A Trace Formula for Multidimensional Schrِdinger Operators

We prove multidimensional analogs of the trace formula obtained previously for one-dimensional Schrödinger operators. For example, letVbe a continuous function on [0, 1]ν⊂Rν. ForA⊂{1, …, ν}, let −ΔAbe the Laplace operator on [0, 1]νwith mixed Dirichlet–Neumann boundary conditions[formula]Let |A|= number of points inA. Then weʹll prove that[formula]with ⦠V⦔ the average ofVat the 2νcorners of [0, 1]ν.