| Summary: | In the conventional (so-called Schrödinger-picture) formulation of quantum theory the operators of observables are chosen self-adjoint and time-independent. In the recent innovation of the theory, the operators can be not only non-Hermitian but also time-dependent. The formalism (called non-Hermitian interaction-picture, NIP) requires a separate description of the evolution of the time-dependent states <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ψ</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> (using Schrödinger-type equations) as well as of the time-dependent observables <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>Λ</mo><mi>j</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>K</mi></mrow></semantics></math></inline-formula> (using Heisenberg-type equations). In the unitary-evolution dynamical regime of our interest, both of the respective generators of the evolution (viz., in our notation, the Schrödingerian generator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> and the Heisenbergian generator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Σ</mo><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>) have, in general, complex spectra. Only the spectrum of their superposition remains real. Thus, only the observable superposition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>G</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mo>Σ</mo><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> (representing the instantaneous energies) should be called Hamiltonian. In applications, nevertheless, the mathematically consistent models can be based not only on the initial knowledge of the energy operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> (forming a “dynamical” model-building strategy) but also, alternatively, on the knowledge of the Coriolis force <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Σ</mo><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> (forming a “kinematical” model-building strategy), or on the initial knowledge of the Schrödingerian generator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> (forming, for some reason, one of the most popular strategies in the literature). In our present paper, every such choice (marked as “one”, “two” or “three”, respectively) is shown to lead to a construction recipe with a specific range of applicability.
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