Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitons

In this paper, we study harmonic and caloric functions of polynomial growth on a complete non-compact gradient shrinking Ricci soliton. On one hand, when the scalar curvature satisfies at least quadratic decay, prove that space with fixed degree is finite dimensional. We also analogous results for ancient functions. other without any condition, sharp dimensional estimates Schrödinger degree.