‘Supersymmetrizing’ an arbitrary quantum-mechanical potential To my understanding, it is not possible to $``\text{supersymmetrize}"$ an arbitrary quantum-mechanical system unless one knows how to represent the corresponding Hamiltonian in the form
$$
H = A^\dagger A \quad,
$$
which can be as difficult as solving the Schrödinger equation. (I guess, in QM we can generate SUSY potentials by playing with arbitrary superpotentials $W(x)$.)
So, I realize that my question is not very rigorous. Anyways,
Is there ANY way to construct a supersymmetric Hamiltonian $H_S$ from a given one, $H$? (without the necessity to solve Schrödinger equation for $H$ or doing some equally hard things)
What I mean here by $``\text{construct}"$ is not necessarily what people typically mean by this word in SUSY QM (see e.g. here). The only thing I want is SOME SUSY potential whose construction would somehow involve using $H$.
P.S.
Note that we know how to $``\text{supersymmetrize}"$ nearly any field theory. This suggests that a similar thing should be possible in QM.
 A: *

*Let there be given a 1D self-adjoint Hamiltonian of the form
$$ H~=~\frac{p^2}{2m}+V(x), \qquad p~=~\frac{\hbar}{i}\frac{d}{dx}.\tag{1}$$


*A necessary condition for supersymmetrization (in the sense of Refs. 1-2) is that the self-adjoint Hamiltonian $H\geq C$ is bounded from below, where $C\in\mathbb{R}$ can be interpreted as a choice of zero-point energy.


*Case of spontaneously unbroken SUSY: Here we assume that the system has a normalizable ground state $\psi_0(x)$ with energy-eigenvalue $E_0$. Since $\psi_0(x)$ is the ground state, it should have no nodes.
By shifting the energy-level, we may & shall assume that $E_0=0$ from now on. We can now choose a superpotential
$$W~:=~-\frac{\hbar}{\sqrt{2m}}\frac{d}{dx}\ln|\psi_0|.\tag{2}$$


*Both cases of spontaneously unbroken & broken SUSY: More generally, we are looking for a global solution $W$ to the Riccati equation
$$ V~=~W^2-\frac{\hbar W^{\prime}}{\sqrt{2m}}, \tag{3}$$
possibly after shifting the the energy-level.
It follows that the Hamiltonian is semipositive
$$H~\stackrel{(1)+(3)}{=}~A^{\dagger}A, \qquad A~:=~\frac{ip}{\sqrt{2m}}+W(x).\tag{4}$$
[It is easy to see that eq. (2) satisfies the Riccati eq. (3). Conversely, given a $W$-solution to the Riccati eq. (3), we can reverse eq. (2) to defined a lowest energy eigen-function $\psi_0$, which may not be normalizable. The unnormalizable case corresponds to spontaneously broken SUSY, cf. Refs. 1-2.]


*The construction of the SUSY QM model proceed in the standard fashion:
$$ H_{\rm SUSY}~:=~\begin{pmatrix} A^{\dagger}A & 0 \cr 0 & AA^{\dagger} \end{pmatrix}~\stackrel{(6)}{=}~\{Q,Q^{\dagger} \}_+~\stackrel{(4)}{=}~(\frac{p^2}{2m}+W^2)\mathbb{1}_{2\times 2}-\frac{\hbar W^{\prime}}{\sqrt{2m}}\sigma_z ,\tag{5} $$
with supercharge
$$ Q~:=~ \begin{pmatrix} 0 & 0 \cr A & 0 \end{pmatrix}~=~A\sigma_-,\qquad  \sigma_{\pm}:= \frac{\sigma_x\pm i\sigma_y}{2},\qquad Q^2~=~0,\tag{6}$$
cf. Refs. 1-2. Here we have rewritten the $2\times2$ matrices in terms of Pauli matrices.


*Alternatively, we may replace the $2\times 2$ Pauli matrices with a Grassmann-odd/fermionic creation & annilation-operator
$$ H_{\rm SUSY}~\stackrel{(5)+(8)}{\leftrightarrow}~(\frac{p^2}{2m}+W^2)\mathbb{1}-\frac{\hbar W^{\prime}}{\sqrt{2m}}[b^{\dagger},b]_-, \qquad Q ~\stackrel{(6)+(8)}{\leftrightarrow}~Ab,\tag{7} $$
under the identification
$$ \sigma_-~\leftrightarrow~b, \qquad \sigma_+~\leftrightarrow~b^{\dagger}, \qquad \sigma_z~\leftrightarrow~[b^{\dagger},b]_-, \qquad \mathbb{1}_{2\times 2}~\leftrightarrow~\mathbb{1}~=~\{b,b^{\dagger}\}_+ .\tag{8}$$
References:

*

*F. Cooper, A. Khare, & U. Sukhatme, Supersymmetry and Quantum Mechanics, Phys. Rept. 251 (1995) 267, arXiv:hep-th/9405029; Chapter 2.


*P. Uttayarat, Supersymmetrizing a QM System, J. Phys.: Conf. Ser.
901 (2017) 012052.
