Spectral solutions for the Schrödinger equation with a regular singularity

This manuscript provides a modified Bethe like ansatz to study quantum mechanical problems with regular singularity and provides an exact quantisation condition.

The manuscript can be trimmed to focus on the problem.

Authors provide an exact quantisation condition for quantum mechanical problems with regular singularity at origin. While the procedure uses the Thermodynamic Bethe ansatz(TBA) technique, the manuscript contains a section on the WKB and exact WKB method which is not quite directly relevant except for the Voros symbols that appear in the TBA. The details of (exact) WKB has appeared in many articles and I believe the section can be pruned and relevant formulae can be added in the TBA section.

I recommend publication of shortened version in SciPost Physics Core

1. In section 2 when authors analytically continue $p(x)$ to $p(z)$, they should do the same with $\psi(x)$ to $\psi(z)$ and $V(x)$ to $V(z)$. Similarly, the table 1 should have $z$ and not $x$. Similarly, in eq.(2.9) the residue should be at $z_j$ and not $x_j$.
2. $V_eff$ in 2.13 has a spurious $r$ in the numerator.
3. The sum is over different variable in eq.(3.3).
4. In eq.(3.4) and (3.5), $\Pi_{\gamma,n}$ cannot be a function of $\hbar$.

Ask for minor revision

1. The paper is well written and self explanatory with sufficient technical details.

2. It contains useful pedagogical reviews of Bethe-like Ansatz and WKB method.

3. The paper addresses an important question on Exact Quantisation Condition in the presence of singular potential. A novel proposal for modification of Bethe-like Ansatz for obtaining spectrum of Hydrogen atom is presented in the paper.

1. Few equations in the paper contain minor typographical errors.

In this paper the authors have studied spectral solution for one-dimensional Schrödinger equation with potentials which have regular singularity at the origin. They have proposed a modification for Bethe-like Ansatz applicable to Hydrogen atom which has a potential of the form $\frac{1}{r}$. The authors have also presented Exact Quantisation Condition for potentials with single and double poles at the origin. The results obtained in this paper are potentially useful in the studies of integrable models. I recommend this paper to be published in SciPost Physics Core. I would like to point out some probable typos in some places, which the authors should verify and rectify them before publication:

1. In the equation below Eq.(2.4) there will be a factor of $\frac{i}{\hbar}$ in side the exponential and the full equation should be $\psi(z)\propto\exp\left(\frac{i}{\hbar}\int p(z)\mathrm{d}z\right) = \exp\left(-\frac
{i}{\hbar}\frac{\left(z-a\right)^{-n+1}}{n-1}\right)$.

2. Similar $\frac{i}{\hbar}$ factor should be present in the first equality of Eq.(2.13).

3. Numerator of the first term in Eq.(2.14) will not have $r$.

4. To be consistent with Eq.(2.14), $b$ in Eq.(2.16) should be $b= -\frac{e}{4\pi\epsilon_{0}}$. The authors are requested to check the sign of $b$ given below Eq.(2.16).

5. The index in the summation symbol appearing in Eq.(3.3) should be $i$ instead of $j$.

6. I am confused with the upper limit of the integration appearing in Eq.(3.10). $\phi_{\infty}$ needs to be defined.

7. In Eq.(4.5) the subscript on the left hand side of the equality will be $a$ instead of $i$.

8. There will be no $\hbar$ in the denominator in the last equality of Eq.(4.10).

In addition, the authors may choose to address the following queries if they wish:

1. While taking the classical limit in Eq.(2.2), the term $i\hbar p'$ is dropped. However from Eq.(2.3) we see that expression of $p(x)$ contains $\hbar$. It will be better if some clarification regarding scalings of $\psi(x)$, $E$ and $V(x)$ appearing in Eq. (2.1) with respect to $\hbar$ is provided.

2. In the naive TBA approach for $|x|$ potential, as the authors have shown, the spectrum matches well with the expected answer for $n\ge 5$. Is there any reason why the mismatch is more for lower values of $n$ only and the analogy of EQC between $|x|$ and harmonic oscillator holds in the large $n$ regime?