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.

Critical strength of attractive central potentials

“…This relation yields a lower limit on the critical value g ( ) c , by making a minimization over p 1, which was shown to be very accurate (see, for example, [9,16,28]). …”

Upper and lower bounds on the mean square radius and criteria for the occurrence of quantum halo states

“…This effect will be attenuated (suppressed in practice) if we use upper limits on g c N . Accurate upper and lower limits on the value of g c 1 and on g c NϾ1 , involving only the potential, can be found in the literature [3,[11][12][13][14][15][16][17][38][39][40].…”

“…Indeed, taking the limit n → ϱ of the relation (38), and taking into account the reduced mass of the system, we obtain for a square well E H Х −1.6͓͑A +1͒ / A͔A −2/3 MeV (in agreement with a previous result [34] Table I, to test the criterion (28) applied to nuclear halos, we compare the value of the energy E H given by the formula (49) with the exact energy E ex at which ͗r 2 ͘ 1/2 = r 0 , with = 2. We also give the exact value of the ratio ͗r 2 ͘ 1/2 / r 0 at energy E H .…”

Bound states and critical behavior of the Yukawa potential

“…Last few decades have seen a lot of ongoing analytical and numerical efforts [2,5,6,7,8,9,10,11,12,13] on exploring the properties of the Yukawa potential. Up to very recently, this model still receives great attention [14,15,16,17,18], for it plays an important role not only in particle/nuclear physics, but also in many other branches: atomic physics, chemical physics, gravitational plasma physics, and solid-state physics.…”

“…Critical phenomena exist not only in Quantum Chromodynamics (QCD) at finite temperature/chemical potential [19,20,21,22], but also in the Yukawa potential in Quantum Mechanics (QM) [3,4,5,10,11,12,13]. For α = 0, the Yukawa potential reduces to the Coulomb potential, and it is known to have infinite number of bound states.…”