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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.11157 |
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| _version_ | 1866912333354237952 |
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| author | Schweiger, W. Klink, W. H. |
| author_facet | Schweiger, W. Klink, W. H. |
| contents | This paper deals with the partial solution of the energy-eigenvalue problem for one-dimensional Schrödinger operators of the form $H_N=X_0^2+V_N$, where $V_N=X_N^2+αX_{N-1}$ is a polynomial potential of degree $(2N-2)$ and $X_i$ are the generators of an irreducible representation of a particular nilpotent group $\mathcal{G}_N$. Algebraization of the eigenvalue problem is achieved for eigenfunctions of the form $\sum_{k=0}^M a_k X_2^k \exp(-\int dx\, X_N)$. It is shown that the overdetermined linear system of equations for the coefficients $a_k$ has a nontrivial solution, if the parameter $α$ and $(N-3)$ Casimir invariants satisfy certain constraints. This general setting works for even $N\geq 2$ and can also be applied to odd $N\geq 3$, if the potential is symmetrized by considering it as function of $|x|$ rather than $x$. It provides a unified approach to quasi-exactly solvable polynomial interactions, including the harmonic oscillator, and extends corresponding results known from the literature. Explicit expressions for energy eigenvalues and eigenfunctions are given for the quasi-exactly solvable sextic, octic and decatic potentials. The case of $E=0$ solutions for general $N$ and $M$ is also discussed. As physical application, the movement of a charged particle in an electromagnetic field of pertinent polynomial form is shortly sketched. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11157 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polynomial potentials and nilpotent groups Schweiger, W. Klink, W. H. Mathematical Physics This paper deals with the partial solution of the energy-eigenvalue problem for one-dimensional Schrödinger operators of the form $H_N=X_0^2+V_N$, where $V_N=X_N^2+αX_{N-1}$ is a polynomial potential of degree $(2N-2)$ and $X_i$ are the generators of an irreducible representation of a particular nilpotent group $\mathcal{G}_N$. Algebraization of the eigenvalue problem is achieved for eigenfunctions of the form $\sum_{k=0}^M a_k X_2^k \exp(-\int dx\, X_N)$. It is shown that the overdetermined linear system of equations for the coefficients $a_k$ has a nontrivial solution, if the parameter $α$ and $(N-3)$ Casimir invariants satisfy certain constraints. This general setting works for even $N\geq 2$ and can also be applied to odd $N\geq 3$, if the potential is symmetrized by considering it as function of $|x|$ rather than $x$. It provides a unified approach to quasi-exactly solvable polynomial interactions, including the harmonic oscillator, and extends corresponding results known from the literature. Explicit expressions for energy eigenvalues and eigenfunctions are given for the quasi-exactly solvable sextic, octic and decatic potentials. The case of $E=0$ solutions for general $N$ and $M$ is also discussed. As physical application, the movement of a charged particle in an electromagnetic field of pertinent polynomial form is shortly sketched. |
| title | Polynomial potentials and nilpotent groups |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2412.11157 |