arXiv subjects

Selim Sukhtaiev

Publications and source records attributed to Selim Sukhtaiev.

20 records · Page 2Linked to original sources

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

A Bound for the Eigenvalue Counting Function for Higher-Order Krein Laplacians on Open Sets

For an arbitrary nonempty, open set $Ω\subset \mathbb{R}^n$, $n \in \mathbb{N}$, of finite (Euclidean) volume, we consider the minimally defined higher-order Laplacian $(- Δ)^m\big|_{C_0^{\infty}(Ω)}$, $m \in \mathbb{N}$, and its Krein--von Neumann extension $A_{K,Ω,m}$ in $L^2(Ω)$. With $N(λ,A_{K,Ω,m})$, $λ> 0$, denoting the eigenvalue counting function corresponding to the strictly positive eigenvalues of $A_{K,Ω,m}$, we derive the bound $$ N(λ,A_{K,Ω,m}) \leq (2 π)^{-n} v_n |Ω| \{1 + [2m/(2m+n)]\}^{n/(2m)} λ^{n/(2m)}, \quad λ> 0, $$ where $v_n := π^{n/2}/Γ((n+2)/2)$ denotes the (Euclidean) volume of the unit ball in $\mathbb{R}^n$. The proof relies on variational considerations and exploits the fundamental link between the Krein--von Neumann extension and an underlying (abstract) buckling problem.

math.SP↗