  Recently, bootstrap methods from conformal field theory have been adapted for
studying the energy spectrum of various quantum mechanical systems. In this
paper, we consider the application of these methods in obtaining the spectrum
from the Schr\"odinger equation with periodic potentials, paying particular
attention to the Kronig-Penney model of a particle in a one-dimensional
lattice. With an appropriate choice of operator basis involving position and
momenta, we find that the bootstrap approach efficiently computes the band gaps
of the energy spectrum but has trouble effectively constraining the minimum
energy. We show how applying more complex constraints involving higher powers
of momenta can potentially remedy such a problem. We also propose an approach
for analytically constructing the dispersion relation associated with the Bloch
momentum of the system.
