We apply the generalized Jordan sets techniques to reduce partial differential-operator equations with the Fredholm operator in the main expression to regular problems. In addition this techniques has been exploited to prove a theorem of existence and uniqueness of a singular initial problem, as well as to construct the left and right regularizators of singular operators in Banach spaces and to...

The aim of the book is to provide an analysis of the boundary element method for the numerical solution of Laplacian eigenvalue problems. The representation of Laplacian eigenvalue problems in the form of boundary integral equations leads to nonlinear eigenvalue problems for related boundary integral operators. The solution of boundary element discretizations of such eigenvalue problems require...

Abstract We analyse the global convergence of a Whittaker-type iterative method and obtain restricted domains, so that we can locate separate solutions Fredholm–Nemytskii nonlinear integral equations by means balls. For this, use two techniques, one based on well-known fixed point theorem other system recurrence relations. In both operator involved auxiliary functions.

and this index is constant on continuous families of such L. Suppose that D is an elliptic operator of order m on sections of a vector bundle E± over a smooth, compact manifold M . Let H (Γ (M,E±)) denote the Sobolev s-norm completion of the space of sections Γ (M,E), with respect to a chosen metric. Then D can be extended to be a bounded linear operator Ds : H s (Γ (M,E)) → Hs−m (Γ (M,E−)) tha...

It is well known that a linear operator equation of the first kind, with an operator having nonclosed range, is ill-posed, that is, the solution depends discontinuously on the data. Tikhonov's method for approximating the solution depends on the choice of a positive parameter which effects a trade-off between fidelity and regularity in the approximate solution. If the parameter is chosen accord...

An equality between the spectral flow of a family $A$ self-adjoint Fredholm operators and index associated differential operator $\frac{d}{dt}-iA$ with Atiyah-Patodi-Singer-style boundary conditions is shown. This generalizes result by C.B\"ar A. Strohmaier used to show their Lorentzian version Atiyah-Patodi-Singer Index Theorem.

Short-ranged and line-gapped non-Hermitian Hamiltonians have strong topological invariants given by an index of associated Fredholm operator. It is shown how these can be accessed via the signature a suitable spectral localizer. This numerical technique implemented in example with relevance to design photonic systems, such as lasers.

We consider eigenvalue problems for dielectric cylindrical scatterers of arbitrary cross with generalized conditions at infinity that enable one to take into account complex eigenvalues. The existence resonance (scattering) frequencies associated these eigenvalues is proved. technique involves the determination characteristic numbers (CNs) Fredholm operator-valued functions constructed using Gr...

An elliptic Fourier integral operator of order 0, associated to a homogeneous canonical diffeomorphism, on a compact manifold is Fredholm on L2. The index may be expressed as the sum of a term, which we call the contact degree, associated to the canonical diffeomorphism and a term, computable by the Atiyah-Singer theorem, associated to the symbol. The contact degree is shown to be defined for a...