arXiv subjects
Publications and source records attributed to Konstantin Pankrashkin.
We provide a new proof of the following inequality: the maximum curvature $k_\mathrm{max}$ and the enclosed area $A$ of a smooth Jordan curve satisfy $k_\mathrm{max}\ge \sqrt{π/A}$. The feature of our proof is the use of the curve shortening flow.
We consider the Laplacian on a class of smooth domains $Ω\subset \mathbb{R}^ν$, $ν\ge 2$, with attractive Robin boundary conditions: \[ Q^Ω_αu=-Δu, \quad \dfrac{\partial u}{\partial n}=αu \text{ on } \partialΩ, \ α>0, \] where $n$ is the outer unit normal, and study the asymptotics of its eigenvalues $E_{j}(Q^Ω_α)$ as well as some other spectral properties for $α\to+\infty$ We work with both compact domains and non-compact ones with a suitable behavior at infinity. For domains with compact $C^2$ boundaries and fixed $j$, we show that \[ E_{j}(Q^Ω_α)=-α^2+μ_j(α)+{\mathcal O}(\log α), \] where $μ_j(α)$ is the $j^{\mbox{th}}$ eigenvalue, as soon as it exists, of $-Δ_{S}-(ν-1)αH$ with $(-Δ_{S})$ and $H$ being respectively the positive Laplace-Beltrami operator and the mean curvature on $\partialΩ$. Analogous results are obtained for a class of domains with non-compact boundaries. In particular, we discuss the existence of eigenvalues in non-compact domains and the existence of spectral gaps for periodic domains. We also show that the remainder estimate can be improved under stronger regularity assumptions. The effective Hamiltonian $-Δ_{S}-(ν-1)αH$ enters the framework of semi-classical Schrödinger operators on manifolds, and we provide the asymptotics of its eigenvalues in the limit $α\to+\infty$ under various geometrical assumptions. In particular, we describe several cases for which our asymptotics provides gaps between the eigenvalues of $Q^Ω_α$ for large $α$.
Let $Ω\subset \mathbb{R}^2$ be the exterior of a convex polygon whose side lengths are $\ell_1,...,\ell_M$. For $α>0$, let $H^Ω_α$ denote the Laplacian in $Ω$, $u\mapsto -Δu$, with the Robin boundary conditions $\partial u/\partialν=αu$, where $ν$ is the exterior unit normal at the boundary of $Ω$. We show that, for any fixed $m\in\mathbb{N}$, the $m$th eigenvalue $E^Ω_m(α)$ of $H^Ω_α$ behaves as \[ E^Ω_m(α)=-α^2+μ^D_m +\mathcal{O}\Big(\dfrac{1}{\sqrtα}\Big) \quad {as $α$ tends to $+\infty$}, \] where $μ^D_m$ stands for the $m$th eigenvalue of the operator $D_1\oplus...\oplus D_M$ and $D_n$ denotes the one-dimensional Laplacian $f\mapsto -f"$ on $(0,\ell_n)$ with the Dirichlet boundary conditions.
We consider a singular Schrödinger operator in $L^2(\mathbb{R}^2)$ written formally as $-Δ- βδ(x-γ)$ where $γ$ is a $C^4$ smooth open arc in $\mathbb{R}^2$ of length $L$ with regular ends. It is shown that the $j$th negative eigenvalue of this operator behaves in the strong-coupling limit, $β\to +\infty$, asymptotically as \[ E_j(β)=-\frac{β^2}{4} +μ_j +\mathcal{O}\Big(\dfrac{\logβ}β\Big), \] where $μ_j$ is the $j$th Dirichlet eigenvalue of the operator \[ -\frac{d^2}{ds^2} -\frac{κ(s)^2}{4}\, \] on $L^2(0,L)$ with $κ(s)$ being the signed curvature of $γ$ at the point $s\in(0,L)$.
We consider the Laplacian with attractive Robin boundary conditions, \[ Q^Ω_αu=-Δu, \quad \dfrac{\partial u}{\partial n}=αu \text{ on } \partialΩ, \] in a class of bounded smooth domains $Ω\in\mathbb{R}^ν$; here $n$ is the outward unit normal and $α>0$ is a constant. We show that for each $j\in\mathbb{N}$ and $α\to+\infty$, the $j$th eigenvalue $E_j(Q^Ω_α)$ has the asymptotics \[ E_j(Q^Ω_α)=-α^2 -(ν-1)H_\mathrm{max}(Ω)\,α+{\mathcal O}(α^{2/3}), \] where $H_\mathrm{max}(Ω)$ is the maximum mean curvature at $\partial Ω$. The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of $H_\mathrm{max}$. In particular, we show that the ball is the strict minimizer of $H_\mathrm{max}$ among the smooth star-shaped domains of a given volume, which leads to the following result: if $B$ is a ball and $Ω$ is any other star-shaped smooth domain of the same volume, then for any fixed $j\in\mathbb{N}$ we have $E_j(Q^B_α)>E_j(Q^Ω_α)$ for large $α$. An open question concerning a larger class of domains is formulated.
We study the Robin Laplacian in a domain with two corners of the same opening, and we calculate the asymptotics of the two lowest eigenvalues as the distance between the corners increases to infinity.
We establish a link between the basic properties of the discriminant of periodic second-order differential equations and an elementary analysis of Herglotz functions. Some generalizations are presented using the language of self-adjoint extensions.
We establish several new relations between the discrete transition operator, the continuous Laplacian and the averaging operator associated with combinatorial and metric graphs. It is shown that these operators can be expressed through each other using explicit expressions. In particular, we show that the averaging operator is closely related with the solutions of the associated wave equation. The machinery used allows one to study a class of infinite graphs without assumption on the local finiteness.
Let $Ω\subset \RR^2$ be a domain having a compact boundary $Σ$ which is Lipschitz and piecewise $C^4$ smooth, and let $ν$ denote the inward unit normal vector on $Σ$. We study the principal eigenvalue $E(β)$ of the Laplacian in $Ω$ with the Robin boundary conditions $\partial f/\partialν+βf=0$ on $Σ$, where $β$ is a positive number. Assuming that $Σ$ has no convex corners we show the estimate $E(β)=-β^2- γ_\mxβ+ O\big(β^\{2}{3}\big)$ as $β\to+\infty$, where $γ_\mx$ is the maximal curvature of the boundary.
We consider small perturbations of the Laplace operator in a multi-dimensional cylindrical domain by second order differential operators with periodic coefficients. We show that under certain non-degeneracy conditions such perturbations can open a gap in the continuous spectrum and give the leading asymptotic terms for the gap edges. We also estimate the values of quasi-momentum at which the spectrum edges are attained. The general machinery is illustrated by several new examples in two- and three-dimensional structures.
The spectral problem for self-adjoint extensions is studied using the machinery of boundary triplets. For a class of symmetric operators having Weyl functions of a special type we calculate explicitly the spectral projections in the form of operator-valued integrals. This allows one to give a constructive proof of the fact that, in certain intervals, the resulting self-adjoint extensions are unitarily equivalent to a certain parameterizing operator acting in a smaller space, and one is able to provide an explicit form the associated unitary transform. Applications to differential operators on metric graphs and to direct sums are discussed.
We develop the machinery of boundary triplets for one-dimensional operators generated by formally self-adjoint quasi-differential expression of arbitrary order on a finite interval. The technique are then used to describe all maximal dissipative, accumulative and self-adjoint extensions of the associated minimal operator and its generalized resolvents in terms of the boundary conditions. Some specific classes are considered in greater detail.
We consider a class of self-adjoint extensions using the boundary triple technique. Assuming that the associated Weyl function has the special form $M(z)=\big(m(z)\Id-T\big) n(z)^{-1}$ with a bounded self-adjoint operator $T$ and scalar functions $m,n$ we show that there exists a class of boundary conditions such that the spectral problem for the associated self-adjoint extensions in gaps of a certain reference operator admits a unitary reduction to the spectral problem for $T$. As a motivating example we consider differential operators on equilateral metric graphs, and we describe a class of boundary conditions that admit a unitary reduction to generalized discrete laplacians.
The spectral problem on a periodic domain with cracks is studied. An asymptotic form of dispersion relations is calculated under assumption that the opening of the cracks is small.
We study Laplace-type operators on hybrid manifolds, i.e. on configurations consisting of closed two-dimensional manifolds and one-dimensional segments. Such an operator can be constructed by using the Laplace-Beltrami operators on each component with some boundary conditions at the points of gluing. The large spectral parameter expansion of the trace of the second power of the resolvent is obtained. Some questions of the inverse spectral theory are adressed.
The spectral properties of the Laplacian on a class of quantum graphs with random metric structure are studied. Namely, we consider quantum graphs spanned by the simple $\ZZ^d$-lattice with $δ$-type boundary conditions at the vertices, and we assume that the edge lengths are randomly independently identically distributed. Under the assumption that the coupling constant at the vertices does not vanish, we show that the operator exhibits the Anderson localization at the bottom of the spectrum almost surely. We also study the case of other spectral edges.
We study quantum graphs corresponding to isotropic lattices with quasiperiodic coupling constants given by the same expressions as the coefficients of the discrete surface Maryland model. The absolutely continuous and the pure point spectra are described. It is shown that the transition between them is governed by the Hill operator corresponding to the edge potential.
The two-dimensional Schroedinger operator with a uniform magnetic field and a periodic zero-range potential is considered. For weak magnetic fields we reduce the spectral problem to the semiclassical analysis of one-dimensional Harper-like operators. This shows the existence of parts of Cantor structure in the spectrum for special values of the magnetic flux.