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Fractional powers of the algebraic sum of normal operators. (English) Zbl 1092.47027
The author establishes sufficient conditions governing the following fact: if \(A\) and \(B\) are unbounded normal linear operators on a (complex) Hilbert space \(H\), then for each \(\alpha \in (0,1)\) \[ {D}(({\overline{A+B}})^{\alpha}) = {D}(A^{\alpha})\cap {D}(B^{\alpha})= {D}(({\overline{A+B}})^{*\alpha}), \] where \(\overline{A+B}\) denotes the closure of the algebraic sum \(A+B\) of \(A\) and \(B\). This result is applied to characterize the domains of fractional powers of a large class of the Hamiltonians with singular potentials arising in quantum mechanics through the study of the Schrödinger equation.

MSC:
47B25 Linear symmetric and selfadjoint operators (unbounded)
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47B44 Linear accretive operators, dissipative operators, etc.
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