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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...

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Autores principales: Chadan, K., Khuri, N.N., Martin, A., Wu, Tai Tsun
Lenguaje:eng
Publicado: 2002
Materias:
Acceso en línea:https://dx.doi.org/10.1063/1.1532538
http://cds.cern.ch/record/575271
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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.
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institution Organización Europea para la Investigación Nuclear
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publishDate 2002
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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
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AT khurinn boundstatesinoneandtwospatialdimensions
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