Self-adjoint Dirac Operators Research Articles

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.

On the Dirac operator for a test electron in a Reissner–Weyl–Nordström black hole spacetime

The present paper studies the Dirac Hamiltonian of a test electron with a domain of bi-spinor wave functions supported on the static region inside the Cauchy horizon of the subextremal RWN black hole spacetime, respectively inside the event horizon of the extremal RWN black hole spacetime. It is found that this Dirac Hamiltonian is not essentially self-adjoint, yet has infinitely many self-adjoint extensions. Including a sufficiently large anomalous magnetic moment interaction in the Dirac Hamiltonian restores essential self-adjointness; the empirical value of the electron's anomalous magnetic moment is large enough. The spectrum of the subextremal self-adjoint Dirac operator with anomalous magnetic moment is purely absolutely continuous and consists of the whole real line; in particular, there are no eigenvalues. The same is true for the spectrum of any self-adjoint extension of the Dirac operator without anomalous magnetic moment interaction, in the subextremal black hole context. In the extremal black hole sector the point spectrum, if non-empty, consists of a single eigenvalue, which is identified.

Poisson wave trace formula for Dirac resonances at spectrum edges and applications

We study the self-adjoint Dirac operators D = D0 + V (x), where D0 is the free three-dimensional Dirac operator and V (x) is a smooth compactly supported Hermitian matrix potential. We define resonances of D as poles of the meromorphic continuation of its cut-off resolvent. By analyzing the resolvent behaviour at the spectrum edges ±m, we establish a generalized Birman-Krein formula, taking into account possible resonances at ±m. As an application of the new Birman-Krein formula we establish the Poisson wave trace formula in its full generality. The Poisson wave trace formula links the resonances with the trace of the difference of the wave groups. The Poisson wave trace formula, in conjunction with asymptotics of the scattering phase, allows us to prove that, under certain natural assumptions on V , the perturbed Dirac operator has infinitely many resonances; a result similar in nature to Melrose’s classic 1995 result for Schr¨odinger operators.

Self-Adjoint Dirac Operators on Domains in mathbb {R}^3

In this paper, the spectral and scattering properties of a family of self-adjoint Dirac operators in L^2(Omega ; mathbb {C}^4), where Omega subset mathbb {R}^3 is either a bounded or an unbounded domain with a compact C^2-smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with boundary conditions as of Robin type. Our approach is based on abstract boundary triple techniques from extension theory of symmetric operators and a thorough study of certain classes of (boundary) integral operators, that appear in a Krein-type resolvent formula. The analysis of the perturbation term in this formula leads to a description of the spectrum and a Birman–Schwinger principle, a qualitative understanding of the scattering properties in the case that Omega is an exterior domain, and corresponding trace formulas.

Poisson wave trace formula for perturbed Dirac operator

We consider self-adjoint Dirac operators D = D0 + V(x), where D0 is the free three-dimensional Dirac operator and V(x) is a smooth compactly supported Hermitian matrix. We define resonances of D as poles of the meromorphic continuation of its cut-off resolvent. An upper bound on the number of resonances in disks, an estimate on the scattering determinant and the Lifshits-Krein trace formula then leads to a global Poisson wave trace formula for resonances of D.

Isospectral Dirac operators

We give the description of self-adjoint regular Dirac operators, on $[0, \pi]$, with the same spectra.

Dirac spectral flow on contact three manifolds I: Eigensection estimates and spectral asymmetry

Let Y be a compact, oriented 3-manifold with a contact form a and a metric ds. Suppose that F → Y is a principal bundle with structure group U(2) = SU(2) ×{±1} S such that F/S is the principal SO(3) bundle of orthonormal frames for TY . A unitary connection A0 on the Hermitian line bundle F ×det U(2) C determines a self-adjoint Dirac operator D0 on the C-bundle F ×U(2) C. The contact form a can be used to perturb the connection A0 by A0− ira. This associates a one parameter family of Dirac operators Dr for r ≥ 0. When r >> 1, we establish a sharp supnorm estimate on the eigensections of Dr with small eigenvalues. The sup-norm estimate can be applied to study the asymptotic behavior of the spectral flow from D0 to Dr. In particular, it implies that the subleading order term of the spectral flow is strictly smaller than O(r 3 2 ). We also relate the η-invariant of Dr to certain spectral asymmetry function involving only the small eigenvalues of Dr.

Regularized trace of the Dirac operator

Formulas for the regularized trace of the one-dimensional non—self-adjoint Dirac operator with L2-potential are obtained. The cases of periodic and antiperiodic boundary conditions as well as of the Dirichlet boundary conditions are considered. The formulas are obtained by using the method of similar operators on the basis of results from the papers [1] and [2].

Inverse Spectral Problems for Dirac Operators with Summable Matrix-Valued Potentials

We solve the direct and inverse spectral problems for self-adjoint Dirac operators on (0,1) with matrix-valued potentials whose entries belong to \({L_p(0,1), p \in [1,\infty)}\). We give a complete description of the spectral data (eigenvalues and suitably introduced norming matrices) for the operators under consideration, show that the spectral data determine the operator uniquely and suggest how to find the operator given the spectral data.

Dirac operators on cobordisms: Degenerations and surgery

We investigate the Dolbeault operator on a pair of pants, i.e., an elementary cobordism between a circle and the disjoint union of two circles. This operator induces a canonical selfadjoint Dirac operator Dt on each regular level set Ct of a fixed Morse function defining this cobordism. We show that as we approach the critical level set C0 from above and from below these operators converge in the gap topology to (different) selfadjoint operators D± that we describe explicitly. We also relate the Atiyah-Patodi-Singer index of the Dolbeault operator on the cobordism to the spectral flows of the operators Dt on the complement of C0 and the Kashiwara-Wall index of a triplet of finite dimensional lagrangian spaces canonically determined by C0. CONTENTS Introduction 1 1. A model degeneration 4 2. The Atiyah-Patodi-Singer theorem 12 3. Dolbeault operators on two-dimensional cobordisms 18 4. The Kashiwara-Wall index 26 References 31

On the Dirac and Pauli Operators with Several Aharonov–Bohm Solenoids

We study the self-adjoint Pauli operators that can be realized as the square of a self-adjoint Dirac operator and correspond to a magnetic field consisting of a finite number of Aharonov-Bohm solenoids and a regular part, and prove an Aharonov-Casher type formula for the number of zero-modes for these operators. We also see that essentially only one of the Pauli operators are spin-flip invariant, and this operator does not have any zero-modes.

Asymptotic morphisms, K-homology and Dirac operators

Using asymptotic morphisms between graded C*-algebras, we construct for every open m-dimensional spin manifold M a fundamental class in the m-th analytic K-homology group of M. This class is associated to the not necessarily essentially self-adjoint Dirac operator on M. A careful treatment is given of the main properties of these fundamental K-homology classes.

Eigenfunction Expansion for the Three-Dimensional Dirac Operator

We consider any selfadjoint Dirac operator with discrete spectrum defined on a three-dimensional domain. We show the local uniform convergence of the corresponding eigenfunction expansion if the function expanded is sufficiently smooth. Then we derive the Riemann's localization principle for these expansions.

Dirac Operators on Hermitian Spin Surfaces

We prove the conformal invariance of the dimension of thekernel of any of the self-adjoint Dirac operators associated to thecanonical Hermitian connections on Hermitian spin surface. In the caseof a surface of nonnegative conformal scalar curvature we estimate thefirst eigenvalue of the self-adjoint Dirac operator associated to theChern connection and list the surfaces on which its kernel isnontrivial.

Continuity and analyticity of families of self-adjoint Dirac operators on a manifold with boundary

Continuity and analyticity of families of self-adjoint Dirac operators on a manifold with boundary

Commutation properties of anticommuting self-adjoint operators, spin representation and Dirac operators

LetA andB be anticommuting self-adjoint operators in a Hilbert space ℋ. It is proven thatiAB is essentially self-adjoint on a suitable domain and its closureC(A, B) anticommutes withA andB. LetU s be the partial isometry associated with the self-adjoint operatorsS, i.e., the partial isometry defined by the polar decompositionS=U S |S|. LetP S be the orthogonal projection onto (KerS)⊥. Then the following are proven: (i) The operatorsU A ,U B ,U C(A,B) ,P A ,P B , andP A P B multiplied by some constants satisfy a set of commutation relations, which may be regarded as an extension of that satisfied by the standard basis of the Lie algebra\(\mathfrak{s}\mathfrak{u}(2,\mathbb{C})\) of the special unitary groupSU(2); (ii) There exists a Lie algebra\(\mathfrak{M}\) associated with those operators; (iii) If ℋ is separable andA andB are injective, then\(\mathfrak{M}\) gives a completely reducible representation of\(\mathfrak{s}\mathfrak{u}(2,\mathbb{C})\) with each irreducible component being the spin representation of the Clifford algebra associated with ℝ3; This result can be extended to the case whereA andB are not necessarily injective. Moreover, some properties ofA+B are discussed. The abstract results are applied to Dirac operators.

Some essentially self-adjoint Dirac operators with spherically symmetric potentials

It is shown that the one electron Dirac operator in a stationary electric field is essentially self-adjoint, on the domain of infinitely differentiable functions of compact support, for a class of spherically symmetric potentials including the Coulomb potential, for atomic numbers less than or equal to 118. In addition, the domain of the closure of the perturbed operator is the same as the domain of the closure of the unperturbed operator. We also give an abstract theorem on domain-preserving essential self-adjointness for perturbed operators, which is perhaps of independent interest.

Some essentially self-adjoint one-electron dirac operators

A class of smooth functions is introduced and it is shown that, the one-electron Dirac operator corresponding to an element with atomic number less than 102, is essentially self-adjoint on this class of functions. The proof makes essential use of the fact that each of these operators admits a complete family of reducing subspaces. At the same time we use an explicit formula for the resolvent kernel of the part of these operators over such a reducing subspace.