Spectrum Of Dirac Operator Research Articles

Deforming the point spectra of one-dimensional Dirac operators

We provide a method of inserting and removing any finite number of prescribed eigenvalues into spectral gaps of a given one-dimensional Dirac operator. This is done in such a way that the original and deformed operators are unitarily equivalent when restricted to the complement of the subspace spanned by the newly inserted eigenvalue. Moreover, the unitary transformation operator which links the original operator to its deformed version is explicitly determined.

Universality of random matrices in the microscopic limit and the Dirac operator spectrum

We prove the universality of correlation functions of chiral unitary and unitary ensembles of random matrices in the microscopic limit. The essence of the proof consists in reducing the three-term recursion relation for the relevant orthogonal polynomials into a Bessel equation governing the local asymptotics around the origin. The possible physical interpretation as the universality of the soft spectrum of the Dirac operator is briefly discussed.

On the spectrum of Dirac operators with the unbounded potential at infinity

On the spectrum of Dirac operators with the unbounded potential at infinity

Remarks on the spectrum of Dirac operators

This paper concerns the number of eigenvalues of the Dirac operator in ]-1,1[ and gives the number of linear independent eigenfunctions with total angular momentum k in [μ, 1[.

Dense point spectrum for the one-dimensional Dirac operator with an electrostatic potential

For the one-dimensional Dirac operator, examples of electrostatic potentials with decay behaviour arbitrarily close to Coulomb decay are constructed for which the operator has a prescribed set of eigenvalues dense in the whole or part of its essential spectrum. A simple proof that the essential spectrum of one-dimensional Dirac operators with electrostatic potentials is never empty is given in the appendix.

DIRAC DESCRIPTION OF ENERGY LEVELS OF POLARONS IN POLYANILINE

In this paper an exact method for determining energy levels of polarons in systems with configuration spaces of the order parameter being the SU(2) group is presented. The base of the method being the evaluation of the spectrum for the Dirac operator on this manifold is shown. Conditions for a substance under which a polaron configuration space is SU(2) are formulated. Satisfaction of these conditions by polyaniline is proved. The representation of the UV-Vis polyaniline spectrum applying the Dirac operator spectrum is given. Applicability of the spectrum to other synthetic metals with SU(2) as the configuration space for the order parameter is discussed.

Resolvent estimates and spectrum of the Dirac operator with periodic potential

Some estimates are given of the norm of the resolvent of the Dirac operator on ann-dimensional torus (n ≥ 2) for complex values of the quasimomentum. It is shown that the spectrum of the periodic Dirac operator with potential $$V \in L_{loc}^\beta (R^3 ),\beta > 3$$ , β>3, is absolutely continuous.

AN O(3) GLOBAL ANOMALY IN 0+1 DIMENSION

We present a simple exactly solvable quantum mechanical example of the global anomaly in an O(3) model with an odd number of fermionic triplets coupled to a gauge field on a circle. Because the fundamental group is non-trivial, π1(O(3))=Z2, fermionic level crossing—circling occurs in the eigenvalue spectrum of the one-dimensional Dirac operator under continuous external field transformations. They are shown to be related to the presence of an odd number of normalizable zero modes in the spectrum of an appropriate two-dimensional Dirac operator. We argue that fermionic degrees of freedom in the presence of an infinitely large external field violate perturbative decoupling.

The level crossing phenomenon with Yukawa interactions

In the minimal model of electroweak interactions we carefully investigate the spectrum of the massive euclidean Dirac operator in three, four and five dimensions ( D) in the presence of topologically non-trivial external fields. More specifically we study the cases of the instanton ( D = 4), sphaleron ( D = 3) as well as that of the ( D = 5) Dirac operator that pertains to the existence of the global SU(2) anomaly. We establish the existence of normalizable massive fermion zero modes in all three cases. We give closed form expressions which relate the massive with the massless zero modes. As a consequence the level crossing phenomenon is shown to be manifest and generic in the presence of Yukawa interactions.

On the essential spectrum of Dirac operators with spherically symmetric potentials

On the essential spectrum of Dirac operators with spherically symmetric potentials

Riemannian foliationin N = 1,D = 11 supergravity

We propose a G-foliated model of the seven-dimensional internal manifoldM7 resulting from compactifying theN = 1,D = 11 model of supergravity of Cremmer, Julia and Scherk. This gives a generalisation of the usualG/H homogeneous-space approach. As a result, there is a natural way of expressing the internal spinor bundle as a graded tensor product. Under coordinate transformations mixing the tangential geometry with the transversal geometry of the foliation, spinors in this bundle with different internal helicities transform differently. We also argue that the ground-state compactifieation of the 11-dimensional manifoldM11 intoM7 ×M4 is the result of a global submersion ofM11 intoM4, giving rise to a foliation of dimension 7 and codimension 4. With these additional structures in place, we show that it is possible to incorporate physical chiral spinors into the original model while retaining the Riemannian geometry. There is also a canonical way of defining a Dirac operator on the internal spinor bundle, so that the mass spectrum of the internal Dirac operator is the sum of a tangential spectrum and a transversal spectrum.

Spectrum of Dirac operator and role of winding number in QCD.

We show that very general considerations based on the properties of the partition function of QCD allow one to extract information about the eigenvalues of the Dirac operator in vacuum gauge fields. In particular, we demonstrate that the familiar suppression of field configurations with a nontrivial topology occurring for small quark masses is a finite size effect which disappears if the four-dimensional volume [ital V] is large enough. The formation of a quark condensate is connected with the occurrence of small eigenvalues of order [lambda][sub [ital n]][proportional to]1/[ital V].

The frequency-dependent convergence properties of some integrals with oscillatory integrands

is divergent; consequently the oscillation in the numerator of the integrand is then necessary for convergence. The principal feature of these problems is the curious manner in which the conditional convergence of the integrals may be affected by the presence in the denominator of the integrand of a bounded oscillatory perturbation. Integrals such as these arise in the analysis of the discrete spectra of Dirac Operators [2]. In the present paper we prove the following theorem which extends some of the earlier results, allowing now for a frequency parameter I in the principal oscillation of the integrand.

Absolutely continuous spectrum of Dirac operators for long-range potentials

Using the resolvent method and the technique of weighted L 2-estimates we deduce the non-existence of the singular continuous spectrum of the Dirac operator τ + P( x) on the interval (1, ∞). We assume that the hermitian matrix potential P( x) = P 1( x) + P 2( x) is divided into a long-range part P 1(x) = O(¦x¦ −ε), [r ∂ rP 1(x)] − = O(¦x¦ −ε) and a short-range part P 2(x) = O(¦x¦ −1 − ε) (¦x¦ → ∞) with local singularities P(x) = O(¦x − a j¦ −1) of Coulomb type for N nuclei a 1,…,a N∈ R 3.

On the point spectrum of Dirac operators

We study the decay of eigenfunctions and we give conditions for the absence of eigenvalues embedded in the continuous spectrum for Dirac Hamiltonians with long range and locally singular potential.

1-forms over the moduli space of irreducible connections defined by the spectrum of Dirac operators

Let M ( P) be the moduli space of irreducible connections of a G-principal bundle P over a closed Riemannian spin manifold M. Let D A be the Dirac operator of M coupled to a connection A of P and f a smooth function on M. We consider a smooth variation A( u) of A with tangent vector ω and denote T ω:= d du ( D A ( u) − f) ( u=0. The coefficients of the asymptotic expansion of trace ( T ω · e - t( D A − f) 2 ) near t=0 define 1-forms a ( k) f , k=0, 1, 2, … on M ( P). In this paper we calculate aa (0) f , a (1) f , a (2) f and study some of their properties. For instance using the 1-form a (2) f for suitable functions f we obtain a foliation of codimension 5 of the space of G-instantons of S 4.

Remnants of the index theorem on the lattice

We investigate the spectrum of lattice Dirac operators in two dimensions in external U(1) gauge fields as well as the properties and renormalization of the quantity m Tr γ 5/(¶ + m) which provides a definition of topological charge based on axial-vector divergence relations.

Dense point spectra of Schr�dinger and Dirac operators

Dense point spectra of Schr�dinger and Dirac operators

On Essential Selfadjointness, Distinguished Selfadjoint Extension and Essential Spectrum of Dirac Operators with Matrix Valued Potentials

(1.2) ajak+akaj=2djkl (/, fe = l, 2, 3, 4). We denote by HQ the operator H with We denote by and | | the usual inner product and norm in C, respectively, and by ( , ) and || || the inner product and norm in the Hilbert space «#=[L2CR )II, respectively. We also denote by | | and || || the operator norm of a 4x4 matrix and a bounded linear operator in «#, respectively. We denote by / the 4x4 identity matrix, which at times implies the 2x2 identity matrix, but no confusion will occur. For a closable operator T in JC, we denote by T its closure. For an (formal) operator T, we denote by T the restriction of T to the domain

A gap in the energy spectrum of the one-dimensional Dirac operator

One considers the one-dimensional Dirac operator with a slowly oscillating potential $$H = \left( {\begin{array}{*{20}c} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right)\frac{d}{{dx}} + q\left( {\begin{array}{*{20}c} {\cos z(x)} & {\sin z(x)} \\ {\sin z(x)} & { - \cos z(x)} \\ \end{array} } \right)_, x \in ( - \infty ,\infty ),q - const,$$ (1) where . The following statement holds. The double absolutely continuous spectrum of the operator (1) fills the intervals (−∞,−¦q¦), (¦q¦, ∞). The interval (−¦q¦, ¦q¦) is free from spectrum. The operator has a simple eigenvalue only for singn C+=sign C−, situated either at the point (under the condition C+>0) or at the point λ=−¦q¦ (under the condition). The proof is based on the investigation of the coordinate asytnptotics of the corresponding equation.