On ground state of non local Schrodinger operators

TL;DR

This paper investigates the ground state of non-local Schrödinger operators linked to population density evolution, revealing that small perturbations can induce ground states even in high dimensions.

Contribution

It introduces a novel effect where minor positive perturbations lead to ground state emergence in high-dimensional non-local Schrödinger operators.

Findings

Small positive perturbations induce ground states in high dimensions

Ground states exist under inhomogeneous mortality rates

Analysis conducted in both $C_b (R^d)$ and $L^2(R^d)$ spaces

Abstract

We study a ground state of a non local Schrodinger 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 $C_b (R^d)$ and in the Hilbert space $L^2(R^d)$.