اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدAn approach to constructing super oscillatory functions
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A recipe is presented for constructing band-limited superoscillating functions that exhibit arbitrarily high frequencies over arbitrarily long intervals.
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In the recent literature there has been a resurgence of interest in the fourth-order field-theoretic model of Pais-Uhlenbeck cite {Pais-Uhlenbeck 50 a}, which has not had a good reception over the last half century due to the existence of {em ghosts}
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This work is divide in two cases. In the first case, we consider a spin manifold $M$ as the set of fixed points of an $S^{1}$-action on a spin manifold $X$, and in the second case we consider the spin manifold $M$ as the set of fixed points of an $S^
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In this paper we give a new and constructive approach to stationary scattering theory for pairs of self-adjoint operators $H_0$ and $H_1$ on a Hilbert space $mathcal H$ which satisfy the following conditions: (i) for any open bounded subset $Delta$ o
f $mathbb R,$ the operators $F E_Delta^{H_0}$ and $F E_Delta^{H_1}$ are Hilbert-Schmidt and (ii) $V = H_1- H_0$ is bounded and admits decomposition $V = F^*JF,$ where $F$ is a bounded operator with trivial kernel from $mathcal H$ to another Hilbert space $mathcal K$ and $J$ is a bounded self-adjoint operator on $mathcal K.$ An example of a pair of operators which satisfy these conditions is the Schrodinger operator $H_0 = -Delta + V_0$ acting on $L^2(mathbb R^ u),$ where $V_0$ is a potential of class $K_ u$ (see B.,Simon, {it Schrodinger semigroups,} Bull. AMS 7, 1982, 447--526) and $H_1 = H_0 + V_1,$ where $V_1 in L^infty(mathbb R^ u) cap L^1(mathbb R^ u).$ Among results of this paper is a new proof of existence and completeness of wave operators $W_pm(H_1,H_0)$ and a new constructive proof of stationary formula for the scattering matrix. This approach to scattering theory is based on explicit diagonalization of a self-adjoint operator $H$ on a sheaf of Hilbert spaces $EuScript S(H,F)$ associated with the pair $(H,F)$ and with subsequent construction and study of properties of wave matrices $w_pm(lambda; H_1,H_0)$ acting between fibers $mathfrak h_lambda(H_0,F)$ and $mathfrak h_lambda(H_1,F)$ of sheaves $EuScript S(H_0,F)$ and $EuScript S(H_1,F)$ respectively. The wave operators $W_pm(H_1,H_0)$ are then defined as direct integrals of wave matrices and are proved to coincide with classical time-dependent definition of wave operators.
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