Exact minimal effective amounts of three 1D continuous functions and their use in Anderson transitions

A conceptual quantity—the minimal effective amount of a quantum state ϕ ( r j ) in d-dimensional systems, defined by N ∗ = ∑ j = 1 N min { N | ϕ ( r j ) | 2 , 1 } , is newly proposed, where system sizes N = L d . The effective dimension d IR can be calculated by N ∗ = h ∗ ( L ) L d I R , where h ∗ ( L ) does not change faster than any nonzero power. However, the nature of h ∗ ( L ) is unknown priori in any given model, but is at the same time very important for its numerical analysis. Hence, analytical results can provide insights on h ∗ ( L ) in more complex situations. In this paper, we get exact results of 1D continuous sine functions, exponential decay functions and power-law decay functions. They are used to distinguish extended and localized phases in the 1D uniform potential model, Anderson model and HMP (hopping rates modulated by a power-law function) model.