## Spectral Theory of Schrödinger Type Operator on Spider Type Quantum Graphs

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## Abstract

The dissertation consists of chapter 1: Introduction, this chapter contains some definitions and examples of quantum graphs, symplectic analysis and its representation on spider graph. Chapter 2-Brownian motion on the spider like quantum graph, this chapter contains the definition of Brownian motion on the N-legged spider graph with infinite legs and Kirchhoff's gluing conditions at the origin and calculation ofthe transition probability of this process. In addition, we study several important Markov moments, for instance the first exit time $\tau_{L}$ from the spider with the length L of all legs. The calculations give not only the moments of $\tau_{L}$ but also the distribution density for $\tau_L$. All results of this section are new ones. Chapter 3- A brief review on the classical spectral theory. This chapter contains the elements of the spectral theory on spider graph. We start from the classical Strum-Liouville theory on the full line $\mathbb{R}^1$ (for the case of the bounded from below potential) and explain how this theory can be generalized to the case of canonical system in $\mathbb{R}^2d$ \begin{align*} J \overrightarrow{\psi}'= (V+\lambda Q) \overrightarrow{\psi} && \overrightarrow{\psi} =\begin{bmatrix}\psi \\ \psi ' \end{bmatrix} \end{align*} The spectral measure for the canonical system is constructed (like in the Strum-Liouville case) by passing to the limit from the discrete spectral measure on the spider with the finite length of all legs and (say) Dirichlet boundary condition at the outer end points of the legs. The corresponding results (expecting the particular details related to specific case of the spider graphs) are not new. Chapter 4- spectral theory of the Schr\"{o}dinger operator on the spider like quantum graph, this chapter contains the main results of the dissertation. We start by constructing the spectral analysis on the finite interval of a three-legged spider graph and then pass it to infinity. Spectral analysis is performed for three different types of potentials. The fast-decreasing potentials, the fast-increasing potentials, mixed potentials, and its spectral theory. The details contain, the absolute continuous spectrum of multiplicity $3$ and its construction using the reflection-transmission coefficients on each leg for the fast-decreasing potential, Bohr's asymptotic formula for $N(\lambda)$ (the negative eigenvalues), instability of the discrete spectrum for the mixed potential on each leg of the spider graph.