Kernel and trace formula for the exponential of the Laplace-Beltrami operator on a decorated graph |
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Authors: | A. A. Tolchennikov |
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Affiliation: | (1) Department of Mechanics and Mathematics, Moscow State University, Vorob’evy Gory, Moscow, 119991, Russia |
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Abstract: | For the kernel of the Laplace operator ΔΛ with potential Σ j=1 k c j δ q j (x) on a manifold, (the operator is given by a Lagrangian plane Λ ⊂ ℂ k ⊕ ℂ k ), an isomorphism Γ: ker ΔΛ → Λ ∩ L is described, where L is a special Lagrangian plane (whose explicit form is evaluated). A similar assertion holds for the Laplace operator on a decorated graph; for such a graph (obtained by decorating a connected finite graph with n edges and v vertices) with “continuity” conditions, the inequality 1 ≤ dimker ≤ n − v + 2 is obtained. It is also proved that the quantity n − v + 1-dim ker cannot reduce when adding new edges and manifolds. The first terms of the expansion of Tr(exp(-tH Λ)) are found. Dedicated to the memory of V. A. Geyler |
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