Convergence Radii for Eigenvalues of Tri--diagonal Matrices

Abstract

Consider a family of infinite tri--diagonal matrices of the form $L+zB,$ where the matrix $L$ is diagonal with entries $L_{kk}=k^2,$ and the matrix $B$ is off--diagonal, with nonzero entries $B_{k,k+1}=B_{k+1,k}=k^{\alpha },0\le \alpha <2.$ The spectrum of $L+zB$ is discrete. For small $|z|$ the $n$-th eigenvalue $E_n(z),E_n(0)=n^2,$ is a well--defined analytic function. Let $R_n$ be the convergence radius of its Taylor's series about $z=0.$ It is proved that $$R_n\le C(\alpha )n^{2-\alpha }\text{if}0\le \alpha <11/6.$$

References in corpus (2)

• Analytic continuation of eigenvalues of a quartic oscillator
• Irreducibility of some spectral determinants

Cited by in corpus (3)

• Differential operators admitting various rates of spectral projection growth
• Local form-subordination condition and Riesz basisness of root systems
• Irreducibility of some spectral determinants