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.
Highlights
Most real quantum mechanical systems can not be solved exactly and we usually apply some approximation methods, the most common one being perturbation theory, to get information about the energy levels and scattering amplitudes
We explicitly demonstrate for a class of singular potential problems that the splitting in the energy levels due to the tunneling can be realized by developing some kind of perturbation theory
The paper is organized as follows: In section 2, we formally summarize the resolvent formulae, called Krein’s formulae, for Hamiltonians perturbed by singular potentials including Dirac delta potentials supported by points and curves
Summary
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 path-integral, WKB, and 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. Brazil Yaroslav Zolotaryuk, Bogolyubov Institute for Theoretical Physics (NAN Ukraine), Ukraine. Specialty section: This article was submitted to Mathematical Physics, a section of the journal.
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