Hierarchical Schr "odinger-type operators: the case of potentials with\n local singularities

2020/06/02 by Alexander Bendikov, Alexander Grigor'yan, Bendikov, Alexander +4
Computer Science · Mathematics · #35P05 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2006.01821

openalex publication_date 2020/06/02 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

The goal of this paper is twofold. We prove that the operator H=L+V , a\nperturbation of the Taibleson-Vladimirov multiplier L= mathfrakD^\α\nby a potential V(x)=b \Vert x \Vert ^-\α, b\≥ b_\∗, is\nessentially self-adjoint and non-negative definite (the critical value\nb_\∗ depends on \α and will be specified later). While the operator\nH is non-negative definite the potential V(x) may well take negative\nvalues, e.g. b_\∗<0 for all 0<\α<1. The equation Hu=v admiits a\nGreen function g_H(x,y), the integral kernel of the operator H^-1. We\nobtain sharp lower- and upper bounds on the ratio of the functions g_H(x,y)\nand g_L(x,y). Examples illustrate our exposition.\n

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