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Bound States in one and two Spatial Dimensions
In this paper we study the number of bound states for potentials in one and two spatial dimensions. We first show that in addition to the well-known fact that an arbitrarily weak attractive potential has a bound state, it is easy to construct examples where weak potentials have an infinite number of...
Autores principales: | , , , |
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Lenguaje: | eng |
Publicado: |
2002
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Materias: | |
Acceso en línea: | https://dx.doi.org/10.1063/1.1532538 http://cds.cern.ch/record/575271 |
_version_ | 1780899369495560192 |
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author | Chadan, K. Khuri, N.N. Martin, A. Wu, Tai Tsun |
author_facet | Chadan, K. Khuri, N.N. Martin, A. Wu, Tai Tsun |
author_sort | Chadan, K. |
collection | CERN |
description | In this paper we study the number of bound states for potentials in one and two spatial dimensions. We first show that in addition to the well-known fact that an arbitrarily weak attractive potential has a bound state, it is easy to construct examples where weak potentials have an infinite number of bound states. These examples have potentials which decrease at infinity faster than expected. Using somewhat stronger conditions, we derive explicit bounds on the number of bound states in one dimension, using known results for the three-dimensional zero angular momentum. A change of variables which allows us to go from the one-dimensional case to that of two dimensions results in a bound for the zero angular momentum case. Finally, we obtain a bound on the total number of bound states in two dimensions, first for the radial case and then, under stronger conditions, for the non-central case. |
id | cern-575271 |
institution | Organización Europea para la Investigación Nuclear |
language | eng |
publishDate | 2002 |
record_format | invenio |
spelling | cern-5752712023-03-14T20:18:16Zdoi:10.1063/1.1532538http://cds.cern.ch/record/575271engChadan, K.Khuri, N.N.Martin, A.Wu, Tai TsunBound States in one and two Spatial DimensionsMathematical Physics and MathematicsIn this paper we study the number of bound states for potentials in one and two spatial dimensions. We first show that in addition to the well-known fact that an arbitrarily weak attractive potential has a bound state, it is easy to construct examples where weak potentials have an infinite number of bound states. These examples have potentials which decrease at infinity faster than expected. Using somewhat stronger conditions, we derive explicit bounds on the number of bound states in one dimension, using known results for the three-dimensional zero angular momentum. A change of variables which allows us to go from the one-dimensional case to that of two dimensions results in a bound for the zero angular momentum case. Finally, we obtain a bound on the total number of bound states in two dimensions, first for the radial case and then, under stronger conditions, for the non-central case.In this paper we study the number of bound states for potentials in one and two spatial dimensions. We first show that in addition to the well-known fact that an arbitrarily weak attractive potential has a bound state, it is easy to construct examples where weak potentials have an infinite number of bound states. These examples have potentials which decrease at infinity faster than expected. Using somewhat stronger conditions, we derive explicit bounds on the number of bound states in one dimension, using known results for the three-dimensional zero angular momentum. A change of variables which allows us to go from the one-dimensional case to that of two dimensions results in a bound for the zero angular momentum case. Finally, we obtain a bound on the total number of bound states in two dimensions, first for the radial case and then, under stronger conditions, for the non-central case.math-ph/0208011CERN-TH-2002-128LPT-ORSAY-02-57CERN-TH-2002-128oai:cds.cern.ch:5752712002-08-06 |
spellingShingle | Mathematical Physics and Mathematics Chadan, K. Khuri, N.N. Martin, A. Wu, Tai Tsun Bound States in one and two Spatial Dimensions |
title | Bound States in one and two Spatial Dimensions |
title_full | Bound States in one and two Spatial Dimensions |
title_fullStr | Bound States in one and two Spatial Dimensions |
title_full_unstemmed | Bound States in one and two Spatial Dimensions |
title_short | Bound States in one and two Spatial Dimensions |
title_sort | bound states in one and two spatial dimensions |
topic | Mathematical Physics and Mathematics |
url | https://dx.doi.org/10.1063/1.1532538 http://cds.cern.ch/record/575271 |
work_keys_str_mv | AT chadank boundstatesinoneandtwospatialdimensions AT khurinn boundstatesinoneandtwospatialdimensions AT martina boundstatesinoneandtwospatialdimensions AT wutaitsun boundstatesinoneandtwospatialdimensions |