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We analyze spectral properties of the Hilbert $L$-matrix $$left(frac{1}{max(m,n)+ u}right)_{m,n=0}^{infty}$$ regarded as an operator $L_{ u}$ acting on $ell^{2}(mathbb{N}_{0})$, for $ uinmathbb{R}$, $ u eq0,-1,-2,dots$. The approach is based on a spectral analysis of the inverse of $L_{ u}$, which is an unbounded Jacobi operator whose spectral properties are deducible in terms of the unit argument ${}_{3}F_{2}$-hypergeometric functions. In particular, we give answers to two open problems concerning the operator norm of $L_{ u}$ published by L. Bouthat and J. Mashreghi in [Oper. Matrices 15, No. 1 (2021), 47--58]. In addition, several general aspects concerning the definition of an $L$-operator, its positivity, and Fredholm determinants are also discussed.
We consider operators of the form H+V where H is the one-dimensional harmonic oscillator and V is a zero-order pseudo-differential operator which is quasi-periodic in an appropriate sense (one can take V to be multiplication by a periodic function fo
We give a short proof of a recently established Hardy-type inequality due to Keller, Pinchover, and Pogorzelski together with its optimality. Moreover, we identify the remainder term which makes it into an identity.
We solve the inverse spectral problem associated with periodic conservative multi-peakon solutions of the Camassa-Holm equation. The corresponding isospectral sets can be identified with finite dimensional tori.
The Neumann-Poincare operator is a boundary-integral operator associated with harmonic layer potentials. This article proves the existence of eigenvalues within the essential spectrum for the Neumann-Poincare operator for certain Lipschitz curves in
We investigate the instability index of the spectral problem $$ -c^2y + b^2y + V(x)y = -mathrm{i} z y $$ on the line $mathbb{R}$, where $Vin L^1_{rm loc}(mathbb{R})$ is real valued and $b,c>0$ are constants. This problem arises in the study of stab