Singular Hahn–Hamiltonian systems

Abstract

In this work, we study a Hahn–Hamiltonian system in the singular case. For this system, the Titchmarsh–Weyl theory is established. In this context, the first part provides a summary of the relevant literature and some necessary fundamental concepts of the Hahn calculus. To pass from the Hahn difference expression to operators, we define the Hilbert space $L_{\omega,q,W} ^{2}((\omega_{0},\infty);\mathbb{C}^{2n})$ in the second part of the work. The corresponding maximal operator $L_{\max}$ are introduced. For the Hahn–Hamiltonian system, we proved Green formula. Then we introduce a regular self-adjoint Hahn–Hamiltonian system. In the third part of the work, we study Titchmarsh-Weyl functions $M(\lambda)$ and circles $\mathcal{C}(a,\lambda)$ for this system. These circles proved to be embedded one to another. The number of square-integrable solutions of the Hahn–Hamilton system is studied. In the fourth part of the work, we obtain boundary conditions in the singular case. Finally, we define a self-adjoint operator in the fifth part of the work.