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A non-Hermitian operator <i>H</i> defined in a Hilbert space with inner product <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">〈</mo> <mo>·</mo> <mo stretchy="false">|</mo> <mo>·</mo> <mo stretchy="false">〉</mo> </mrow> </semantics> </math> </inline-formula> may serve as the Hamiltonian for a unitary quantum system if it is <inline-formula> <math display="inline"> <semantics> <mi>η</mi> </semantics> </math> </inline-formula>-pseudo-Hermitian for a metric operator (positive-definite automorphism) <inline-formula> <math display="inline"> <semantics> <mi>η</mi> </semantics> </math> </inline-formula>. The latter defines the inner product <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">〈</mo> <mo>·</mo> <mo stretchy="false">|</mo> <mi>η</mi> <mo>·</mo> <mo stretchy="false">〉</mo> </mrow> </semantics> </math> </inline-formula> of the physical Hilbert space <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">H</mi> <mi>η</mi> </msub> </semantics> </math> </inline-formula> of the system. For situations where some of the eigenstates of <i>H</i> depend on time, <inline-formula> <math display="inline"> <semantics> <mi>η</mi> </semantics> </math> </inline-formula> becomes time-dependent. Therefore, the system has a non-stationary Hilbert space. Such quantum systems, which are also encountered in the study of quantum mechanics in cosmological backgrounds, suffer from a conflict between the unitarity of time evolution and the unobservability of the Hamiltonian. Their proper treatment requires a geometric framework which clarifies the notion of the energy observable and leads to a geometric extension of quantum mechanics (GEQM). We provide a general introduction to the subject, review some of the recent developments, offer a straightforward description of the Heisenberg-picture formulation of the dynamics for quantum systems having a time-dependent Hilbert space, and outline the Heisenberg-picture formulation of dynamics in GEQM.