Momentum autocorrelation function of an impurity in a classical oscillator chain with alternating masses III. Some limiting cases

• Momentum autocorrelation function of a mass impurity in a diatomic chain.
• 6 limiting cases where any of three masses approaches to zero or infinity.
• The cases m0→0m0→0 and m0→(2m2)+m0→(2m2)+ are closely related to each other.
• A quite different case m2→0m2→0 and its ergodicity.

The momentum autocorrelation function of a mass impurity in a classic diatomic chain is studied using the recurrence relations method. General expressions for the contributions of branch cuts and resonant poles have been derived and illustrated in previous papers I and II, respectively. In the present paper a series of limiting cases that any one of the three masses m0,m1,m2m0,m1,m2 approaches to zero or infinity are analyzed. It is found that the cases m0→0m0→0 and →(2m2)+→(2m2)+ are closely related to each other and that the general expressions for the amplitudes are valid also in the limits λ→0λ→0 and ∞∞. The ergodicity in the case m2→0m2→0 is studied and the ratio of two specific infinite products is obtained.