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This is a short survey on the connection between general extension theoriesand the study of realizations of elliptic operators A on smooth domains in R^n,n > 1. The theory of pseudodifferential boundary problems has turned out to bevery useful here, not only as a formulational framework, but also for thesolution of specific questions. We recall some elements of that theory, andshow its application in several cases (including recent results), namely to thelower boundedness question, and the question of spectral asymptotics fordifferences between resolvents.

Extension theory for elliptic partial differential operators with pseudodifferential methods