Determinant of differential operators & Holomorphy

omg!

hi,

i would appreciate ideas for the proof that

[tex]\det\left(-\frac{d^2}{dx^2}+P(x)-\lambda\right)[/tex]

is a holomorphic function in [tex]\lambda[/tex], if the spectrum of the operator is discrete.

thanks!

sorry, should have posted this here in the first place