Bound states in one and two spatial dimensions

Abstract

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.

Upper and lower limits on the number of bound states in a central potential

“…It is well known that conditions (3) and (4) on the potential V (r) are sufficient to guarantee that the (singular) Sturm-Liouville problem characterized by the ODE (10) with the boundary conditions (11) and (12) has a finite (possibly vanishing) number of discrete eigenvalues κ 2 ,n . To count them one notes that for sufficiently large (for definiteness, positive) values of κ the solution u (κ; r) of the radial Schrödinger equation (10) with the boundary condition (11) (which characterizes the solution uniquely up to a multiplicative constant) has no zeros in the interval 0 < r < ∞ and diverges as r → ∞ (proportionally to exp(κr)), because for sufficiently large values of κ the quantity in the square brackets on the right-hand side of the radial Schrödinger equation (10) is positive for all values of r, hence the solution u (κ; r) of the second-order ODE (10) is everywhere convex.…”

On the Negative Spectrum of the Two-Dimensional Schrödinger Operator with Radial Potential

“…Here R > 0 can be taken arbitrary, and one can minimize the estimate with respect to R. The results in [9,6] are formulated for α = 1, but the general case follows immediately by the substitution F → αF . The proofs in both papers make use of the Lieb-Thirring estimate for operators in dimension one.…”

“…It is well-known that in R 2 any negative potential generates at least one negative bound state, and therefore, a similar estimate for b = 0 cannot be valid. Three years later, in the paper [6], the following important estimate was obtained for the Hamiltonian (1.1): if d = 2 and V (x) = F (|x|), then…”

“…We show this in Subsection 5.2. The possibility of this removal was formulated in [6] as a conjecture. Remark 1.6.…”

Stabilized vortices in layered Kerr media

“…Following Ref. [18] we can bound the number of ℓ-wave bound states in this potential, by N ℓ < (a 12 g/ℓ) R + r|u 1 | 2 dr . Vortex-type solutions with smallest topological charge are those with ℓ = 1.…”