# Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Self-Adjoint Extensions: Determination of Their Eigenvalues

- (i)
- Those which preserve time reversal;
- (ii)
- Those which preserve parity;
- (iii)
- Those preserving positivity.

- The eigenvector $(A,B)$ of $\mathcal{N}\left(s\right)$ with 0 eigenvalue is given by$$\begin{array}{ccc}\hfill \phantom{\rule{-56.9055pt}{0ex}}A\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}& \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}& \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\left[i+{e}^{i\psi}(i{m}_{0}+{m}_{1}-i{m}_{2}+{m}_{3})\right]sin\frac{s}{2}+s\left[1+{e}^{i\psi}({m}_{0}+i{m}_{1}+{m}_{2}+i{m}_{3})\right]cos\frac{s}{2},\hfill \end{array}$$$$\begin{array}{ccc}\hfill \phantom{\rule{-56.9055pt}{0ex}}B\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}& \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}& \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}s\left[-1+{e}^{i\psi}({m}_{0}+i{m}_{1}+{m}_{2}-i{m}_{3})\right]sin\frac{s}{2}+\left[i+{e}^{i\psi}(i{m}_{0}-{m}_{1}+i{m}_{2}+{m}_{3})\right]cos\frac{s}{2}.\hfill \end{array}$$
- The extensions preserving time reversal invariance, are given by$${m}_{2}=0\phantom{\rule{0.166667em}{0ex}}.$$
- The parity preserving extensions of ${H}_{0}$ are those for which the eigenfunctions $\varphi \left(x\right)$ verify:$${\left|\varphi \left(x\right)\right|}^{2}={\left|\varphi (-x)\right|}^{2}\u27f9{\left|\varphi \left(a\right)\right|}^{2}={\left|\varphi (-a)\right|}^{2}\phantom{\rule{0.166667em}{0ex}}.$$

#### Parity Preserving Extensions of ${H}_{0}$

## 3. Spectrum of the Free Particle on a Finite Interval

#### 3.1. The Angular Representation of the Self-Adjoint Extensions of ${H}_{0}$

#### 3.2. Some Simple Cases

#### 3.2.1. Parity and Time Reversal Invariance: ${m}_{2}={m}_{3}=0$

#### 3.2.2. Parity Preserving Extensions Fulfilling $sins=0$

#### 3.2.3. Parity and Time Reversal Invariance Extensions Fulfilling (20c)

#### 3.3. About the Negative and Zero Energies

## 4. Supersymmetric Partners for the Simplest Extensions

#### 4.1. First Order SUSY Partners

#### 4.2. Second Order SUSY Partners

## 5. Supersymmetric Self-Adjoint Extensions of the Infinite Well at ℓ-Order

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1.

#### Appendix A.2. Trigonometric Expansion of P ℓ n -itans 0 y 2a

#### Appendix A.3. Trigonometric Expansion of Q ℓ n (-itans 0 y 2a)

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**Figure 1.**Two plots of the implicit Equation (13) with the parametrization (22) allow us to see the variation of the parameter s (remember that $E={s}^{2}/{\left(2a\right)}^{2}$) as a function of $\psi $ and $sin{\theta}_{0}$: on the left for ${\theta}_{1}=\pi /4$, on the right for ${\theta}_{1}=4\pi /3$.

**Figure 2.**Energy levels ($E={s}^{2}/{\left(2a\right)}^{2}$) for odd parity (blue) and even parity (yellow) solution for ${m}_{2}={m}_{3}=0$, coming from (23).

**Figure 3.**First order supersymmetric (SUSY) states ${\varphi}_{n}^{\left(2\right)}\left(x\right)$ from (41) when the ground state of the original system is either purely even, that is $B=0$ (plot on the left), or purely odd, that is $A=0$ (plot on the right). Note that the quantum number n of the Legendre function in (41) is the number of the nodes of the function.

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Gadella, M.; Hernández-Muñoz, J.; Nieto, L.M.; San Millán, C.
Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian. *Symmetry* **2021**, *13*, 350.
https://doi.org/10.3390/sym13020350

**AMA Style**

Gadella M, Hernández-Muñoz J, Nieto LM, San Millán C.
Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian. *Symmetry*. 2021; 13(2):350.
https://doi.org/10.3390/sym13020350

**Chicago/Turabian Style**

Gadella, Manuel, José Hernández-Muñoz, Luis Miguel Nieto, and Carlos San Millán.
2021. "Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian" *Symmetry* 13, no. 2: 350.
https://doi.org/10.3390/sym13020350