Extremal values of the sojourn time
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Date
2010
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Abstract
Consider a self-adjoint operator H on a separable Hilbert space H with non-trivial absolutely continuous component. We study the general properties of the real-valued functional, tau(H) (psi) = f(R) vertical bar(e(-itH)psi, psi)vertical bar(2) dt, which in quantum mechanics represents the sojourn time (or life time) of an initial state psi is an element of H. We characterize the critical points of the sojourn time, tau(X), of the operator multiplication by x in L(2)(R), and prove that it attains a global maximum in the unit sphere of the Sobolev space W(1,2)(R).