Kondej, Sylwia; Veselić, Ivan Lower bounds on the lowest spectral gap of singular potential Hamiltonians. (English) Zbl 1113.81054 Ann. Henri Poincaré 8, No. 1, 109-134 (2007). Basically, the paper studies quantum Hamiltonians with singular potentials, also called singular interactions. Namely, it is considered the following Hamilatonian \(-\Delta-\alpha\delta(x-\Gamma)\), where \(\alpha>0\) and \(\Gamma\)-submanifold of \(\mathbb R^d\). Authors extend the results of W. Kirsch and B. Simon [J. Funct. Anal. 75, 396–410 (1987; Zbl 0661.35062)] for singular potentials. The main results of the paper can be formulated as follows: Let \(2R\) be the diameter of \(\Gamma\). Then for the first lowest eigenvalues \(E_1\), \(E_0\) of the Schrödinger operator, the following lower bound holds \[ E_1 - E_0\geq \kappa^2_1\mu_{\Gamma,\alpha}(\rho,\kappa_0)e^{-C_0\rho}, \] with \(\rho:= \kappa_0R\), where \(\kappa_i=\sqrt{-E_i}\) and \(C_0\) is a constant. The function \(\mu_{\Gamma,\alpha}\) is explicitly written in the paper. To obtain the result it is generalized techniques of the mentioned paper to singular potentials to estimate the first spectral gap. Moreover, the behavior of eigenfunctions is analyzed and estimates for gradients of those, in particular near the support of the singular potential. Reviewer: Farruh Mukhamedov (Kuantan) Cited in 11 Documents MSC: 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 35J10 Schrödinger operator, Schrödinger equation 35P15 Estimates of eigenvalues in context of PDEs Keywords:lower bounds; lowest spectral gap; singular potential Citations:Zbl 0661.35062 PDFBibTeX XMLCite \textit{S. Kondej} and \textit{I. Veselić}, Ann. Henri Poincaré 8, No. 1, 109--134 (2007; Zbl 1113.81054) Full Text: DOI arXiv