# A Perturbative Approach to the Tunneling Phenomena

## TL;DR

This paper demonstrates that perturbative methods can be used to calculate tunneling-induced energy splittings in quantum systems, challenging the common belief that such phenomena are only accessible through non-perturbative approaches.

## Contribution

It introduces a perturbative approach to compute tunneling effects and energy splittings, including in systems with singular potentials, which is a novel application.

## Key findings

- Perturbative methods can successfully compute tunneling splittings.
- Explicit examples with singular potentials validate the approach.
- The method extends to relativistic and singular potential systems.

## Abstract

The double-well potential is a good example, where we can compute the splitting in the bound state energy of the system due to the tunneling effect with various methods, namely WKB or instanton calculations. All these methods are non-perturbative and there is a common belief that it is difficult to find the splitting in the energy due to the barrier penetration from a perturbative analysis. However, we will illustrate by explicit examples containing singular potentials (e.g., Dirac delta potentials supported by points and curves and their relativistic extensions)that it is possible to find the splitting in the bound state energies by developing some kind of perturbation method.