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How do you find for what range of energies the absolute value potential has bound and unbound states?

What I have understood from my previous Intro to Quantum lectures are that in order to derive the allowed energies, one must solve the TISE for that given potential, and through the $k$ value ($k=sqrt(2m(E-V(x)))/(hbar)$ in this case) one can find the allowed energies. Moreover, a given state would be bound if $E<V(x)$ and unbound if $E>V(x)$; please correct me if I'm wrong.

And this makes sense to me when analyzing a square well potential or a barrier potential, but I am rather confused as how to apply this to an absolute value potential. Any guidance of sorts would be greatly appreciated.

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Are you referring to $V(x)= \lambda |x|$? If so the wavefunctions are are Airy functions. The eigenvalues are determined by the zeros of ${\rm Ai}'(x)$ and ${\rm Ai}(x)$ respectivly.

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