On the Existence and Uniqueness of Self-Adjoint Realizations of Discrete (Magnetic) Schrödinger Operators

@inproceedings{Schmidt2020OnTE,
  title={On the Existence and Uniqueness of Self-Adjoint Realizations of Discrete (Magnetic) Schr{\"o}dinger Operators},
  author={Marcel Schmidt},
  year={2020}
}
Discrete Laplacians and discrete magnetic Schrödinger operators feature in many different areas of mathematics. They are used in combinatorics and computer science, appear as discretizations of (pseudo-)differential operators on Riemannian manifolds, serve as toy models for Hamiltonians in mathematical physics and play an important role in the study of random walks just to name a few. Even though discrete operators are used for very different means, their basic structure is always the same… 
View via Publisher
arxiv.org
5 Citations
Background Citations
3

5 Citations

Citation Type

Citation Type

Laplacians on infinite graphs: discrete vs continuous
There are two main notions of a Laplacian operator associated with graphs: discrete graph Laplacians and continuous Laplacians on metric graphs (widely known as quantum graphs). Both objects have a…
Self-adjointness of perturbed bi-Laplacians on infinite graphs
A Glazman–Povzner–Wienholtz theorem on graphs
The generalized porous medium equation on graphs: existence and uniqueness of solutions with $\ell^1$ data
We study solutions of the generalized porous medium equation on infinite graphs. For nonnegative or nonpositive integrable data, we prove the existence and uniqueness of mild solutions on any graph.…
Agmon estimates for Schr\"odinger operators on graphs
We prove decay estimates for generalized eigenfunctions of discrete Schrödinger operators on weighted infinite graphs in the spirit of Agmon. 2000 Mathematics Subject Classification. Primary 39A12,…

References

SHOWING 1-10 OF 52 REFERENCES
Spectral reciprocity and matrix representations of unbounded operators
Essential self-adjointness for combinatorial Schr\"odinger operators III- Magnetic fields
We define the magnetic Schr\"odinger on an infinite graph by the data of a magnetic field, some weights on vertices and some weights on edges . We discuss essential self-adjointness of this operator…
Essential self-adjointness of the graph-Laplacian
We study the operator theory associated with such infinite graphs G as occur in electrical networks, in fractals, in statistical mechanics, and even in internet search engines. Our emphasis is on the…
Maximal Accretive Extensions of Schrödinger Operators on Vector Bundles over Infinite Graphs
Given a Hermitian vector bundle over an infinite weighted graph, we define the Laplacian associated to a unitary connection on this bundle and study a perturbation of this Laplacian by an…
Self-Adjoint Extensions of Discrete Magnetic Schrödinger Operators
Using the concept of an intrinsic metric on a locally finite weighted graph, we give sufficient conditions for the magnetic Schrödinger operator to be essentially self-adjoint. The present paper is…
Heat Kernel and Essential Spectrum of Infinite Graphs
We study the existence and uniqueness of the heat kernel on infinite, locally finite, connected graphs. For general graphs, a uniqueness criterion, shown to be optimal, is given in terms of the…
A Feynman–Kac–Itô formula for magnetic Schrödinger operators on graphs
In this paper we prove a Feynman–Kac–Itô formula for magnetic Schrödinger operators on arbitrary weighted graphs. To do so, we have to provide a natural and general framework both on the operator…
A Sears-type self-adjointness result for discrete magnetic Schr\"odinger operators
Global properties of Dirichlet forms in terms of Green’s formula
We study global properties of Dirichlet forms such as uniqueness of the Dirichlet extension, stochastic completeness and recurrence. We characterize these properties by means of vanishing of a…
Uniqueness of form extensions and domination of semigroups
...
1
2
3
4
5
...