Analysis of the heat kernel of the Dirichlet-to-Neumann operator (Q466234)

Let \(\Omega \subset \mathbb{R}^d\) be a bounded domain with boundary of class \(C^\infty \). Let \(V\in L_\infty (\Omega )\) and suppose that \(V\geq 0\). Let \(N_V\) be the Dirichlet-to-Neumann operator corresponding to the Dirichlet problem \((-\Delta +V)u=0\) in \(\Omega \), \(u=g\) on \(\partial \Omega \). It is proved that \(N_V\) is a positive self-adjoint operator in \(L_2(\partial \Omega )\). Denote by \(K^V\) the kernel of the semigroup generated by \(-N_V\). Estimates of \(K^V(x,y)\) are given.