Schrödinger Operators and the Zeros of Their Eigenfunctions

Schrödinger operators with a complex valued potential

“…If one assumes V is real valued, then our present result yields the theorem at the end of [2] in the case where the manifold has no boundary.…”

“…It is known that one can always find a coordinate patch about p in which Λ = dx 1 ∧• • •∧dx n . In [2] it is proven that if D ∇u denotes Lie differentiation with respect to the vector field ∇u, then the imaginary part of (Δψ/ψ)e 2k Λ equals D ∇u (e 2k Λ). A computation in [2] shows that in our special coordinate system,…”

“…It should be mentioned that the principal part of [2] is devoted to proving a theorem of the present type in the case in which M n is complete rather than compact. To compensate for the absence of compactness, one assumes that our real valued potential V is bounded below.…”