# Fundamental existence theorems

# Fundamental existence theorems

This chapter deals with the fundamental existence theorems of time-dependent density-functional theory. The first section introduces some important quantum mechanical concepts of time-dependent many-body systems, such as the time-dependent Schrödinger equation, time evolution operators, the continuity equation, and local conservation laws. In the next section, the Runge-Gross theorem and the van Leeuwen theorem are proved. These theorems establish the one-to-one correspondence between time-dependent densities and time-dependent potentials, for given initial states. Therefore, the many-body Hamiltonian, and all quantities following from it, are formally expressible as functionals of the time-dependent density. The proofs of the Runge-Gross and the van Leeuwen theorems are discussed in detail, including some subtle points such as the question of v-representability and the Taylor expansion about the initial time.

*Keywords:*
time-dependent Schrödinger equation, time evolution, continuity equation, local conservation laws, Runge-Gross theorem, van Leeuwen theorem, one-to-one correspondence, time-dependent densities

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