Higher Order Terms of the Spectral Heat Content for Killed Subordinate and Subordinate Killed Brownian Motions Related to Symmetric α-Stable Processes in ${\mathbb {R}}$

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We investigate the 3rd term of the spectral heat content for killed subordinate and subordinate killed Brownian motions on a bounded open interval D = (a,b) in a real line when the underlying subordinators are stable subordinators with index α ∈ (1, 2) or α = 1. We prove that in the 3rd term of the spectral heat content, one can observe the length b − a of the interval D.