Invertible harmonic oscillator

Question

Let $T=-\frac{d^2}{dx^2}-x^2$ be the invertible harmonic oscillator.

What is the explicite formula of $(T-i\xi)^{-1}$ with $\xi\in\Bbb{R}^*$.

Any help will be appreciated a lot of.

Answer 1

$T$ is a self-adjoint operator (provided you defined its domain in a sensible way), so yes, the operator $(T-i\mu)^{-1}$ exists for any $0\neq\mu\in\mathbb{R}$. But I don't think there is a nice formula for it other than simply writing "$(T-i\mu)^{-1}$": If $f=(T-i\mu)^{-1}g$, then $f$ is a solution of the differential equation $$(-\frac{d^2}{dx^2}-x^2-i\mu)f(x)=g(x)$$ for example for $\mu=1$ this can be solved by wolfram alpha (you can fix the constants $c_1$ and $c_2$ by boundary conditions $f(\pm\infty)=0$). But its not a beautiful expression, so I doubt that this is useful for you.