dorsal/arxiv
View SchemaUniversal Superpositions of Coherent States and Self-Similar Potentials
| Authors | V. Spiridonov |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9601030 |
| URL | https://arxiv.org/abs/quant-ph/9601030 |
| DOI | 10.1103/PhysRevA.52.1909 |
| Journal | Phys.Rev. A52 (1995) 1909-1935 |
Abstract
A variety of coherent states of the harmonic oscillator is considered. It is formed by a particular superposition of canonical coherent states. In the simplest case, these superpositions are eigenfunctions of the annihilation operator $A=P(d/dx+x)/\sqrt2$, where $P$ is the parity operator. Such $A$ arises naturally in the $q\to -1$ limit for a symmetry operator of a specific self-similar potential obeying the $q$-Weyl algebra, $AA^\dagger-q^2A^\dagger A=1$. Coherent states for this and other reflectionless potentials whose discrete spectra consist of $N$ geometric series are analyzed. In the harmonic oscillator limit the surviving part of these states takes the form of orthonormal superpositions of $N$ canonical coherent states $|\epsilon^k\alpha\rangle$, $k=0, 1, \dots, N-1$, where $\epsilon$ is a primitive $N$th root of unity, $\epsilon^N=1$. A class of $q$-coherent states related to the bilateral $q$-hypergeometric series and Ramanujan type integrals is described. It includes a curious set of coherent states of the free nonrelativistic particle which is interpreted as a $q$-algebraic system without discrete spectrum. A special degenerate form of the symmetry algebras of self-similar potentials is found to provide a natural $q$-analog of the Floquet theory. Some properties of the factorization method, which is used throughout the paper, are discussed from the differential Galois theory point of view.
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"abstract": "A variety of coherent states of the harmonic oscillator is considered. It is\nformed by a particular superposition of canonical coherent states. In the\nsimplest case, these superpositions are eigenfunctions of the annihilation\noperator $A=P(d/dx+x)/\\sqrt2$, where $P$ is the parity operator. Such $A$\narises naturally in the $q\\to -1$ limit for a symmetry operator of a specific\nself-similar potential obeying the $q$-Weyl algebra, $AA^\\dagger-q^2A^\\dagger\nA=1$. Coherent states for this and other reflectionless potentials whose\ndiscrete spectra consist of $N$ geometric series are analyzed. In the harmonic\noscillator limit the surviving part of these states takes the form of\northonormal superpositions of $N$ canonical coherent states\n$|\\epsilon^k\\alpha\\rangle$, $k=0, 1, \\dots, N-1$, where $\\epsilon$ is a\nprimitive $N$th root of unity, $\\epsilon^N=1$. A class of $q$-coherent states\nrelated to the bilateral $q$-hypergeometric series and Ramanujan type integrals\nis described. It includes a curious set of coherent states of the free\nnonrelativistic particle which is interpreted as a $q$-algebraic system without\ndiscrete spectrum. A special degenerate form of the symmetry algebras of\nself-similar potentials is found to provide a natural $q$-analog of the Floquet\ntheory. Some properties of the factorization method, which is used throughout\nthe paper, are discussed from the differential Galois theory point of view.",
"arxiv_id": "quant-ph/9601030",
"authors": [
"V. Spiridonov"
],
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"doi": "10.1103/PhysRevA.52.1909",
"journal_ref": "Phys.Rev. A52 (1995) 1909-1935",
"title": "Universal Superpositions of Coherent States and Self-Similar Potentials",
"url": "https://arxiv.org/abs/quant-ph/9601030"
},
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