Inverse spectral problems for Dirac operator with eigenvalue dependent boundary and jump conditions

We deal with the Dirac operator with eigenvalue dependent boundary and jump conditions. Properties of eigenvalues, eigenfunctions and the resolvent operator are studied. Moreover, uniqueness theorems of the inverse problem according to the Weyl functions and the spectral data (the sets of eigenvalues and norming constants; two different eigenvalues sets) are proved.     

Inverse nodal problem for p–laplacian dirac system

In this study, we solve an inverse nodal problem for p‐Laplacian Dirac system with boundary conditions depending on spectral parameter. Asymptotic formulas of eigenvalues, nodal points and nodal lengths are obtained by using modified Prüfer substitution. The key step is to apply modified Prüfer substitution to derive a detailed asymptotic estimate for eigenvalues. Furthermore, we have shown that the functions r(x) and q(x) in Dirac system can be established uniquely by using nodal parameters with the method used by Wang et al. Obtained results are more general than the classical Dirac system. Copyright © 2016 John Wiley & Sons, Ltd.     

Sinc-based computations of eigenvalues of Dirac systems

The celebrated classical sampling theorem is used to compute approximate values of the eigenvalues of Dirac systems with eigenvalue parameter in the boundary conditions. We deal with problems with an eigenparameter in one or two boundary conditions. The error analysis is established considering both truncation and amplitude errors associated with the sampling theorem. We indicate the role of the amplitude error as well as other parameters in the method via illustrative examples. AMS subject classification (2000)  34L16, 65L15, 94A20     

Determination of Dirac operator with eigenvalue-dependent boundary and jump conditions

Dirac operator with eigenvalue-dependent boundary and jump conditions is studied. Uniqueness theorems of the inverse problems from either Weyl function or the spectral data (the sets of eigenvalues and norming constants except for one eigenvalue and corresponding norming constant; two sets of different eigenvalues except for two eigenvalues) are proved. Finally, we investigate two applications of these theorems and obtain analogues of a theorem of Hochstadt-Lieberman and a theorem of Mochizuki-Trooshin.     

Uniform asymptotics of the eigenvalues and eigenfunctions of the Dirac system with an integrable potential

We consider the Dirac operator on a finite interval with a potential belonging to some set X completely bounded in the space L₁[0, π] and with strongly regular boundary conditions. We derive asymptotic formulas for the eigenvalues and eigenfunctions of the operator; moreover, the constants occurring in the estimates for the remainders depend on the boundary conditions and the set X alone.     

Spectral Problems for the Dirac System with Spectral Parameter in Local Boundary Conditions

We consider a spectral boundary value problem in a 3-dimensional bounded domain for the Dirac system that describes the behavior of a relativistic particle in an electromagnetic field. The spectral parameter is contained in a local boundary condition. We prove that the eigenvalues of the problem have finite multiplicities and two points of accumulation, zero and infinity and indicate the asymptotic behavior of the corresponding series of eigenvalues. We also show the existence of an orthonormal basis on the boundary consisting of two-dimensional parts of the four-dimensional eigenfunctions.     

Absolutely continuous spectra for matrix Dirac systems

This paper investigates the matrix Dirac systems. Under some conditions on the potential matrices, it is shown that the spectrum of the Dirac operator is purely absolutely continuous outside the gaps containing the eigenvalues of q.     

一类Dirac方程特征值问题的迹恒等式

讨论了带有非局部边界条件的一维Dirac方程BdY/dx+P(x)Y=λY的特征值问题,其中首先建立了问题的特征值集合与一个整函数u(λ)零点集合的对应,并对Dirac算子的特征值进行了估计,然后借助于一个积分恒等式,采用留数方法,得到了该问题的特征值的迹恒等式.     

On the Spectrum of the Dirac Operator and the Existence of Discrete Eigenvalues for the Defocusing Nonlinear Schrödinger Equation

We revisit the scattering problem for the defocusing nonlinear Schrödinger equation with constant, nonzero boundary conditions at infinity, i.e., the eigenvalue problem for the Dirac operator with nonzero rest mass. By considering a specific kind of piecewise constant potentials we address and clarify two issues, concerning: (i) the (non)existence of an area theorem relating the presence/absence of discrete eigenvalues to an appropriate measure of the initial condition; and (ii) the existence of a contribution to the asymptotic phase difference of the potential from the continuous spectrum.     

Direct and inverse problems for the Dirac operator with a spectral parameter linearly contained in a boundary condition

We investigate a problem for the Dirac differential operators in the case where an eigenparameter not only appears in the differential equation but is also linearly contained in a boundary condition. We prove uniqueness theorems for the inverse spectral problem with known collection of eigenvalues and normalizing constants or two spectra.     

The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to ± or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic fibers. Using the Hopf fibration we use this fact to compute the Dirac eigenvalues on complex projective space including the multiplicities.Finally, we show that there are 1-parameter families of Riemannian nilmanifolds such that the Laplacian on functions and the Dirac operator for certain spin structures have constant spectrum while the Laplacian on 1-forms and the Dirac operator for the other spin structures have nonconstant spectrum. The marked length spectrum is also constant for these families.     

Eigenvalue Estimates for Non-Selfadjoint Dirac Operators on the Real Line

We show that the non-embedded eigenvalues of the Dirac operator on the real line with complex mass and non-Hermitian potential V lie in the disjoint union of two disks, provided that the L ¹-norm of V is bounded from above by the speed of light times the reduced Planck constant. The result is sharp; moreover, the analogous sharp result for the Schrödinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on V implies the absence of non-real eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials.     

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.     

On the approximation of isolated eigenvalues of ordinary differential operators

We extend a result of Stolz and Weidmann on the approximation of isolated eigenvalues of singular Sturm-Liouville and Dirac operators by the eigenvalues of regular operators.

     

Asymptotic expansions of the zeta-determinants of Dirac Laplacians on a compact manifold with boundary

In this article we discuss the asymptotic expansions of the zeta-determinants of Dirac Laplacians on a compact manifold with boundary when the boundary part is stretched. In [12] the author studied the same question under the assumption of no existence of L² - and extended L² -solutions of Dirac operators when the boundary part is stretched to infinite length. Therefore, the results in this article generalize those in [12]. Using the main results we obtain the formula describing the ratio of two zeta-determinants of Dirac Laplacians with the APS boundary conditions associated with two unitary involutions σ₁ and σ₂ on ker B satisfying Gσ_i = -σ_i G. We also prove the adiabatic decomposition formulas for the zeta-determinants of Dirac Laplacians on a closed manifold when the Dirichlet or the APS boundary condition is imposed on partitioned manifolds, which generalize the results in [10] and [11].     

A Minimax Principle for the Eigenvalues in Spectral Gaps

A minimax principle is derived for the eigenvalues in the spectralgap of a possibly non-semibounded self-adjoint operator. Itallows the nth eigenvalue of the Dirac operator with Coulombpotential from below to be bound by the nth eigenvalue of asemibounded Hamiltonian which is of interest in the contextof stability of matter. As a second application it is shownthat the Dirac operator with suitable non-positive potentialhas at least as many discrete eigenvalues as the Schrödingeroperator with the same potential.     

A boundary-value problem for the stationary vlasov-poisson equations: The plane diode

The stationary Vlasov-Poisson boundary value problem in a spatially one-dimensional domain is studied. The equations describe the flow of electrons in a plane diode. Existence is proved when the boundary condition (the cathode emission distribution) is a bounded function which decays super-linearly or a Dirac mass. Uniqueness is proved for (physically realistic) boundary conditions which are decreasing functions of the velocity variable. It is shown that uniqueness does not always hold for the Dirac mass boundary conditions.     

A mathematical aspect of a tunnel-junction for spintronic qubit

We consider the Dirac particle that lives in the 1-dimensional configuration space consisting of two quantum wires and a junction between the two. We regard the spin of a Dirac particle as spintronic qubit. We give concrete formulae explicitly expressing the one-to-one correspondence between every self-adjoint extension of the minimal Dirac operator and its corresponding boundary condition of the wave functions of the Dirac particle. We then show that all the boundary conditions can be classified into just two types. The two types are characterized by whether the electron passes through the junction or not. We also show how the tunneling produces its own phase factor and what is the relation between the phase factor and the spintronic qubit in the tunneling boundary condition.     

Atiyah—Patodi—Singer type index theorems for manifolds with splitting of $ \eta $-invariants

We construct self-adjoint extensions of Dirac operators on manifolds with corners of codimension 2, which generalize the Atiyah—Patodi—Singer boundary conditions. The boundary conditions are related to geometric constructions, which convert problems on manifolds with corners into problems on manifolds with boundary and wedge singularities. In the case, where the Dirac bundle is a super-bundle, we prove two general index theorems, which differ by the splitting formula for -invariants. Further we work out the de Rham, signature and twisted spin complex in closer detail. Finally we give a new proof of the splitting formula for the -invariant. Submitted: October 1999, Revised version: March 2001.     

Multiplicities and relative position of eigenvalues of a quadratic pencil of Sturm-Liouville operators

For boundary value problems generated by a second-order differential equation with regular nonseparated boundary conditions, criteria for the eigenvalues to be multiple are given and the relative position of the eigenvalues is studied. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 369–381, March, 2000.