On Dirac Systems With Multi-Eigenparameter-Dependent Transmission Conditions
Abstract
In this work, we investigate a Dirac system which has discontinuities at finite interior points and contains eigenparameter in both boundary and transmission conditions.
By defining a suitable Hilbert space $\mathfrak{H}$ associated with the problem, we generate a self-adjoint operator $\mathcal{T}$ such that the eigenvalues of the considered problem coincide with those of $\mathcal{T}$. We construct the fundamental system of solutions of the problem and get the asymptotic formulas for the fundamental solutions, eigenvalues and eigen-vector-functions. Also, we examine the asymptotic behaviour for the norm of eigenvectors corresponding to the operator $\mathcal{T}$. We construct Green's matrix, and derive the resolvent of the operator $\mathcal{T}$ in terms of Green's matrix. Finally, we estimate the norm of resolvent of the operator $\mathcal{T}$. In the special case, when our problem has no eigenparameter in both boundary and transmission conditions, the obtained results coincide with the corresponding
results in Tharwat (Boundary Value Problems, DOI:10.1186/s13661-015-0515-1, 2016).
By defining a suitable Hilbert space $\mathfrak{H}$ associated with the problem, we generate a self-adjoint operator $\mathcal{T}$ such that the eigenvalues of the considered problem coincide with those of $\mathcal{T}$. We construct the fundamental system of solutions of the problem and get the asymptotic formulas for the fundamental solutions, eigenvalues and eigen-vector-functions. Also, we examine the asymptotic behaviour for the norm of eigenvectors corresponding to the operator $\mathcal{T}$. We construct Green's matrix, and derive the resolvent of the operator $\mathcal{T}$ in terms of Green's matrix. Finally, we estimate the norm of resolvent of the operator $\mathcal{T}$. In the special case, when our problem has no eigenparameter in both boundary and transmission conditions, the obtained results coincide with the corresponding
results in Tharwat (Boundary Value Problems, DOI:10.1186/s13661-015-0515-1, 2016).
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