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Discreteness of the spectrum for some differential operators with unbounded coefficients in R n

Giorgio MetafuneDiego Pallara — 2000

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We give sufficient conditions for the discreteness of the spectrum of differential operators of the form A u = - u + F , u in L μ 2 R n where d μ x = e - F x d x and for Schrödinger operators in L 2 R n . Our conditions are also necessary in the case of polynomial coefficients.

The domain of the Ornstein-Uhlenbeck operator on an L p -space with invariant measure

Giorgio MetafuneJan PrüssAbdelaziz RhandiRoland Schnaubelt — 2002

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We show that the domain of the Ornstein-Uhlenbeck operator on L p ( N , μ d x ) equals the weighted Sobolev space W 2 , p ( N , μ d x ) , where μ d x is the corresponding invariant measure. Our approach relies on the operator sum method, namely the commutative and the non commutative Dore-Venni theorems.

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