Quantum Systems at the Brink: Existence of Bound States, Critical Potentials and Dimensionality (original) (raw)
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It is well known that N-electron atoms undergoes unbinding for a critical charge of the nucleus Z c , i.e. the atom has eigenstates for the case Z > Z c and it has no bound states for Z < Z c. In the present paper we derive upper bound for the bound state for the case Z = Z c under the assumption Z c < N − K where K is the number of electrons to be removed for atom to be stable for Z = Z c without any change in the ground state energy. We show that the eigenvector decays faster as exp −C |x| k where we sum K largest values of |x j |, j ∈ {1,. .. , N }. Our method do not require Born-Oppenheimer approximation.
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