This is either 

Helffer-Sjostrand

https://www.tandfonline.com/doi/abs/10.1080/03605308408820335

or Barry Simon

https://www.jstor.org/stable/pdf/2007072.pdf?refreqid=excelsior%3A258084917fff9e0c10088abbb2679c55

PS: I cannot resist pointing out that the last paper is in Annals of Mathematics proving, as you notice yourself, an exercise in Landau/Lifshitz.