On ground state of some non local Schrödinger operators

Abstract

We study a ground state of some non-local Schrödinger operator associated with an evolution equation for the density of population in the stochastic contact model in continuum with inhomogeneous mortality rates. We found a new effect in this model, when even in the high-dimensional case the existence of a small positive perturbation of a special form (so-called, small paradise) implies the appearance of the ground state. We consider the problem in the Banach space of bounded continuous functions and in the Hilbert space .

Keywords:

• Dispersal kernel
• discrete spectrum
• spectral radius
• compact operator
• contact model

AMS Subject Classifications:

• 47A10
• 60J35
• 47D06

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work is partially supported by Universitat Bielefeld [SFB 701]. The research of Sergey Pirogov and Elena Zhizhina (Sections 2–3) was supported by the Russian Science Foundation [project number 14-50-00150]; Stanislav Molchanov was supported by NSF [grant number DMS 1008132] (USA).