We give a necessary and sufficient condition for the maximum principle of Schr\"{o}dinger operators in terms of the bottom of the spectrum of time-changed processes. As a corollary, we obtain a sufficient condition for the Liouville property of Schr\"{o}dinger operators.
Backward Volterra equations with time-changed Lévy noise and maximum principles
Heat content and small time asymptotics for Schrödinger operators on $R^d$
Existence and construction of quasi-stationary distributions for one-dimensional diffusions
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