  We develop a new method to calculate eigenvalues in frustrated quantum spin
models. It is based on the stochastic state selection (SSS) method, which is an
unconventional Monte Carlo technique we have investigated in recent years. We
observe that a kind of equilibrium is realized under some conditions when we
repeatedly operate a Hamiltonian and a random choice operator, which is defined
by stochastic variables in the SSS method, to a trial state. In this
equilibrium, which we call the SSS equilibrium, we can evaluate the lowest
eigenvalue of the Hamiltonian using the statistical average of the
normalization factor of the generated state.
  The SSS equilibrium itself has been already observed in un-frustrated models.
Our study in this paper shows that we can also see the equilibrium in
frustrated models, with some restriction on values of a parameter introduced in
the SSS method. As a concrete example, we employ the spin-1/2 frustrated J1-J2
Heisenberg model on the square lattice. We present numerical results on the
20-, 32-, 36-site systems, which demonstrate that statistical averages of the
normalization factors reproduce the known exact eigenvalue in good precision.
  Finally we apply the method to the 40-site system. Then we obtain the value
of the lowest energy eigenvalue with an error less than 0.2%.
