral density is derived, and it is shown that, under certain conditions on the potential, the spectral function is convex for large values of the spectral parameter. This settles a question considered in earlier work by M. S. P. Eastham and the author.     

Spectral Properties of Classical Dirac Operators and Operators with Involution in Homogeneous Function Spaces

Differential Equations - We present spectrum localization theorems for Dirac operators and operators with involution in a class of function spaces introduced in [Contemp. Math., 2018, vol. 706, pp....     

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.     

Fischer Decomposition for Difference Dirac Operators

We establish the basis of a discrete function theory starting with a Fischer decomposition for difference Dirac operators. Discrete versions of homogeneous polynomials, Euler and Gamma operators are obtained. As a consequence we obtain a Fischer decomposition for the discrete Laplacian.     

To the Spectral Theory of One-Dimensional Matrix Dirac Operators with Point Matrix Interactions

We investigate one-dimensional (2p × 2p)-matrix Dirac operators D_X,α and D_X,β with point matrix interactions on a discrete set X. Several results of [4] are generalized to the case of (p × p)-matrix interactions with p > 1. It is shown that a number of properties of the operators D_X,α and D_X,β (self-adjointness, discreteness of the spectrum, etc.) are identical to the corresponding properties of some Jacobi matrices B_X,α and B_X,β with (p × p)-matrix entries. The relationship found is used to describe these properties as well as conditions of continuity and absolute continuity of the spectra of the operators D_X,α and D_X,β. Also the non-relativistic limit at the velocity of light c → ∞ is studied.     

Sharp Spectral Multipliers for a New Class of Grushin Type Operators

We describe weighted restriction (Plancherel) type estimates and sharp Hebisch-Müller-Stein type spectral multiplier result for a new class of Grushin type operators. We also discuss the optimal exponent for Bochner-Riesz summability in this setting.     

Twisted Higher Spin Dirac Operators

In this paper, we define twisted higher spin Dirac operators and explain how these invariant differential operators can be used to define more general higher spin Dirac operators acting on functions $f({\underline{x}})$ on $\mathbb{R }^m$ which then take values in general half-integer representations for the spin group.     

Scattering Matrix,Phase Shift,Spectral Shift and Trace Formula for One-dimensional Dissipative Schrödinger-type Operators

The paper is devoted to Schr?dinger operators with dissipative boundary conditions on bounded intervals. In the framework of the Lax-Phillips scattering theory the asymptotic behaviour of the phase shift is investigated in detail and its relation to the spectral shift is discussed. In particular, the trace formula and the Birman-Krein formula are verified directly. The results are exploited for dissipative Schr?dinger-Poisson systems. In friendship dedicated to P. Exner on the occasion of his 60^th birthday This work was supported by DFG, Grant 1480/2.     

Discrete Dirac Operators in Clifford Analysis

We develop a constructive framework to define difference approximations of Dirac operators which factorize the discrete Laplacian. This resulting notion of discrete monogenic functions is compared with the notion of discrete holomorphic functions on quad-graphs. In the end Dirac operators on quad-graphs are constructed.