When a particle in one dimension have discrete spectrum?

[fab333]

What are the conditions for which it can be concluded that a system has discrete energy levels?
For example a system in one dimension with the potential
$V(x)=b|x|$
has only a discrete spectrum. How I can prove it?
My book says moreover that the energy eigenvalues have to satisfy the condition
$\lmoustache_{x_1}^{x_2} dx \sqrt{2m[E- \lambda |x|]} = (n+1/2) \pi \hbar$
why?

thanks for help.

 

[ardie]

In general, in cases where the minimum potential is bigger than the energy of the particle, you find that the general solution will be complex, and the energy takes a continuous form. on the other hand when the energy is bigger than the minimum potential, the solutions take discrete values as we have a bound particle.