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Gaussian variational bound for an attractive Gaussian well

For

H = - \frac{d ^{2}}{d x ^{2}} - V_{0} e^{- x^{2}}

and the trial state

ψ_{a} (x) = e^{- a x^{2} /2}

,

E_{0} \leq E (a) = \frac{a}{2} - V_{0} \frac{a}{1 + a} .

This is negative for some

a > 0

for every

V_{0} > 0

, proving that the attractive well has a bound state. For small

V_{0}

, taking

a = V_{0}^{2}

gives

- V_{0} < E_{0} \leq - \frac{1}{2} V_{0}^{2} + O (V_{0}^{4}) .

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