Gesztesy F., Holden H., Simon B., Zhao Z.
A Trace Formula for Multidimensional Schrodinger Operators
(37K, AMSTeX)

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 
$[0, 1]^{\nu}$.