[Physics] Bound States in a Double Delta Function Potential

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Let $V(x) = −u \delta(x) – v \delta(x − a)$ where $u, v > 0$ correspond to a potential with two $\delta$ wells. Let $v > u$. If $a$ is very large, there is certainly a bound state: the particle sits in the $\delta$-well. As $a$ decreases to a certain critical value, the bound state disappears. I need help finding that value.

My idea was: Before the bound state disappears, its energy approaches $0$. I'm trying to assume that the energy $E$ is a very small negative number, solve the Schrodinger equation, and find the suitable value of $a$, but I'm having trouble doing this.

Would someone be able to help me with this problem?

Answer 1

I'm not going to answer your exact question, but this is a good example (from an old copy of Griffith's that my loser chem bro uses [real women and men of physics use Shankar and Sakurai]

Consider the double delta-function potential $$V(x)=-\alpha[\delta(x+a)+\delta(x-a)]$$ where $a$ and $\alpha$ are positive constants. Hope this helps! -Dylan