On the Growth of the Number of Bound States with Increase in Potential Strength

Abstract

For a wide class of potentials, it is shown that N(λ), the number of bound states (including multiplicity) of −Δ + λV, obeys the conditions Aλ^(3/2) < N(λ) < Bλ^(3/2) for λ sufficiently large. A and B are positive finite numbers. In the centrally symmetric cases, a related growth condition on l_(max)(λ), the largest l channel with bound states, is also obtained, namely, aλ^(1/2) < l_(max)(λ) < bλ^(1/2). Finally, we discuss analogous results for a larger class of central potentials and for the many‐body case.