NoemaNicolussi UniversityofVienna OperatorTheoryandKreinSpaces,Vienna ...

Self-adjoint extensions of Infinite Quantum Graphs

Noema Nicolussi University of Vienna

Operator Theory and Krein Spaces, Vienna 20 December 2019

A. Kostenko, D. Mugnolo, and N. Nicolussi, Self-adjoint and Markovian extensions of infinite quantum graphs, submitted, arXiv:1911.04735 (2019).

Noema Nicolussi

20 December 2019 1 / 14

Metric Graphs

Definition:

A graph is a (countable) set of vertices V together with a set of edges E. If every edge e E is assigned a finite edge length 0 < |e| < , we call G = (V, E, | ? |) a metric graph.

G = Z2

Noema Nicolussi

20 December 2019 2 / 14

Metric Graphs

Definition:

A graph is a (countable) set of vertices V together with a set of edges E. If every edge e E is assigned a finite edge length 0 < |e| < , we call G = (V, E, | ? |) a metric graph.

G = Z2

Roughly, a metric graph is a 1-dimensional complex of finite intervals glued together at certain endpoints (a "wire").

Noema Nicolussi

20 December 2019 2 / 14

Quantum Graphs

Assumptions

G is connected G is simple (no loops or multiple edges)

Noema Nicolussi

20 December 2019 3 / 14

Quantum Graphs

Assumptions

G is connected G is simple (no loops or multiple edges) G is locally finite, i.e. all vertices have only finitely many neighbors

Noema Nicolussi

20 December 2019 3 / 14

Quantum Graphs

Assumptions

G is connected G is simple (no loops or multiple edges) G is locally finite, i.e. all vertices have only finitely many neighbors

An (informal) Definition

Quantum graphs = Laplacian-type differential operators on metric graphs

Noema Nicolussi

20 December 2019 3 / 14

Quantum Graphs

Assumptions

G is connected G is simple (no loops or multiple edges) G is locally finite, i.e. all vertices have only finitely many neighbors

An (informal) Definition

Quantum graphs = Laplacian-type differential operators on metric graphs

Motivation: Bridge between manifolds and graphs (random walks vs. Brownian motion)

Noema Nicolussi

20 December 2019 3 / 14

Quantum Graphs

Assumptions

G is connected G is simple (no loops or multiple edges) G is locally finite, i.e. all vertices have only finitely many neighbors

An (informal) Definition

Quantum graphs = Laplacian-type differential operators on metric graphs

Motivation: Bridge between manifolds and graphs (random walks vs. Brownian motion) Applications: chemistry/physics/biology (simpler models for PDEs)

Noema Nicolussi

20 December 2019 3 / 14

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