On the Solutions of the Equation Arising from the Singular Limit of Some Eigen Problems
Shuenn‐Jyi Sheu, Alexander D. Wentzell · Birkhäuser Boston eBooks · 1999
We consider a family of matrices \(A^\varepsilon = \left( {A^\varepsilon \left( {i,j} \right)} \right)_{i,j = 1}^M ,\) indexed by ε > 0, having each element the asymptotics \(A^\varepsilon \left( {i,j} \right) \approx \,\exp \,\left( {\varepsilon ^{ - 1} V\left( {i,j} \right)} \right)\). We study the limiting behavior of the principal eigenvalue με and the corresponding normalized eigenvector ϕε for A ε. Under some conditions, we can write a limiting equation for λε = ε log με, W ε(i) = ε log ϕε (i) as ε → 0;. We study it solutions. We show also by some examples that the limit of W ε, when it exists, may depend on the higher order asymptotics of A ε(i,j).