Pointwise Properties of Eigenfunctions and Heat Kernels of Dirichlet–Schrödinger Operators

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Let D ⊂ ℝd be an open domain, H0 = ∑i,j=1d∂i (ai,j(x)∂j) a second order elliptic operator with continuous coefficients, and let H=H0+q (q: D ↦ [0,∞)) be a Schrödinger operator associated with H0, acting on L2(D), with Dirichlet boundary conditions. We provide in this paper both L2 and pointwise bounds for the eigenfunctions of H, in term of the Agmon's metric of q and of the quasi-hyperbolic geometry of D. At least when H0 = −Δ, we show that the pointwise bounds obtained for the ground state eigenfunction of H are qualitatively sharp either when q diverges sufficiently fast at the boundary, or in planar regular domains. We also give applications to the intrinsic ultracontractivity of H. Finally, we prove a result concerning the pointwise decay at the boundary of the heat kernel of H in Δ-regular domains.

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