Local exponential estimates for \(h\)-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators
The author deals with matrix \(h\)-pseudodifferential operators \(O_{P_{h}} (a)\) with symbols \(a = (a_{i j})\) for which \(a_{i j} \in C^{\infty} (\mathbb R_x^n \times \mathbb R_{\xi}^n) \otimes C (0,1]\) satisfying \(|\partial_{x}^{\beta} \partial_{\xi}^{\alpha} a _{i j} (x,\xi,h)| \leq C_{\alpha \beta} \langle \xi \rangle^m\) \((i,j = 1,\dots,N\); \(\langle \xi \rangle = (1 + |\xi|^2)^{1/2})\) for all multi-indices \(\alpha\) and \(\beta\). Under some suitable conditions, the author proves local exponential estimates for the solutions of considered \(h\)-pseudodifferential equations. The results are applied to study the behavior of eigenfunctions of matrix Schrödinger, Dirac and square-root Klein-Gordon operators.
scientific article; zbMATH DE number 5628077
scientific article