- #1

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[tex]

E = E(\epsilon) = E_0 + \epsilon E_1 + \epsilon^2 E_2 + \ldots

[/tex]

converges, where [itex]E_0[/itex] is a discrete eigenvalue of [itex]H_0[/itex] and we are considering a Hamiltonian [itex]H = H_0 + \epsilon H_1[/itex]. Does this mean that we know

[tex]

E - E_0 = O(\epsilon)

[/tex]

as [itex]\epsilon \to 0[/itex] in the precise sense that we know there exists a [itex]\delta > 0[/itex] and a [itex]C > 0[/itex] such that if [itex]|\epsilon| < \delta[/itex], then [itex]|E - E_0| \leq C|\epsilon|[/itex]?