Given a Hermitian line bundle $L$ over a flat torus $M$, a connection $\nabla$ on $L$, and a function $Q$ on $M$, one associates a Schrödinger operator acting on sections of $L$; its spectrum is denoted $Spec(Q;L,\nabla )$. Motivated by work of V. Guillemin in dimension two, we consider line bundles over tori of arbitrary even dimension with “translation invariant” connections $\nabla$, and we address the extent to which the spectrum $Spec(Q;L,\nabla )$ determines the potential $Q$. With a genericity condition, we show that if the connection is invariant under...

We study the special Lagrangian Grassmannian $SU\left(n\right)/SO\left(n\right)$, with $n\ge 3$, and its reduced space, the reduced Lagrangian Grassmannian $X$. The latter is an irreducible symmetric space of rank $n-1$ and is the quotient of the Grassmannian $SU\left(n\right)/SO\left(n\right)$ under the action of a cyclic group of isometries of order $n$. The main result of this paper asserts that the symmetric space $X$ possesses non-trivial infinitesimal isospectral deformations. Thus we obtain the first example of an irreducible symmetric space of arbitrary rank $\ge 2$, which is...