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Let us say that an $n$-sided polygon is semi-regular if it is circumscriptible and its angles are all equal but possibly one, which is then larger than the rest. Regular polygons, in particular, are semi-regular. We prove that semi-regular polygons are spectrally determined in the class of convex piecewise smooth domains. Specifically, we show that if $Omega$ is a convex piecewise smooth planar domain, possibly with straight corners, whose Dirichlet or Neumann spectrum coincides with that of an $n$-sided semi-regular polygon $P_n$, then $Omega$ is congruent to $P_n$.
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We obtain asymptotic formulae for the Steklov eigenvalues and eigenfunctions of curvilinear polygons in terms of their side lengths and angles. These formulae are quite precise: the errors tend to zero as the spectral parameter tends to infinity. The Steklov problem on planar domains with corners is closely linked to the classical sloshing and sloping beach problems in hydrodynamics; as we show it is also related to quantum graphs. Somewhat surprisingly, the arithmetic properties of the angles of a curvilinear polygon have a significant effect on the boundary behaviour of the Steklov eigenfunctions. Our proofs are based on an explicit construction of quasimodes. We use a variety of methods, including ideas from spectral geometry, layer potential analysis, and some new techniques tailored to our problem.
This paper studies the inverse Steklov spectral problem for curvilinear polygons. For generic curvilinear polygons with angles less than $pi$, we prove that the asymptotics of Steklov eigenvalues obtained in arXiv:1908.06455 determines, in a constructive manner, the number of vertices and the properly ordered sequence of side lengths, as well as the angles up to a certain equivalence relation. We also present counterexamples to this statement if the generic assumptions fail. In particular, we show that there exist non-isometric triangles with asymptotically close Steklov spectra. Among other techniques, we use a version of the Hadamard--Weierstrass factorisation theorem, allowing us to reconstruct a trigonometric function from the asymptotics of its roots.
Consider a regular $d$-dimensional metric tree $Gamma$ with root $o$. Define the Schroedinger operator $-Delta - V$, where $V$ is a non-negative, symmetric potential, on $Gamma$, with Neumann boundary conditions at $o$. Provided that $V$ decays like $x^{-gamma}$ at infinity, where $1 < gamma leq d leq 2, gamma eq 2$, we will determine the weak coupling behavior of the bottom of the spectrum of $-Delta - V$. In other words, we will describe the asymptotical behavior of $inf sigma(-Delta - alpha V)$ as $alpha to 0+$
We examine the spectrum of a family of Sturm--Liouville operators with regularly spaced delta function potentials parametrized by increasing strength. The limiting behavior of the eigenvalues under this spectral flow was described in a previor result of the last two authors with Berkolaiko, where it was used to study the nodal deficiency of Laplacian eigenfunctions. Here we consider the eigenfunctions of these operators. In particular, we give explicit formulas for the limiting eigenfunctions, and also characterize the eigenfunctions and eigenvalues for all values for the spectral flow parameter (not just in the limit). We also develop spectrally accurate numerical tools for comparison and visualization.
Smoothing (and decay) spacetime estimates are discussed for evolution groups of self-adjoint operators in an abstract setting. The basic assumption is the existence (and weak continuity) of the spectral density in a functional setting. Spectral identities for the time evolution of such operators are derived, enabling results concerning best constants for smoothing estimates. When combined with suitable comparison principles (analogous to those established in our previous work), they yield smoothing estimates for classes of functions of the operators . A important particular case is the derivation of global spacetime estimates for a perturbed operator $H+V$ on the basis of its comparison with the unperturbed operator $H.$ A number of applications are given, including smoothing estimates for fractional Laplacians, Stark Hamiltonians and Schrodinger operators with potentials.
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