TY - JOUR

T1 - Exact, E = 0, quantum solutions for general power-law potentials

AU - Daboul, Jamil

AU - Nieto, Michael Martin

PY - 1996/1/1

Y1 - 1996/1/1

N2 - For zero energy, E = 0, we derive exact, quantum solutions for all power-law potentials, V(r) = -γ/rν, with γ > 0 and -∞ < ν < ∞. The solutions are, in general, Bessel functions of powers of r. For ν > 2 and l ≥ 1 the solutions are normalizable. Surprisingly, the solutions for ν < -2, which correspond to highly repulsive potentials, are also normalizable, for all l ≥ 0. For these |ν| > 2 the partial-wave Hamiltonians, Hl have overcomplete sets of normalizable eigensolutions. We discuss how to obtain self-adjoint extensions of Hl such that the above E = 0 solutions become included in their domains. When 2 > ν ≥ -2 the E = 0 solutions are not square-integrable. The ν = 2 solutions are also unnormalizable, but are exceptional solutions. We also find that, by increasing the dimension of the Schrödinger equation beyond 4, an effective centrifugal barrier is created which is sufficient to cause binding when E = 0 and ν > 2, even for l = 0. We discuss the physics of the above solutions and compare them to the corresponding classical solutions, which are derived elsewhere.

AB - For zero energy, E = 0, we derive exact, quantum solutions for all power-law potentials, V(r) = -γ/rν, with γ > 0 and -∞ < ν < ∞. The solutions are, in general, Bessel functions of powers of r. For ν > 2 and l ≥ 1 the solutions are normalizable. Surprisingly, the solutions for ν < -2, which correspond to highly repulsive potentials, are also normalizable, for all l ≥ 0. For these |ν| > 2 the partial-wave Hamiltonians, Hl have overcomplete sets of normalizable eigensolutions. We discuss how to obtain self-adjoint extensions of Hl such that the above E = 0 solutions become included in their domains. When 2 > ν ≥ -2 the E = 0 solutions are not square-integrable. The ν = 2 solutions are also unnormalizable, but are exceptional solutions. We also find that, by increasing the dimension of the Schrödinger equation beyond 4, an effective centrifugal barrier is created which is sufficient to cause binding when E = 0 and ν > 2, even for l = 0. We discuss the physics of the above solutions and compare them to the corresponding classical solutions, which are derived elsewhere.

UR - http://www.scopus.com/inward/record.url?scp=0008003639&partnerID=8YFLogxK

U2 - 10.1142/S0217751X96001796

DO - 10.1142/S0217751X96001796

M3 - Article

AN - SCOPUS:0008003639

VL - 11

SP - 3801

EP - 3817

JO - International Journal of Modern Physics A

JF - International Journal of Modern Physics A

SN - 0217-751X

IS - 20

ER -