On the Number of Positive Solutions to a Class of Integral Equations

By using the complete discrimination system for polynomials, we study the number of positive solutions in {\small $C[0,1]$} to the integral equation {\small $\phi (x)=\int_0^1k(x,y)\phi ^n(y)dy$}, where {\small $k(x,y)=\phi_1(x)\phi_1(y)+\phi_2(x)\phi_2(y), \phi_i(x)>0, \phi_i(y)>0, 0<x,y<1, i=1,2,$} are continuous functions on {\small $[0,1]$}, {\small $n$} is a positive integer. We prove the following results: when {\small $n= 1$}, either there does not exist, or there exist infinitely many positive solutions in {\small $C[0,1]$}; when {\small $n\geq 2$}, there exist at least {\small 1}, at most {\small $n+1$} positive solutions in {\small $C[0,1]$}. Necessary and sufficient conditions are derived for the cases: 1) {\small $n= 1$}, there exist positive solutions; 2) {\small $n\geq 2$}, there exist exactly {\small $m(m\in \{1,2,...,n+1\})$} positive solutions. Our results generalize the existing results in the literature, and their usefulness is shown by examples presented in this paper.Comment: 9 page

Quantum Phase Transition in the Sub-Ohmic Spin-Boson Model: Extended Coherent-state Approach

We propose a general extended coherent state approach to the qubit (or fermion) and multi-mode boson coupling systems. The application to the spin-boson model with the discretization of a bosonic bath with arbitrary continuous spectral density is described in detail, and very accurate solutions can be obtained. The quantum phase transition in the nontrivial sub-Ohmic case can be located by the fidelity and the order-parameter critical exponents for the bath exponents $s<1/2$ can be correctly given by the fidelity susceptibility, demonstrating the strength of the approach.Comment: 4 pages, 3 figure