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.

     

Relative oscillation theory for Sturm-Liouville operators extended

We extend relative oscillation theory to the case of Sturm-Liouville operators Hu=r⁻¹(−^′(pu^′)+qu) with different p's. We show that the weighted number of zeros of Wronskians of certain solutions equals the value of Krein's spectral shift function inside essential spectral gaps.     

Inverse Nodal Problem for Dirac System with Spectral Parameter in Boundary Conditions

An inverse nodal problem lies in constructing operators from the given zeros of their eigenfunctions. In this work, we deal with an inverse nodal problem of reconstructing the Dirac system with the spectral parameter in the boundary conditions. We prove that a set of nodal points of one of the components of the eigenfunctions uniquely determines all the parameters of the boundary conditions and the coefficients of the Dirac equations. We also provide a constructive procedure for solving this inverse nodal problem.     

Renormalized oscillation theory for Dirac operators

Oscillation theory for one-dimensional Dirac operators with separated boundary conditions is investigated. Our main theorem reads: If and if solve the Dirac equation , (in the weak sense) and respectively satisfy the boundary condition on the left/right, then the dimension of the spectral projection equals the number of zeros of the Wronskian of and . As an application we establish finiteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Dirac operators.

     

Complex geometry and real geometry

There is a well-known way to generalize the Riemann-Roch operator for Kahler manifold to that for Hermitian manifold. In this paper we show a slightly different way to get a generalized Riemann-Roch operator, which is just the Dirac operator. The difference between the two operators is that the latter one enables the so-called Pythagoras equalities.     

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.     

Fully nonlinear elliptic equations and the dini condition

Abstract

We investigate transmission problems with strongly Lipschitz interfaces for the Dirac equation by establishing spectral estimates on an associated boundary singular integral operator, the rotation operator. Using Rellich estimates we obtain angular spectral estimates on both the essential and full spectrum for general bi-oblique transmission problems. Specializing to the normal transmission problem, we investigate transmission problems for Maxwell's equations using a nilpotent exterior/interior derivativeoperator. The fundamental commutation properties for this operator with the two basic reflection operators are proved. We show how the L ₂spectral estimates are inherited for the domain of the exterior/interior derivative operator and prove some complementary eigenvalue estimates. Finally we use a general algebraic theorem to prove a regularity property needed for Maxwell's equations.     

Complex geometry and real geometry

There is a well-known way to generalize the Riemann-Roch operator for Kähler manifold to that for Hermitian manifold. In this paper we show a slightly different way to get a generalized Riemann-Roch operator, which is just the Dirac operator. The difference between the two operators is that the latter one enables the so-called Pythagoras equalities.     

Embedded eigenvalues for perturbed periodic Jacobi operators using a geometric approach

We consider the problem of embedding eigenvalues into the essential spectrum of periodic Jacobi operators, using an oscillating, decreasing potential. To do this we employ a geometric method, previously used to embed eigenvalues into the essential spectrum of the discrete Schrödinger operator. For periodic Jacobi operators we relax the rational dependence conditions on the values of the quasi-momenta from this previous work. We then explore conditions that permit not just the existence of infinitely many subordinate solutions to the formal spectral equation but also the embedding of infinitely many eigenvalues.     

On the double commutation method

We provide a complete spectral characterization of the double commutation method for general Sturm-Liouville operators which inserts any finite number of prescribed eigenvalues into spectral gaps of a given background operator. Moreover, we explicitly determine the transformation operator which links the background operator to its doubly commuted version (resulting in extensions and considerably simplified proofs of spectral results even for the special case of Schrödinger-type operators).

     

Reconstruction of the Dirac Operator From Nodal Data

Inverse nodal problems consist in constructing operators from the given nodes (zeros) of their eigenfunctions. In this work, we deal with the inverse nodal problem of reconstructing the Dirac operator on a finite interval. We prove that a dense subset of nodal points uniquely determine the parameters of the boundary conditions, the mass of a particle and the potential function of the Dirac system. We also provide a constructive procedure for the solution of the inverse nodal problem.     

A Cauchy kernel for the Hermitian submonogenic system

Hermitian monogenic functions are the null solutions of two complex Dirac type operators. The system of these complex Dirac operators is overdetermined and may be reduced to constraints for the Cauchy datum together with what we called the Hermitian submonogenic system (see [8], [9]). This last system is no longer overdetermined and it has properties that are similar to those of the standard Dirac operator in Euclidean space, such as a Cauchy–Kowalevski extension theorem and Vekua type solutions. In this paper, we investigate plane wave solutions of the Hermitian submonogenic system, leading to the construction of a Cauchy kernel. We also establish a Stokes type formula that, when applied to the Cauchy kernel provides an integral representation formula for Hermitian submonogenic functions.     

A proximal point method for the sum of maximal monotone operators

In this paper, we concentrate on the maximal inclusion problem of locating the zeros of the sum of maximal monotone operators in the framework of proximal point method. Such problems arise widely in several applied mathematical fields such as signal and image processing. We define two new maximal monotone operators and characterize the solutions of the considered problem via the zeros of the new operators. The maximal monotonicity and resolvent of both of the defined operators are proved and calculated, respectively. The traditional proximal point algorithm can be therefore applied to the considered maximal inclusion problem, and the convergence is ensured. Furthermore, by exploring the relationship between the proposed method and the generalized forward‐backward splitting algorithm, we point out that this algorithm is essentially the proximal point algorithm when the operator corresponding to the forward step is the zero operator. Copyright © 2013 John Wiley & Sons, Ltd.     

Properties of eigenvalues and spectral singularities for impulsive quadratic pencil of difference operators

In this paper, we investigate the spectral analysis of impulsive quadratic pencil of difference operators. We first present a boundary value problem consisting one interior impulsive point on the whole axis corresponding to the above mentioned operator. After introducing the solutions of impulsive quadratic pencil of difference equation, we obtain the asymptotic equation of the function related to the Wronskian of these solutions to be helpful for further works, then we determine resolvent operator and continuous spectrum. Finally, we provide sufficient conditions guarenteeing finiteness of eigenvalues and spectral singularities by means of uniqueness theorems of analytic functions. The main aim of this paper is demonstrating the impulsive quadratic pencil of difference operator is of finite number of eigenvalues and spectral singularities with finite multiplicities which is an uninvestigated problem proposed in the literature.     

Linearized quantum and relativistic Fokker–Planck–Landau equations

After a recent work on spectral properties and dispersion relations of the linearized classical Fokker–Planck–Landau operator [8], we establish in this paper analogous results for two more realistic collision operators: The first one is the Fokker–Planck–Landau collision operator obtained by relativistic calculations of binary interactions, and the second is a collision operator (of Fokker–Planck–Landau type) derived from the Boltzmann operator in which quantum effects have been taken into account. We apply Sobolev–Poincaré inequalities to establish the spectral gap of the linearized operators. Furthermore, the present study permits the precise knowledge of the behaviour of these linear Fokker–Planck–Landau operators including the transport part. Relations between the eigenvalues of these operators and the Fourier‐space variable in a neighbourhood of 0 are then investigated. This study is a first natural step when one looks for solutions near equilibrium and their hydrodynamic limit for the full non‐linear problem in all space in the spirit of several works [3, 6, 20, 2] on the non‐linear Boltzmann equation. Copyright © 2000 John Wiley & Sons, Ltd.     

Essential spectra of spectral operators

The various essential spectra of a linear operator have been surveyed byB. Gramsch andD. Lay [4]. In this paper we characterize the essential spectra and the related quantities nullity, defect, ascent and descent of bounded spectral operators. It is shown that a number of these spectra coincide in the case of a spectral or a scalar type operator. Some results known for normal operators in Hilbert space are extended to spectral operators in Banach space.     

Nonlinear impulsive systems on infinite dimensional spaces

In this paper we consider two different classes of nonlinear impulsive systems one driven purely by Dirac measures at a fixed set of points and the second driven by signed measures. The later class is easily extended to systems driven by general vector measures. The principal nonlinear operator is monotone hemicontinuous and coercive with respect to certain triple of Banach spaces called Gelfand triple. The other nonlinear operators are more regular, non-monotone continuous operators with respect to suitable Banach spaces. We present here a new result on compact embedding of the space of vector-valued functions of bounded variation and then use this result to prove two new results on existence and regularity properties of solutions for impulsive systems described above. The new embedding result covers the well-known embedding result due to Aubin.     

On an inverse scattering problem for a class of Dirac operators with spectral parameter in the boundary condition

In this work, we consider the inverse scattering problem for a class of one dimensional Dirac operators on the semi-infinite interval with the boundary condition depending polynomially on a spectral parameter. The scattering data of the given problem is defined and its properties are examined. The main equation is derived, its solvability is proved and it is shown that the potential is uniquely recovered in terms of the scattering data. A generalization of the Marchenko method is given for a class of Dirac operator.     

Joint seminormality and Dirac operators

In this article, an approach to joint seminormality based on the theory of Dirac and Laplace operators on Dirac vector bundles is presented. To eachn-tuple of bounded linear operators on a complex Hilbert space we first associate a Dirac bundle furnished with a metric-preserving linear connection defined in terms of thatn-tuple. Employing standard spin geometry techniques we next get a Bochner type and two Bochner-Kodaira type identities in multivariable operator theory. Further, four different classes of jointly seminormal tuples are introduced by imposing semidefiniteness conditions on the remainders in the corresponding Bochner-Kodaira identities. Thus we create a setting in which the classical Bochner's method can be put into action. In effect, we derive some vanishing theorems regarding various spectral sets associated with commuting tuples. In the last part of this article we investigate a rather general concept of seminormality for self-adjoint tuples with an even or odd number of entries.     

An unusual self-adjoint linear partial differential operator

In an American Mathematical Society Memoir, published in 2003, the authors Everitt and Markus apply their prior theory of symplectic algebra to the study of symmetric linear partial differential expressions, and the generation of self-adjoint differential operators in Sobolev Hilbert spaces. In the case when the differential expression has smooth coefficients on the closure of a bounded open region, in Euclidean space, and when the region has a smooth boundary, this theory leads to the construction of certain self-adjoint partial differential operators which cannot be defined by applying classical or generalized conditions on the boundary of the open region.

This present paper concerns the spectral properties of one of these unusual self-adjoint operators, sometimes called the ``Harmonic' operator.

The boundary value problems considered in the Memoir (see above) and in this paper are called regular in that the cofficients of the differential expression do not have singularities within or on the boundary of the region; also the region is bounded and has a smooth boundary. Under these and some additional technical conditions it is shown in the Memoir, and emphasized in this present paper, that all the self-adjoint operators considered are explicitly determined on their domains by the partial differential expression; this property makes a remarkable comparison with the case of symmetric ordinary differential expressions.

In the regular ordinary case the spectrum of all the self-adjoint operators is discrete in that it consists of a countable number of eigenvalues with no finite point of accumulation, and each eigenvalue is of finite multiplicity. Thus the essential spectrum of all these operators is empty.

This spectral property extends to the present partial differential case for the classical Dirichlet and Neumann operators but not to the Harmonic operator. It is shown in this paper that the Harmonic operator has an eigenvalue of infinite multiplicity at the origin of the complex spectral plane; thus the essential spectrum of this operator is not empty.

Both the weak and strong formulations of the Harmonic boundary value problem are considered; these two formulations are shown to be equivalent.

In the final section of the paper examples are considered which show that the Harmonic operator, defined by the methods of symplectic algebra, has a domain that cannot be determined by applying either classical or generalized local conditions on the boundary of the region.