On the absence of necessary conditions for linear evolution operators

Goldstein, Jerome A.

doi:10.1090/S0002-9939-1977-0500284-2

On the absence of necessary conditions for linear evolution operators
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Proc. Amer. Math. Soc. 64 (1977), 77-80 Request permission

Abstract:

There exist selfadjoint operators $A(t)(t \geqslant 0)$, whose resolvents depend smoothly on t, such that the initial value (Schrödinger) problem $du(t)/dt = iA(t)u(t),u(0) = f$ has a unique $({C^\infty })$ solution u for each f in a dense set in the underlying Hilbert space, with u depending continuously on f, and yet the intersection over $t \geqslant 0$ of the domain of $A(t)$ is {0}.

References

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Functional analysis