Theoretical calculations are presented to optimize modulation period for maximum total peak capacity in comprehensive two-dimensional HPLC (2D-HPLC) taking into account the effect of modulation on the apparent peak capacity of the first-dimension (1D) separation. Results indicate that modulation periods are most favorable when they are adjusted to approximately 2.2-4 times the standard deviation of a 1D peak in order to avoid excessively short run times at the second dimension (2D). Data are presented that effective peak capacities of several thousand in 60 min can be expected for practical 2D-HPLC conditions, utilizing 1D gradient elution followed by 2D isocratic elution, that remain at approximately 50-70% of the theoretical maximum peak capacity. This work suggests that lower modulation frequencies and longer 2D separation times than previously proposed are favorable under realistic chromatographic conditions, alleviating some practical problems associated with 2D-HPLC.

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The integrated density of states (IDS) for the Schrödinger operators is defined in two ways: by using the counting function of eigenvalues of the operator restricted to bounded regions with appropriate boundary conditions or by using the spectral projection of the whole space operator. A sufficient condition for the coincidence of the two definitions above is given. Moreover, a sufficient condition for the coincidence of the IDS for the Dirichlet boundary conditions and the IDS for the Neumann boundary conditions is given. The proof is based only on the fundamental items in functional analysis, such as the min-max principle, etc.