  We present a semidefinite program (SDP) algorithm to find eigenvalues of
Schr\"{o}dinger operators within the bootstrap approach to quantum mechanics.
The bootstrap approach involves two ingredients: a nonlinear set of constraints
on the variables (expectation values of operators in an energy eigenstate),
plus positivity constraints (unitarity) that need to be satisfied. By fixing
the energy we linearize all the constraints and show that the feasability
problem can be presented as an optimization problem for the variables that are
not fixed by the constraints and one additional slack variable that measures
the failure of positivity. To illustrate the method we are able to obtain
high-precision, sharp bounds on eigenenergies for arbitrary confining
polynomial potentials in 1-D.
