Let $$L(t) = - \mathrm{div} \left( A(x,t) \nabla _x \right) $$ for $$t \in (0, \tau )$$ be a uniformly elliptic operator with boundary conditions on domain $$\Omega of $$\mathbb {R}^d$$ and $$\partial \frac{\partial }{\partial t}$$ . Define the parabolic $${{\mathcal {L}}}= \partial + L$$ $$L^2(0, , L^2(\Omega ))$$ by $$({{\mathcal {L}}}u)(t) := u(t)}{\partial t} L(t)u(t)$$ We assume very littl...

We study fluctuations of the matrix coefficients for the quantized cat map. We consider the sum of matrix coefficients corresponding to eigenstates whose eigenphases lie in a randomly chosen window, assuming that the length of the window shrinks with Planck’s constant. We show that if the length of the window is smaller than the square root of Planck’s constant, but larger than the separation b...