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Register a new userAn estimate for the Steklov zeta function of a planar domain derived from a first variation formula
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We consider the Steklov zeta function $zeta$ $Omega$ of a smooth bounded simply connected planar domain $Omega$ $subset$ R 2 of perimeter 2$pi$. We provide a first variation formula for $zeta$ $Omega$ under a smooth deformation of the domain. On the base of the formula, we prove that, for every s $in$ (--1, 0) $cup$ (0, 1), the difference $zeta$ $Omega$ (s) -- 2$zeta$ R (s) is non-negative and is equal to zero if and only if $Omega$ is a round disk ($zeta$ R is the classical Riemann zeta function). Our approach gives also an alternative proof of the inequality $zeta$ $Omega$ (s) -- 2$zeta$ R (s) $ge$ 0 for s $in$ (--$infty$, --1] $cup$ (1, $infty$); the latter fact was proved in our previous paper [2018] in a different way. We also provide an alternative proof of the equality $zeta$ $Omega$ (0) = 2$zeta$ R (0) obtained by Edward and Wu [1991].
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We consider the zeta function $zeta_Omega$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $Omega$ bounded by a smooth closed curve of perimeter $2pi$. We prove that $zeta_Omega(0)ge zeta_{mathbb{D}}(0)$ with equality if and only if $Omega$ is a disk where $mathbb{D}$ denotes the closed unit disk. We also provide an elementary proof that for a fixed real $s$ satisfying $sle-1$ the estimate $zeta_Omega(s)ge zeta_{mathbb{D}}(s)$ holds with equality if and only if $Omega$ is a disk. We then bring examples of domains $Omega$ close to the unit disk where this estimate fails to be extended to the interval $(0,2)$. Other computations related to previous works are also detailed in the remaining part of the text.
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We consider a system of two coupled ordinary differential equations which appears as an envelope equation in Bose-Einstein Condensation. This system can be viewed as a nonlinear extension of the celebrated model introduced by Landau and Zener. We show how the nonlinear system may appear from different physical models. We focus our attention on the large time behavior of the solution. We show the existence of a nonlinear scattering operator, which is reminiscent of long range scattering for the nonlinear Schrodinger equation, and which can be compared with its linear counterpart.
Formulas relating Poincare-Steklov operators for Schroedinger equations related by Darboux-Moutard transformations are derived. They can be used for testing algorithms of reconstruction of the potential from measurements at the boundary.
For an arbitrary open, nonempty, bounded set $Omega subset mathbb{R}^n$, $n in mathbb{N}$, and sufficiently smooth coefficients $a,b,q$, we consider the closed, strictly positive, higher-order differential operator $A_{Omega, 2m} (a,b,q)$ in $L^2(Omega)$ defined on $W_0^{2m,2}(Omega)$, associated with the higher-order differential expression $$ tau_{2m} (a,b,q) := bigg(sum_{j,k=1}^{n} (-i partial_j - b_j) a_{j,k} (-i partial_k - b_k)+qbigg)^m, quad m in mathbb{N}, $$ and its Krein--von Neumann extension $A_{K, Omega, 2m} (a,b,q)$ in $L^2(Omega)$. Denoting by $N(lambda; A_{K, Omega, 2m} (a,b,q))$, $lambda > 0$, the eigenvalue counting function corresponding to the strictly positive eigenvalues of $A_{K, Omega, 2m} (a,b,q)$, we derive the bound $$ N(lambda; A_{K, Omega, 2m} (a,b,q)) leq C v_n (2pi)^{-n} bigg(1+frac{2m}{2m+n}bigg)^{n/(2m)} lambda^{n/(2m)} , quad lambda > 0, $$ where $C = C(a,b,q,Omega)>0$ (with $C(I_n,0,0,Omega) = |Omega|$) is connected to the eigenfunction expansion of the self-adjoint operator $widetilde A_{2m} (a,b,q)$ in $L^2(mathbb{R}^n)$ defined on $W^{2m,2}(mathbb{R}^n)$, corresponding to $tau_{2m} (a,b,q)$. Here $v_n := pi^{n/2}/Gamma((n+2)/2)$ denotes the (Euclidean) volume of the unit ball in $mathbb{R}^n$. Our method of proof relies on variational considerations exploiting the fundamental link between the Krein--von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of $widetilde A_{2} (a,b,q)$ in $L^2(mathbb{R}^n)$. We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension $A_{F,Omega, 2m} (a,b,q)$ in $L^2(Omega)$ of $A_{Omega, 2m} (a,b,q)$. No assumptions on the boundary $partial Omega$ of $Omega$ are made.
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