×

Lower bounds on the lowest spectral gap of singular potential Hamiltonians. (English) Zbl 1113.81054

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.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35J10 Schrödinger operator, Schrödinger equation
35P15 Estimates of eigenvalues in context of PDEs

Citations:

Zbl 0661.35062
PDFBibTeX XMLCite
Full Text: DOI arXiv