Discretized Laplacians on an Interval and their Renormalization Group

TL;DR

This paper explores how discretized Laplace operators on a lattice can recover all self-adjoint extensions on an interval through a renormalization group flow, revealing fixed points linked to scale-invariant boundary conditions.

Contribution

It demonstrates a method to obtain all self-adjoint extensions of the Laplacian via continuum limits of discretized models and analyzes the associated renormalization flow.

Findings

Infinite fixed points correspond to scale-invariant boundary conditions.

Other self-adjoint extensions are obtained from different trajectories of the flow.

The renormalization flow provides a unifying framework for understanding boundary conditions.

Abstract

The Laplace operator admits infinite self-adjoint extensions when considered on a segment of the real line. They have different domains of essential self-adjointness characterized by a suitable set of boundary conditions on the wave functions. In this paper we show how to recover these extensions by studying the continuum limit of certain discretized versions of the Laplace operator on a lattice. Associated to this limiting procedure, there is a renormalization flow in the finite dimensional parameter space describing the dicretized operators. This flow is shown to have infinite fixed points, corresponding to the self-adjoint extensions characterized by scale invariant boundary conditions. The other extensions are recovered by looking at the other trajectories of the flow.