dorsal/arxiv
View SchemaConnection Conditions and the Spectral Family under Singular Potentials
| Authors | Izumi Tsutsui, Tamas Fulop, Taksu Cheon |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0209110 |
| URL | https://arxiv.org/abs/quant-ph/0209110 |
| DOI | 10.1088/0305-4470/36/1/319 |
| Journal | Journal of Physics A: Mathematical and General 36 (2003) 275-287 |
Abstract
To describe a quantum system whose potential is divergent at one point, one must provide proper connection conditions for the wave functions at the singularity. Generalizing the scheme used for point interactions in one dimension, we present a set of connection conditions which are well-defined even if the wave functions and/or their derivatives are divergent at the singularity. Our generalized scheme covers the entire U(2) family of quantizations (self-adjoint Hamiltonians) admitted for the singular system. We use this scheme to examine the spectra of the Coulomb potential $V(x) = - e^2 / | x |$ and the harmonic oscillator with square inverse potential $V(x) = (m \omega^2 / 2) x^2 + g/x^2$, and thereby provide a general perspective for these models which have previously been treated with restrictive connection conditions resulting in conflicting spectra. We further show that, for any parity invariant singular potentials $V(-x) = V(x)$, the spectrum is determined solely by the eigenvalues of the characteristic matrix $U \in U(2)$.
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"abstract": "To describe a quantum system whose potential is divergent at one point, one\nmust provide proper connection conditions for the wave functions at the\nsingularity. Generalizing the scheme used for point interactions in one\ndimension, we present a set of connection conditions which are well-defined\neven if the wave functions and/or their derivatives are divergent at the\nsingularity. Our generalized scheme covers the entire U(2) family of\nquantizations (self-adjoint Hamiltonians) admitted for the singular system. We\nuse this scheme to examine the spectra of the Coulomb potential $V(x) = - e^2 /\n| x |$ and the harmonic oscillator with square inverse potential $V(x) = (m\n\\omega^2 / 2) x^2 + g/x^2$, and thereby provide a general perspective for these\nmodels which have previously been treated with restrictive connection\nconditions resulting in conflicting spectra. We further show that, for any\nparity invariant singular potentials $V(-x) = V(x)$, the spectrum is determined\nsolely by the eigenvalues of the characteristic matrix $U \\in U(2)$.",
"arxiv_id": "quant-ph/0209110",
"authors": [
"Izumi Tsutsui",
"Tamas Fulop",
"Taksu Cheon"
],
"categories": [
"quant-ph",
"cond-mat.mes-hall",
"math-ph",
"math.MP"
],
"doi": "10.1088/0305-4470/36/1/319",
"journal_ref": "Journal of Physics A: Mathematical and General 36 (2003) 275-287",
"title": "Connection Conditions and the Spectral Family under Singular Potentials",
"url": "https://arxiv.org/abs/quant-ph/0209110"
},
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