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Наноструктуры. Математическая физика и моделирование, 2014, том 11, № 2, 76-105

А.А. Толченников, В.Л. Чернышев, А.И. Шафаревич

Уравнения Шрёдингера на графах и сингулярных пространствах: спектральные свойства и квазиклассическая динамика локализованных пакетов

Ключевые слова: уравнение Шрёдингера, граф, сингулярные пространства


[ Полный текст статьи ]

Nanostructures. Mathematical physics and modelling, 2014, vol. 11, № 2, 76-105

А.А. Tolchennikov, V.L. Chernyshev, А.I. Shafarevich

Schroedinger equation on graphs and singular spaces: spectral properties and semiclassical dynamics of localized packets

Keywords: Schroedinger equation, graph, singular spaces


We study spectral properties and semiclassical dynamics of localized packets on graphs and singular spaces. The structure of the paper is as follows. In Section 2 we give a general definition of the Schrödinger operator on a metric graph and discuss semiclassical properties of eigenvalues and eigenfunctions. In Section 3 we present the results related to the dynamics of localized wave packets on a graph. In particular, we study the statistics of the number of pack ets and establish a connection with some problems in number theory. In Section 4 we define a decorated graph and the Schrödinger operator on it. We formulate several theorems on the structure of the kernel of such operator and on asymptotics of the trace of the resolvent. In Section 5 we describe some facts relating to the dynamics of localized packets on decorated graphs. Section 6 is devoted to the results of computer modelling.

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