arXiv subjects

Alim Sukhtayev

Publications and source records attributed to Alim Sukhtayev.

21 records · Page 2Linked to original sources

arXiv2015

The Morse and Maslov indices for multidimensional Schrödinger operators with matrix-valued potentials

We study the Schrödinger operator $L=-Δ+V$ on a star-shaped domain $Ω$ in $\mathbb{R}^d$ with Lipschitz boundary $\partialΩ$. The operator is equipped with quite general Dirichlet- or Robin-type boundary conditions induced by operators between $H^{1/2}(\partialΩ)$ and $H^{-1/2}(\partialΩ)$, and the potential takes values in the set of symmetric $N\times N$ matrices. By shrinking the domain and rescaling the operator we obtain a path in the Fredholm-Lagrangian-Grassmannian of the subspace of $H^{1/2}(\partialΩ)\times H^{-1/2}(\partialΩ)$ corresponding to the given boundary condition. The path is formed by computing the Dirichlet and Neumann traces of weak solutions to the rescaled eigenvalue equation. We prove a formula relating the number of negative eigenvalues of $L$ (the Morse index), the signed crossings of the path (the Maslov index), the number of negative eigenvalues of the potential matrix evaluated at the center of the domain, and the number of negative squares of a bilinear form related to the boundary operator.

math.SP↗

arXiv2014

The Morse and Maslov Indices for Schrödinger Operators

We study the spectrum of Schrödinger operators with matrix valued potentials utilizing tools from infinite dimensional symplectic geometry. Using the spaces of abstract boundary values, we derive relations between the Morse and Maslov indices for a family of operators on a Hilbert space obtained by perturbing a given self-adjoint operator by a smooth family of bounded self-adjoint operators. The abstract results are applied to the Schrödinger operators with quasi-periodic, Dirichlet and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for the multidimensional Schrödinger operators with periodic potentials. For quasi convex domains in $\mathbb{R}^n$ we recast the results connecting the Morse and Maslov indices using the Dirichlet and Neumann traces on the boundary of the domain.

math.AP↗

arXiv2014

Vakhitov-Kolokolov and energy vanishing conditions for linear instability of solitary waves in models of classical self-interacting spinor fields

We study the linear stability of localized modes in self-interacting spinor fields, analyzing the spectrum of the operator corresponding to linearization at solitary waves. Following the generalization of the Vakhitov--Kolokolov approach, we show that the bifurcation of real eigenvalues from the origin is completely characterized by the Vakhitov--Kolokolov condition $dQ/dω=0$ and by the vanishing of the energy functional. We give the numerical data on the linear stability in the generalized Gross--Neveu model and the generalized massive Thirring model in the charge-subcritical, critical, and supercritical cases, showing the agreement with the Vakhitov--Kolokolov and the energy vanishing conditions.

math-ph↗