Energy spectrum for a step potential

One ought to clarify the kind of step you are referring to, in particular in connection with boundary conditions.

On the real line (extending from $-infty$ to $+infty$), the potential step is defined typically as $$ V(x)=left{begin{array}{cc} 0&hbox{if } xle 0, , V_0&hbox{if } x> 0, . end{array}right. $$ In this case this potential does not accommodate bound states and discrete values of energy for any $V_0$ (positive or negative). The solutions are plane waves with different wave vectors $k_+$ and $k_-$ (depending on $V_0$) on the positive or negative portion of the axis. This setup is typically studied for scattering by the step.

If, on the other hand, $$ V(r)=left{begin{array}{cc} -V_0&hbox{if } r<r_0, , 0&hbox{if } rge 0, ,end{array}right. $$ with $V_0>0$ and $rge 0$, the restriction on $r$ functions as an infinite barrier at $r=0$. This is then a finite potential well, which can accommodate one bound state and possible more depending on the depth $V_0$ and the range $r_0$. This kind of potential is a crude approximation to the Wood-Saxon potential of nuclear physics.