Necessary and sufficient conditions for existence of bound states in a central potential

Abstract

We obtain, using the Birman-Schwinger method, a series of necessary conditions for the existence of at least one bound state applicable to arbitrary central potentials in the context of nonrelativistic quantum mechanics. These conditions yield a monotonic series of lower limits on the "critical" value of the strength of the potential (for which a first bound state appears) which converges to the exact critical strength. We also obtain a sufficient condition for the existence of bound states in a central monotonic potential which yield an upper limit on the critical strength of the potential.

7 pages

References in corpus (3)

• Upper and lower limits for the number of S-wave bound states in an attractive potential
• Upper limit on the number of bound states of the spinless Salpeter equation
• A class of ($\ell$-dependent) potentials with the same number of ($\ell$-wave) bound states

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• Auxiliary field method and analytical solutions of the Schrödinger equation with exponential potentials
• Sufficient conditions for the existence of bound states in a central potential
• Long-range potential scattering: Converting long-range potential to short-range potential by tortoise coordinate
• Critical strength of attractive central potentials
• Existence of mesons after deconfinement
• Further developments for the auxiliary field method
• Upper and lower bounds on the mean square radius and criteria for occurrence of quantum halo states