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Spectral solutions for the Schrödinger equation with a regular singularity
by Pushkar Mohile, Ayaz Ahmed, T.R.Vishnu, Pichai Ramadevi
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Ramadevi Pichai |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202403_00014v2 (pdf) |
| Date accepted: | 2024-05-22 |
| Date submitted: | 2024-05-18 18:48 |
| Submitted by: | Pichai, Ramadevi |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We propose a modification in the Bethe-like ansatz to reproduce the hydrogen atom spectrum and the wave functions. Such a proposal provided a clue to attempt the exact quantization condition (EQC) for the quantum periods associated with potentials V(x) which are of the form V(x) = |x|+a/|x|+ b/|x|2. We validate the EQC proposal by showing that our computed Voros spectrum in the limit a, b →0 is matching well with the true spectrum of the familiar |x| potential. Thus we have given a route to obtain the spectral solution for the one dimensional Schrödinger equation involving potentials with regular singularity at the origin.
Author comments upon resubmission
We have revised the manuscript incorporating the minor suggestions and queries put forth by referee 1 and referee 2. The places where we have made the corrections are indicated in blue color for clarity in the .pdf format of the revised manuscript. Also we have pruned section 3 removing `conventional WKB' subsection as suggested by referee 2.
We hope the revised paper can be accepted for publication in SciPost Physics Core.
List of changes
We have addressed the typo in equation (2.14).
Referee 2 correctly pointed out that $\Pi_{\gamma,n}$ cannot be a function of $\hbar$. We have rectified this mistake accordingly.
Following the observation made by Referee 1, we have corrected the typos in equations (2.5, 2.10,2.14,2.15, 2.30, 2.31, 4.5,4.6 4.10) and the $b$ value in the line after eqn.(2.17)
We have corrected the definition of the parameter \(b\) before eqn.(2.19) to be consistent.
We have cited refs.[6,8] at the end of second paragraph in section 3.1.
We have added a line explaining the limits of integration for the Borel resummation in eqn. 3.5.
We have added a line in the `Acknowledgements' to thank both the referees.
In addition to these corrections, we have substantially reduced the discussion in section 3 on the conventional and exact WKB methods. Instead of discussing many details that are well known and peripheral to the paper as pointed out by Referee 2, we mainly discuss the definitions of the quantum periods and their Borel resummation in order to be self contained.
We have removed Fig. 1 in the previous manuscript related to the WKB theory of the quantum harmonic oscillator also.
We have also changed \(x \to z\) in section 2 wherever required.
We have added footnote 2 in page 15 to address one of the queries of Referee 1.
Published as SciPost Phys. Core 7, 041 (2024)