Beyond the classical Weyl and Colin de Verdière's formulas for Schrödinger operators with polynomial magnetic and electric fields
Generalizations of the classical Weyl and Colin de Verdière's formulas and the orbit method
The goal of this paper is to write down a conjectural formula for the leading term of the spectral asymptotics of a Schrödinger operator on \(L^2(\mathbb{R}^n)\) with quasi-homogeneous polynomial magnetic and electric fields \(B\) and \(V\). The construction is based on the orbit method due to Kirillov. It makes sense for any nilpotent Lie algebra and is related to the geometry of coadjoint orbits, as well as to the growth properties of certain ``algebraic integrals'', studied by Nilsson. The precise version of the conjecture uses the function \(\Psi^*\) constructed by Helffer and Mohamed: \[ \Psi^*(x):= \sum_\alpha|\partial^\alpha V(x)|^{1/(|\alpha|+ 2)}+ \sum_{\alpha,j,k}|\partial^\alpha B_{jk}(x)|^{1/(|\alpha|+ 2)}, \] compared with the function \[ \Phi^*(x):= \sum_\alpha |\partial^\alpha V(x)|^{1/2}+ \sum_{\alpha, j,k} |\partial^\alpha B_{jk}(x)|^{1/2}. \] By using the direct variational method, the authors prove that the formulas give the correct answer not only in the ``regular'' cases where the classical formulas of Weyl or Colin de Verdière are applicable, but in may ``irregular'' cases, such as for the Schrödinger operator in 2D with magnetic tensor \(B(x)= x^k_1 x^\ell_2\) \((k\geq \ell\geq 1)\) and zero electric potential.
scientific article; zbMATH DE number 5169142
scientific article