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Register a new userA remark on the discrete spectrum of non-self-adjoint Jacobi operators
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Authors
Leonid Golinskii
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We study the trace class perturbations of the whole-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we refine the Lieb--Thirring inequality due to Hansmann--Katriel. The spectral enclosure for such operators is also discussed.
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We study the trace class perturbations of the half-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we obtain the Lieb--Thirring inequalities for such operators. The spectral enclosure for the discrete spectrum and embedded eigenvalues are also discussed.
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 stability of solitons for certain nonlinear equations (e.g., the short pulse equation and the generalized Bullough-Dodd equation). We show how to apply the standard approach in the situation under consideration and as a result we provide a formula for the instability index in terms of certain spectral characteristics of the 1-D Schrodinger operator $H_V=-c^2frac{d^2}{dx^2}+b^2 +V(x)$.
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