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The generalized constrained search formalism is used to address the issuesconcerning density-to-potential mapping for excited states in time-independentdensity-functional theory. The multiplicity of potentials for any given densityand the uniqueness in density-to-potential mapping are explained within theframework of unified constrained search formalism for excited-states due toG\"orling, Levy-Nagy, Samal-Harbola and Ayers-Levy. The extensions ofSamal-Harbola criteria and it's link to the generalized constrained searchformalism are revealed in the context of existence and unique construction ofthe density-to-potential mapping. The close connections between the proposedcriteria and the generalized adiabatic connection are further elaborated so asto keep the desired mapping intact at the strictly correlated regime.Exemplification of the unified constrained search formalism is done throughmodel systems in order to demonstrate that the seemingly contradictory resultsreported so far are neither the true confirmation of lack of Hohenberg-Kohntheorem nor valid representation of violation of Gunnarsson-Lundqvist theoremfor excited states. Hence the misleading interpretation of subtle differencesbetween the ground and excited state density functional formalism areexemplified.

Excited-State Density-Functional Theory Revisited: on the Uniqueness, Existence, and Construction of the Density-to-Potential Mapping