On the spectrum discreteness of the quantum graph Hamiltonian with δ-coupling

2nd International School and Conference Saint-Petersburg OPEN on Optoelectronics, Photonics, Engineering and Nanostructures

Smolkina, M.O.^1 ; Popov, I.Yu.^1

ITMO University, Kronverkskiy, 49, St. Petersburg

197101, Russia^1

关键词: Finite number;    Graph vertex;    Hamiltonian spectrum;    Infinite chains;    Quantum graph;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/643/1/012099/pdf
DOI  :  10.1088/1742-6596/643/1/012099

来源: IOP

【 摘 要 】

The condition on the potential ensuring the discreteness of the spectrum of infinite quantum graph Hamiltonian is considered. The corresponding necessary and sufficient condition was obtained by Molchanov 50 years ago. In the present paper, the analogous condition is obtained for a quantum graph. A quantum graph with infinite leads (edges) or/and infinite chains of vertices such that neighbor ones are connected by finite number of edges and with δ-type conditions at the graph vertices is suggested. The Molchanov-type theorem for the quantum graph Hamiltonian spectrum discreteness is proved.