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I am looking for the spectrum of th inverted oscillator $H=-\frac{d^2}{dx^2}-x^2$. Thanks in advance.

Suppose that $X$ is the continuous-time simple symmetric random walk on the lattice $\mathbb Z^d$ (i.e., a simple symmetric random walk with i.i.d. exponential jump times), and let $$u(t,x):=\mathbf E\...