| Stone's Theorem On One-parameter Unitary Groups |
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Information AboutStone's Theorem On One-parameter Unitary Groups |
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: which are Strongly Continuous , that is : and are homomorphisms: : Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups. The theorem is named after Marshall Stone who formulated and proved this theorem in 1932 . The formal statement is as follows: Theorem. Let ''A'' be a self-adjoint operator on a Hilbert space ''H''. Then : is a strongly continuous one-parameter family of unitary operators. The Infinitesimal Generator of {''U''''t''}''t'' is the operator i ''A''. This mapping is a bijective correspondence. ''A'' will be a bounded operator Iff the operator-valued function ''t'' → ''U''''t'' is Norm continuous. Example. The family of translation operators : is a one-parameter unitary group of unitary operators; the infinitesimal generator of this family is an extension of the differential operator : defined on the space of complex-valued continuously differentiable functions of compact support on R. Thus : Stone's theorem has numerous applications in Quantum Mechanics . For instance, given an isolated quantum mechanical system, with Hilbert space of states ''H'', time evolution is a strongly continuous one-parameter unitary group on ''H''. The infinitesimal generator of this group is the system Hamiltonian . The Hille-Yosida Theorem is a generalization of Stone's theorem to strongly continuous one-parameter semigroups of Contraction s on a Banach Space s. REFERENCES
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