Contributed talks session B (Tuesday)Copyright (c) 2021 Washington University in St. Louis All rights reserved.
https://openscholarship.wustl.edu/iwota2016/Contributed/contributedtalksTuesdayB
Recent Events in Contributed talks session B (Tuesday)en-usThu, 15 Apr 2021 09:30:41 PDT3600Sampling for multi-parameter spectral theory
https://openscholarship.wustl.edu/iwota2016/Contributed/contributedtalksTuesdayB/6
https://openscholarship.wustl.edu/iwota2016/Contributed/contributedtalksTuesdayB/6Tue, 19 Jul 2016 17:30:00 PDT
In this talk we extend the sampling method to deal with the multiparameter spectral theory of Sturm-Liouville systems. To do so, we use the 2-dimensional version of the Whittaker-Shannon-Kotelnikov sampling theorem to find a representation for the characteristic function which leads to the computation of the eigencurves of the system.
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Amin BoumenirApproximations of spectra of Schr\"odinger operators with complex potentials on $\mathbb{R}^d$
https://openscholarship.wustl.edu/iwota2016/Contributed/contributedtalksTuesdayB/5
https://openscholarship.wustl.edu/iwota2016/Contributed/contributedtalksTuesdayB/5Tue, 19 Jul 2016 17:00:00 PDT
We study spectral approximations of Schr\"odinger operators $T=-\Delta+Q$ with complex potentials on $\Omega=\mathbb{R}^d$, or exterior domains $\Omega \subset \mathbb{R}^d$, by domain truncation. Our weak assumptions cover wide classes of potentials $Q$ for which $T$ has discrete spectrum, of approximating domains $\Omega_n$, and of boundary conditions on $\partial \Omega_n$ such as mixed Dirichlet/Robin type. In particular, $\Re Q$ need not be bounded from below and $Q$ may be singular. We prove generalized norm resolvent convergence and spectral exactness, i.e. approximation of all eigenvalues of $T$ by those of the truncated operators $T_n$ without spectral pollution. Moreover, we estimate the eigenvalue convergence rate and prove convergence of pseudospectra. Our results are illustrated by numerical computations for several examples, such as complex harmonic and cubic oscillators for $d=1,2,3$. The talk is based on a joint work with S. B\"ogli and C. Tretter.
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Petr SieglRepresentation of solutions to one-dimensional SchrÃ¶dinger and perturbed Bessel equations in terms of Neumann series of Bessel functions
https://openscholarship.wustl.edu/iwota2016/Contributed/contributedtalksTuesdayB/4
https://openscholarship.wustl.edu/iwota2016/Contributed/contributedtalksTuesdayB/4Tue, 19 Jul 2016 16:00:00 PDT
The new representations of solutions to the one-dimensional SchrÃ¶dinger equation $-y''+q(x)y=\omega^{2}y$ and to the perturbed Bessel equation $-y''+\frac{\ell(\ell+1)}{x^2}+q(x)y=\omega^{2}y$ are introduced. For the first equation a pair of linearly independent solutions has the form \[ c(\omega,x) =\cos\omega x+2\sum_{n=0}^{\infty} (-1)^{n}\beta_{2n}(x)j_{2n}(\omega x) \] and \[ s(\omega,x) =\sin\omega x+2\sum_{n=0}^{\infty} (-1)^{n}\beta_{2n+1}(x)j_{2n+1}(\omega x), \] where $j_n$ are the spherical Bessel functions and the coefficients $\beta_j$ can be calculated by the simple recursive procedure. For the second equation the regular solution $u_\ell$ satisfying the asymptotics $u_\ell(\omega,x)\sim x^{\ell+1}$, $x\to 0$ has the form \[ u_\ell(\omega,x) =\frac{2^{\ell+1}\Gamma(\ell+3/2)}{\sqrt{\pi}\omega^\ell} x j_{\ell}(\omega x)+\sum_{n=0}^{\infty} (-1)^{n}\beta_{n}(x)j_{2n}(\omega x), \] where the coefficients $\beta_n$ can be obtained by a similar recursive procedure. The representations are based on the expansion of the integral kernels of the transmutation operators into Fourier-Legendre series and the recent results obtained by the author jointly with V.\ V.\ Kravchenko. It is shown that the partial sums of the series approximate the solutions uniformly with respect to $\omega$. Convergence rate estimates and representations for the derivatives of the solutions are given. The talk is based on the results obtained with V. V. Kravchenko, Luis J. Navarro and R. Castillo-Perez.
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Sergii TorbaOn some stochastic singular integro-partial differential equations and the parabolic transform
https://openscholarship.wustl.edu/iwota2016/Contributed/contributedtalksTuesdayB/3
https://openscholarship.wustl.edu/iwota2016/Contributed/contributedtalksTuesdayB/3Tue, 19 Jul 2016 15:30:00 PDT
Some stochastic singular integro-partial differential equations are studied without any restrictions on the characteristic forms of the partial differential operators. Linear and nonlinear cases are studied. Using the parabolic transform, existence and stability results are obtained. The Cauchy problem of fractional stochastic partial differential equations can be considered as a special case from the obtained results. Key words: Singular integral equations, Stochastic partial differential equations, Existence and stability of solutions, Fractional stochastic partial differential equations.
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Mahmoud El-BoraiOn some fractional nonlocal integrated semi groups
https://openscholarship.wustl.edu/iwota2016/Contributed/contributedtalksTuesdayB/2
https://openscholarship.wustl.edu/iwota2016/Contributed/contributedtalksTuesdayB/2Tue, 19 Jul 2016 15:00:00 PDT
Some classes of fractional abstract differential equations with $\alpha$-integrated semi groups are studied in Banach space. The existence of a unique solution of the nonlocal Cauchy problem is studied. Some properties are given.
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Khairia El-NadiFourth order Birkhoff regular problems with eigenvalue parameter dependent boundary conditions
https://openscholarship.wustl.edu/iwota2016/Contributed/contributedtalksTuesdayB/1
https://openscholarship.wustl.edu/iwota2016/Contributed/contributedtalksTuesdayB/1Tue, 19 Jul 2016 14:30:00 PDT
A regular fourth order differential equation which depends quadratically on the eigenvalue parameter $\lambda$ is considered with classes of separable boundary conditions independent of $\lambda$ or depending on $\lambda$ linearly. Conditions are given for the problems to be Birkhoff regular.
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Bertin Zinsou