Absorption semigroups and Dirichlet boundary conditions (Q1318014)

Given a positive \(C_ 0\)-semigroup \(T\) on \(L^ p(X)\) and an arbitrary non-negative potential \(V\), an absorption semigroup \(T_ V\) is constructed as a \(C_ 0\)-semigroup on \(L^ p(X_ V)\) for a certain subset \(X_ V\) of \(X\). Information is obtained about \(X_ V\). When \(p=1\) and \(T\) is contractive and holomorphic, \(T_ V\) is also holomorphic. It follows that Schrödinger semigroups for arbitrary non- negative potentials are holomorphic on the appropriate part of \(L^ 1({\mathbf R}^ N)\), and the semigroup on \(L^ 1(\Omega)\) generated by the Laplacian with Dirichlet boundary conditions is holomorphic, for arbitrary open sets \(\Omega\).