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.     

Inverse nodal problem for Dirac operator with integral type nonlocal boundary conditions

In this paper, Dirac operator with some integral type nonlocal boundary conditions is studied. We show that the coefficients of the problem can be uniquely determined by a dense set of nodal points. Moreover, we give an algorithm for the reconstruction of some coefficients of the operator.     

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.     

Green and Poisson functions with Wentzell boundary conditions

We discuss the construction and estimates of the Green and Poisson functions associated with a parabolic second order integro-differential operator with Wentzell boundary conditions.     

On the dependence of the reflection operator on boundary conditions for biharmonic functions

The biharmonic equation arises in areas of continuum mechanics including linear elasticity theory and the Stokes flows, as well as in a radar imaging problem. We discuss the reflection formulas for the biharmonic functions u(x,y)∈R² subject to different boundary conditions on a real-analytic curve in the plane. The obtained formulas, generalizing the celebrated Schwarz symmetry principle for harmonic functions, have different structures. In particular, in the special case of the boundary, Γ₀:={y=0}, reflections are point-to-point when the given on Γ₀ conditions are u=n_∂u=0, u=Δu=0 or n_∂u=n_∂Δu=0, and point to a continuous set when u=n_∂Δu=0 or n_∂u=Δu=0 on Γ₀.     

Two-sided estimates for root vector functions of the Dirac operator

We consider the one-dimensional Dirac operator. We derive a shift formula for its root vector functions and prove anti-a priori and two-sided estimates for various L _p -norms of these functions.     

Degenerate boundary conditions for the diffusion operator

We describe all degenerate two-point boundary conditions possible in a homogeneous spectral problem for the diffusion operator. We show that the case in which the characteristic determinant is identically zero is impossible for the nonsymmetric diffusion operator and that the only possible degenerate boundary conditions are the Cauchy conditions. For the symmetric diffusion operator, the characteristic determinant is zero if and only if the boundary conditions are falsely periodic boundary conditions; the characteristic determinant is identically a nonzero constant if and only if the boundary conditions are generalized Cauchy conditions.     

The Chern-Connes character for the Dirac operator on manifolds with boundary

A cyclic cocycle is constructed for the Dirac operator on a compact spin manifold with boundary with the -invariant cochain introduced as the boundary correction term. This cocycle is seen to satisfy certain growth condition weaker than being entire and its pairing with the Chern characters of idempotents as well as the relevant index formulae are studied. The -cochain is a generalization of the Atiyah-Patodi-Singer -invariant and it carries information on the APS -invariants for Dirac operators twisted by bundles. It is also shown that one obtains the transgressed Chern character, defined by Connes and Moscovici, by applying the boundary operatorB in the cyclic bicomplex to the higher components of the -cochain.     

Green function of fourth-order differential operator with eigenparameter-dependent boundary and transmission conditions

We investigate a class of fourth-order regular differential operator with transmission conditions at an interior discontinuous point and the eigenparameter appears not only in the differential equation but also in the boundary conditions. We prove that the operator is symmetric, construct basic solutions of differential equation, and give the corresponding Green function of the operator is given.     

Mean value formula for the root vector functions of the Dirac operator

We consider the one-dimensional Dirac operator on an arbitrary interval and obtain a mean value formula for the root vector functions of this operator.     

Variation of Neumann and Green functions under homotopies of the boundary

We develop variational formulas for certain Neumann and Green functions of multiply connected planar domains, valid for any smooth homotopy of the boundary. The modified Neumann and Green functions under consideration arise in the study of holomorphic functions with single-valued primitives.     

Riesz inequality for systems of root vector functions of the Dirac operator

We establish a criterion for the Riesz property of systems of root vector functions of the one-dimensional Dirac operator.     

On the oscillation of eigenvector functions of the one-dimensional Dirac operator

A spectral problem for the one-dimensional Dirac system is considered. The question of the number of zeros for the components of the eigenvector functions of this problem is studied.     

Half-inverse problem for the Dirac operator

Given the spectrum of the Dirac operator, together with the potential on the half-interval and one boundary condition, this paper provides reconstruction of the potential on the whole interval, and proves the existence conditions of the solution.     

Interpolation and local data for meromorphic matrix and operator functions

The main result of this paper is a generalization of the Mittag-Leffler theorem to matrix and operator valued meromorphic functions. Namely, a meromorphic matrix or operator valued function is constructed when the singular parts of the function and if its inverse are given in all singular points (which are assumed to be isolated). The paper contains also interpolation theorems based on other forms of local data (Jordan chains from left and right of the function and its inverse). An analysis of the local data, which is used in the proofs of these theorems is also included.Dedicated to the memory of D.P. MilmanThe research of this author partially supported by the Fund for Basic Research administrated by the Israel Academy of Science and Humanities.