Discrete spectrum of electromagnetic Dirac operators

We consider the Dirac operators with electromagnetic fields on 2-dimensional Euclidean space. We offer the sufficient conditions for electromagnetic fields that the associated Dirac operator has only discrete spectrum.

     

On the spectrum of the two-dimensional periodic Dirac operator

We prove the absolute continuity of the Dirac operator spectrum inR ² with the scalar potential V and the vector potential A=(A₁, A₂) being periodic functions (with a common period lattice) such that V, A_j≠L _loc ^q (R ²), q>2. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 1, pp. 3–14, January, 1999.     

Eigenvalues of the Dirac operator in the continuous spectrum

Functionsp(x) andq(x) for which the Dirac operator $$Dy = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ { - 1} \\ \end{array} } & {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \\ \end{array} } \right)\frac{{dy}}{{dx}} + \left( {\begin{array}{*{20}c} {p(x) q(x)} \\ {q(x) - p(x)} \\ \end{array} } \right)y = \lambda y, y = \left( {\begin{array}{*{20}c} {y_1 } \\ {y_2 } \\ \end{array} } \right), y_1 (0) = 0,$$ has a countable number of eigenvalues in the continuous spectrum are constructed.     

Asymptotics of the discrete spectrum of a perturbed Hill operator

One considers a perturbed one-dimensional Hill operator. One gives a formula for the asymptotics of the discrete spectrum in the gap of the continuous spectrum and conditions under which this formula is valid.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 188–189, 1985.In conclusion the author expresses his deep gratitude to his scientific adviser M. Sh. Birman for the formulation of the problem and for his constant interest in this note.     

On the discrete spectrum of a perturbed Wiener-Hopf integral operator

Translated from Matematicheskie Zametki, Vol. 51, No. 1, pp. 102–113, January, 1992.     

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

One considers the one-dimensional Dirac operator with a slowly oscillating potential (1) $$H = \left( {\begin{array}{*{20}c} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right)\frac{d}{{dx}} + q\left( {\begin{array}{*{20}c} {\cos z(x)} & {\sin z(x)} \\ {\sin z(x)} & { - \cos z(x)} \\ \end{array} } \right)_, x \in ( - \infty ,\infty ),q - const,$$ 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.     

Discrete spectrum in the gaps of a perturbed pseudorelativistic hamiltonian

The pseudorelativistic Hamiltonian is considered under wide conditions on potentials A(x), W(x). It is assumed that a real point λ is regular for G_1/2. Let G_1/2(α)=G_1/2−αV, where α>0, V(x)≥0, and V ∈L _d(ℝ^d). Denote by N(λ, α) the number of eigenvalues of G_1/2(t) that cross the point λ as t increases from 0 to α. A Weyl-type asymptotics is obtained for N(λ, α) as α→∞. Bibliography: 5 titles. To O. A. Ladyzhenskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997. pp. 102–117. Translated by A. B. Pushnitskii.     

The point spectrum of the Dirac operator on noncompact symmetric spaces

In this note, we consider the Dirac operator on a Riemannian symmetric space  of noncompact type. Using representation theory, we show that has point spectrum iff the -genus of its compact dual does not vanish. In this case, if  is irreducible, then with  odd, and  .

     

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 3$$ " align="middle" border="0"> , >3, is absolutely continuous.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 1, pp. 3–22, April, 1995.     

Discrete spectrum and principal functions of non-selfadjoint differential operator

In this article, we consider the operator L defined by the differential expression in L ₂(–, ), where q is a complex valued function. Discussing the spectrum, we prove that L has a finite number of eigenvalues and spectral singularities, if the condition holds. Later we investigate the properties of the principal functions corresponding to the eigenvalues and the spectral singularities.     

Discrete spectrum of a model operator in Fock space

We consider a model describing a “truncated” operator (truncated with respect to the number of particles) acting in the direct sum of zero-, one-, and two-particle subspaces of a Fock space. Under some natural conditions on the parameters specifying the model, we prove that the discrete spectrum is finite. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 518–527, September, 2007.     

The discrete spectrum in the gaps of a second-order perturbed periodic operator

Leningrad State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 2, pp. 89–92, April–June, 1991.     

On the discrete spectrum of a perturbed periodic Schrödinger operator

A perturbed periodic Schrödinger operator is considered. Conditions on the perturbation are obtained, under which the discrete spectrum in the gap is finite or, on the contrary, is infinite. In the latter case there are given formulas for the asymptotic behavior of the eigenvalues with respect to the index and conditions under which these formulas hold. The results are formulated in terms of a model problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 157–162, 1991.In conclusion the author expresses his gratitude to his scientific adviser M. Sh. Birman.     

Comparison of perturbed Dirac operators

This paper extends the index theory of perturbed Dirac operators to a collection of noncompact even-dimensional manifolds that includes both complete and incomplete examples. The index formulas are topological in nature. They can involve a compactly supported standard index form as well as a form associated with a Toeplitz pairing on a hypersurface.