# Spectral Problem of the Hamiltonian in Quantum Mechanics without Reference to a Potential Function

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## Abstract

**:**

## 1. Introduction

## 2. The Energy Polynomials

- i.
- The set of energies $\left\{{E}_{k}\right\}$ that satisfy: $\underset{n\to \infty}{\mathrm{lim}}{P}_{n}({E}_{k})=0$. That is, ${E}_{k}$ is an asymptotic zero for all energy polynomials in the limit of infinite (large enough) degrees (If for a particular value of the energy, ${P}_{n}(\widehat{E})=0$ for all n (not only asymptotically) then $\widehat{E}$ is not the energy of a bound state. In fact, this property makes the energy polynomials non-orthogonal. If we remove this zero by defining ${P}_{n}(E)=(E-\widehat{E}){Q}_{n-1}(E)$ then the polynomials $\left\{{Q}_{n}(E)\right\}$ will form a true orthogonal sequence of energy polynomials).
- ii.
- The set of eigenvalues of the tridiagonal matrix (7) that lie outside the energy bands and do not change significantly (within the desired accuracy) if we vary the size of the matrix around a large enough size $N\times N$. (It may happen that an eigenvalue $\widehat{E}$ of the matrix (7), which lies isolated outside the energy bands or in an energy gap, does not correspond to a bound state. It is advisable that one evaluates the polynomial at all such eigenvalues and performs the test $\underset{n\to \infty}{\mathrm{lim}}{P}_{n}(\widehat{E})=0$).
- iii.
- The set of energies that make the asymptotic limit ($n\to \infty $) of the polynomial ${P}_{n}(E)$ vanish (Typically, these asymptotics take the form ${P}_{n}(E)\to \frac{1}{{n}^{\alpha}\sqrt{\rho (E)}}\mathrm{cos}\left[{n}^{\beta}\phi (E)+\delta (E)\right]$, where $\alpha $ and $\beta $ are positive real parameters, $\rho (E)$ is the weight function, $\phi (E)$ is an entire function, and $\delta (E)$ is the scattering phase shift. If $\beta \to 0$ then ${n}^{\beta}\to \mathrm{ln}(n)$. As an illustration, we plot ${P}_{n}(E)$ as a function of n for a fixed E from within the bands in Problem II and verify the oscillatory behavior of the asymptotics (we take, for example, $E=$ ${\lambda}^{2}\left\{1.9,-0.5\right\}$). We also show that the asymptotics in fact vanishes at the energy $\mathcal{E}={\lambda}^{2}/2$).

## 3. Problems in the Infinite Domain

#### 3.1. Problem I

#### 3.2. Problem II

#### 3.3. Problem III

#### 3.4. Problem IV

## 4. Problems in the Semi-Infinite Domain

#### 4.1. Problem V

#### 4.2. Problem VI

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Partial Solutions

#### Appendix A.1. Problem I

**Table A1.**The left and right boundaries of the energy band in units of ${\lambda}^{2}$ for different values of N. The exact values are $\pm {\lambda}^{2}$.

N | Left Boundary | Right Boundary |
---|---|---|

10 | −0.959493 | 0.959493 |

20 | −0.988831 | 0.988831 |

50 | −0.998103 | 0.998103 |

100 | −0.999516 | 0.999516 |

200 | −0.999878 | 0.999878 |

**Figure A2.**The un-normalized wavefunction $\tilde{\psi}(x,E)$ for $E=0$ at the middle of the band. The horizontal x-axis is in units of ${\lambda}^{-1}$. We took $N=100$.

**Figure A3.**The un-normalized wavefunction $\tilde{\psi}(x,E)$ for $E=+{\lambda}^{2}/2$. We took $N=100$.

**Figure A4.**The un-normalized wavefunction $\tilde{\psi}(x,E)$ for $E=-{\lambda}^{2}/2$. We took $N=100$.

**Figure A5.**The un-normalized wavefunction $\tilde{\psi}(x,E)$ for $E={\lambda}^{2}$, which is at the right edge of the energy band. We took $N=100$.

**Figure A6.**The un-normalized wavefunction $\tilde{\psi}(x,E)$ for $E=3{\lambda}^{2}/2$, which is a forbidden energy outside the band. We took $N=100$. Note the unbounded oscillations everywhere.

#### Appendix A.2. Problem II

**Table A2.**The boundaries of the left and right energy bands in units of ${\lambda}^{2}$ for different values of N. The exact values are $-{\lambda}^{2}$, 0, ${\lambda}^{2}$, and $2{\lambda}^{2}$.

N | Left Boundaries | Right Boundaries | ||
---|---|---|---|---|

20 | −0.926138 | −0.105799 | 1.105799 | 1.926138 |

50 | −0.973840 | −0.034724 | 1.034724 | 1.97384 |

100 | −0.988118 | −0.014908 | 1.014908 | 1.988118 |

200 | −0.994550 | −0.006517 | 1.006517 | 1.99455 |

400 | −0.997467 | −0.002915 | 1.002915 | 1.997467 |

**Figure A8.**The un-normalized wavefunction $\tilde{\psi}(x,\mathcal{E})$ for the bound state in the middle of the energy gap with $\mathcal{E}={\scriptscriptstyle \frac{1}{2}}{\lambda}^{2}$. The horizontal x-axis is in units of ${\lambda}^{-1}$. We took $N=100$.

**Figure A9.**The un-normalized wavefunction $\tilde{\psi}(x,E)$ for $E=3{\lambda}^{2}/2$ in the right energy band. We took $N=100$.

**Figure A10.**The un-normalized wavefunction $\tilde{\psi}(x,E)$ for $E=-{\lambda}^{2}/2$ in the left energy band. We took $N=100$.

**Figure A11.**The un-normalized wavefunction $\tilde{\psi}(x,E)$ for $E={\lambda}^{2}/4$, which is a forbidden energy in the gap. We took $N=100$. Note the unbounded oscillations everywhere.

#### Appendix A.3. Problem III

**Figure A12.**The zeros of ${P}_{N}(E)$ in units of ${\lambda}^{2}$ for $N=100$. We took $\alpha =1.5$, $\beta =2.3$ and $\gamma =3.7$. Out of the six isolated eigenvalues (shown with red circles) only $E=3{\lambda}^{2}/2$ pass the asymptotic test (2.iii) and corresponds to a bound state.

**Table A3.**The left and right boundaries of the left and right energy bands in units of ${\mathsf{\lambda}}^{2}$ for different values of N. The exact values are in (12).

N | Left Boundaries | Right Boundaries | ||
---|---|---|---|---|

20 | −2.61667 | −1.02433 | 2.02433 | 3.61667 |

50 | −2.65286 | −0.81445 | 1.81445 | 3.65286 |

100 | −2.65775 | −0.760255 | 1.76026 | 3.65775 |

200 | −2.6601 | −0.737958 | 1.73796 | 3.6601 |

300 | −2.6609 | −0.731469 | 1.73147 | 3.6609 |

Exact | −2.66228 | −0.720656 | 1.72066 | 3.66228 |

**Figure A13.**The un-normalized wavefunction $\tilde{\psi}(x,E)$ for the bound state inside the energy gap with $E=3{\lambda}^{2}/2$. The horizontal x-axis is in units of ${\lambda}^{-1}$. We took $N=100$.

#### Appendix A.4. Problem IV

**Figure A14.**The zeros of ${P}_{N}(E)$ in units of ${\lambda}^{2}$ for $N=200$. We took $\alpha =3$, $\beta =2$ and $\gamma =1$. The energy bands extend to infinity and the gap is located in the interval $-2{\lambda}^{2}<E<+3{\lambda}^{2}$. The system has no bound states.

**Table A4.**The left and right boundaries energy gap in units of ${\lambda}^{2}$ for different values of N. We took $\alpha =3$, $\beta =2$ and $\gamma =1$. The exact values are $\frac{{\lambda}^{2}}{2}\left(1\pm \left|2\alpha -1\right|\right)$.

N | Gap Boundaries | |
---|---|---|

20 | −2.54592 | 3.54592 |

50 | −2.25155 | 3.25155 |

100 | −2.13357 | 3.13357 |

200 | −2.06917 | 3.06917 |

300 | −2.04672 | 3.04672 |

Exact | −2.00000 | 3.00000 |

#### Appendix A.5. Problem V

**Figure A15.**The zeros of ${P}_{N}(E)$ in units of ${\lambda}^{2}$ for different values of N. We took $\omega =3/2$, $\mu =5/4$, and $\theta =\pi $. The energies are found to converge to a discrete equally spaced spectrum.

**Figure A16.**The difference between consecutive zeros of ${P}_{N}(E)$ for $N=200$. We took $\omega =1$, $\mu =5$, for different values of θ. The difference is constant, which matches the theoretical value of $2\omega $. However, we get better convergence as θ increases.

**Figure A17.**The kth zero of ${P}_{N}(E)$ vs k for $N=200$. We took $\omega =1$, $\mu =5$, for different values of θ. This shows how the plot have better convergence as θ increases.

#### Appendix A.6. Problem VI

**Figure A18.**The zeros of ${P}_{N}(E)$ in units of ${\lambda}^{2}$ for $N=200$. We took $\gamma =10$ and $\mu =-9.5$. The energy bands for $E\ge 0$ extend to infinity.

**Figure A19.**The zeros of ${P}_{N}(E)$ in units of ${\lambda}^{2}$ for different values of N. The energy bands for $E<0$ are found to converge to a discrete linearly spaced spectrum.

**Figure A20.**The kth zero of ${P}_{N}(E)$ vs k for $N=500$. We took $\gamma =10$ and $\mu =-9.5$. When $E<0$, the zeros of ${P}_{N}(E)$ agrees with Equation (18) up to $k=\lfloor -\mu \rfloor $.

**Figure A21.**The un-normalized wavefunction $\tilde{\psi}(r,{E}_{k})$ for different values of $k$. The horizontal r-axis is in units of ${\lambda}^{-1}$. We took $\mu =-15,\gamma =16$, and $\nu =1$. We can observe each wavefunction having $k$ nodes, and vanishes at the boundaries.

**Figure A22.**The un-normalized wavefunction $\tilde{\psi}(r,E)$ for $N=100$. We took $E=150{\lambda}^{2}$, for $\mu =-15,\gamma =16$, and $\nu =2$. We can see the bounded oscillations that extends to infinity.

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**MDPI and ACS Style**

Al-Yousef, I.F.; Ekhwan, M.; Bahlouli, H.; Alhaidari, A.D.
Spectral Problem of the Hamiltonian in Quantum Mechanics without Reference to a Potential Function. *Axioms* **2023**, *12*, 334.
https://doi.org/10.3390/axioms12040334

**AMA Style**

Al-Yousef IF, Ekhwan M, Bahlouli H, Alhaidari AD.
Spectral Problem of the Hamiltonian in Quantum Mechanics without Reference to a Potential Function. *Axioms*. 2023; 12(4):334.
https://doi.org/10.3390/axioms12040334

**Chicago/Turabian Style**

Al-Yousef, Ibraheem F., Moayad Ekhwan, H. Bahlouli, and A. D. Alhaidari.
2023. "Spectral Problem of the Hamiltonian in Quantum Mechanics without Reference to a Potential Function" *Axioms* 12, no. 4: 334.
https://doi.org/10.3390/axioms12040334