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Filtered subspace iterations can be used to approximate a finite cluster of eigenvalues of a lower semi-bounded selfadjoint operator in a Hilbert space. Prototype examples of such operators are Schrödinger operators with short-range potentials. A rational function (filter) of the operator is constructed such that the eigenspace of interest (eigenvalues below the infimum of the essential spectrum) is its dominant eigenspace, and a subspace iteration procedure is used to approximate this eigenspace. To approximate an operator in an unbounded domain we use a sequence of finitely truncated domains whose union is the whole space. We present an adaptive multispace algorithm based on a posteriori error estimation. Numerical experiments with spectral and finite element approximation methods confirm the theoretical results. We also discuss an application of the results for other moving domain eigenvalue problems.