Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1994-08-09
Int.J.Mod.Phys. A11 (1996) 180
Physics
High Energy Physics
High Energy Physics - Theory
LaTeX, 19 pages, preprint LA-UR-94-2569
Scientific paper
For zero energy, $E=0$, we derive exact, quantum solutions for {\it all} power-law potentials, $V(r) = -\gamma/r^{\nu}$, with $\gamma > 0$ and $-\infty < \nu < \infty$. The solutions are, in general, Bessel functions of powers of $r$. For $\nu > 2$ and $l \ge 1$ the solutions are normalizable; they correspond to states which are bound by the angular-momentum barrier. Surprisingly, the solutions for $\nu < -2$ are also normalizable, They are discrete states but do not correspond to bound states. For $2> \nu \geq -2$ the states are unnormalizable continuum states. The $\nu=2$ solutions are also unnormalizable, but are exceptional solutions. Finally, we find that by increasing the dimension of the \seq beyond 4 an effective centrifugal barrier is created, due solely to the extra dimensions, which is enough to cause binding. Thus, if $D>4$, there are $E=0$ bound states for $\nu > 2$ even for $l=0$. We discuss the physics of the above solutions are compare them to the classical solutions of the preceding paper.
Daboul Jamil
Nieto Michael Martin
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