  Transition states or quantum states of zero energy appear at the boundary
between the discrete part of the spectrum of negative energies and the
continuum part of positive energy states. As such, transition states can be
regarded as a limiting case of a bound state with vanishing binding energy,
emerging for a particular set of critical potential parameters. In this work we
study the properties of these critical parameters for short range central
potentials. To this end we develop two exact methods and also utilize the first
and second order WKB approximations. Using these methods we have calculated the
critical parameters for several widely used central potentials.
  The general analytic expressions for the asymptotic representations of the
critical parameters were derived for cases where either the orbital quantum
number $l$ or the number $n$ of bound states approaches infinity.
  The above mathematical models enable us to answer the following physical
(quantum mechanical) questions: i) what is the number of bound states for a
given central potential and given orbital quantum number $l$; ii) what is the
maximum value of $l$ which can provide a bound state for the given central
potential; iii) what is the order of energy levels for the given form of the
central potential.
  It is revealed that the ordering of energy levels depends on the potential
singularity at the origin.
