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Consider a one-dimensional Hamilton operator of the form

[tex] H = \frac{P^2}{2M} - |v\rangle V \langle v| [/tex]

where the potential strength V is a postive constant and [tex] |v \rangle\langle v| [/tex] is a normalised projector, [tex] \langle v|v \rangle = 1 [/tex]. Determine all negative eigenvalues of H if [tex] |v \rangle [/tex] has the position wave function [tex] \langle x|v \rangle = \sqrt{\kappa} e^{- \kappa |x|} [/tex] with [tex] \kappa > 0 [/tex].

It seems to me the only step i could take is to apply Hamilton H to x bra and v ket:

[tex] \langle x|H|v \rangle = - \frac{\hbar ^2}{2M} \left( \frac{\partial}{\partial x} \right)^2 \langle x|v \rangle -\langle x|v \rangle V [/tex]

which gives [tex] = - \left( \frac{\hbar ^2 \kappa ^2}{2M} + V \right) \langle x|v \rangle [/tex]

So how do I proceed (if i am right thus far)?

Can someone give me some hints as to how to solve this problem?

Thanks.