Maximal-velocity deficit under a finite-support constraint in hard-wall half-line continuous-time quantum walk
Continuous-time quantum walks on a lattice spread ballistically and converge to a limiting distribution for the rescaled position. On the hard-wall half line the boundary reflects the walker but does not change the bulk dispersion, so the ballistic front remains set by the maximal group velocity. We ask how close one can get to this front when the initial state is constrained to occupy only the first $M$ sites. We show that optimizing the moment-generating function of the limiting-velocity distribution over this finite-support class reduces to the principal eigenvalue of an explicit $M\times M$ Hermitian matrix, which yields the optimal state by direct diagonalization. For large $M$, a near-front scaling limit produces a continuum description that controls the entire peak region. In particular, the maximal mean drift approaches the front with an inverse-square deficit in $M$ and a constant prefactor $\pi^2/4$. Numerical results verify both the finite-$M$ spectral formulation and the predicted scaling.