94-75 Gesztesy F., Holden H., Simon B., Zhao Z.

A Trace Formula for Multidimensional Schrodinger Operators (37K, AMSTeX) Mar 30, 94

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Abstract. We prove multidimensional analogs of the trace formula obtained previously for one-dimensional Schr\"odinger operators. For example, let $V$ be a continuous function on $[0, 1]^{\nu}\subset\Bbb R^{\nu}$. For $A\subset\{1,\dots ,\nu\}$, let $-\Delta_{A}$ be the Laplace operator on $[0, 1]^{\nu}$ with mixed Dirichlet-Neumann boundary conditions $$\alignat2 \varphi(x) &=0, &&\qquad x_{j}=0 \text{ or } x_{j}=1 \quad\text{for } j\in A, \\ \frac{\partial\varphi}{\partial x_{j}}(x) &= 0, &&\qquad x_{j}=0 \text{ or } x_{j}=1 \quad\text{for } j\notin A. \endalignat $$ Let $|A|=$ number of points in $A$. Then we'll prove that $$ \text{Tr}\biggl(\sum_{A\subset\{1,\dots ,\nu\}} (-1)^{|A|} e^{-t(- \Delta_{A}+V)}\biggr)=1-t\langle V\rangle +o(t) \quad\text{as }t\downarrow 0 $$ with $\langle V\rangle$ the average of $V$ at the $2^{\nu}$ corners of $[0, 1]^{\nu}$.

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