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Register a new userSpectral identities and smoothing estimates for evolution operators
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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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We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. We demonstrate our findings by considering trees, antitrees and Cayley graphs of finitely generated groups.
We introduce the concept of essential numerical range $W_{!e}(T)$ for unbounded Hilbert space operators $T$ and study its fundamental properties including possible equivalent characterizations and perturbation results. Many of the properties known for the bounded case do emph{not} carry over to the unbounded case, and new interesting phenomena arise which we illustrate by some striking examples. A key feature of the essential numerical range $W_{!e}(T)$ is that it captures spectral pollution in a unified and minimal way when approximating $T$ by projection methods or domain truncation methods for PDEs.
Let $L$ be a non-negative self-adjoint operator acting on the space $L^2(X)$, where $X$ is a metric measure space. Let ${ L}=int_0^{infty} lambda dE_{ L}({lambda})$ be the spectral resolution of ${ L}$ and $S_R({ L})f=int_0^R dE_{ L}(lambda) f$ denote the spherical partial sums in terms of the resolution of ${ L}$. In this article we give a sufficient condition on $L$ such that $$ lim_{Rrightarrow infty} S_R({ L})f(x) =f(x), {rm a.e.} $$ for any $f$ such that ${rm log } (2+L) fin L^2(X)$. These results are applicable to large classes of operators including Dirichlet operators on smooth bounded domains, the Hermite operator and Schrodinger operators with inverse square potentials.
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.
We derive explicit Krein resolvent identities for generally singular Sturm-Liouville operators in terms of boundary condition bases and the Lagrange bracket. As an application of the resolvent identities obtained, we compute the trace of the resolvent difference of a pair of self-adjoint realizations of the Bessel expression $-d^2/dx^2+( u^2-(1/4))x^{-2}$ on $(0,infty)$ for values of the parameter $ uin[0,1)$ and use the resulting trace formula to explicitly determine the spectral shift function for the pair.
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