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Fulvio Zonca - One of the best experts on this subject based on the ideXlab platform.
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shear alfven wave Continuous Spectrum within magnetic islands
Physics of Plasmas, 2010Co-Authors: Alessandro Biancalani, Liu Chen, Francesco Pegoraro, Fulvio ZoncaAbstract:The radial structure of the Continuous Spectrum of shear Alfven waves is calculated in this paper within the separatrix of a magnetic island. Geometrical effects due to the noncircularity of the flux surface's cross section are retained to all orders. On the other hand, only curvature effects responsible for the beta-induced gap in the low-frequency part of the Continuous Spectrum are kept. Modes with different helicity from that of the magnetic island are considered. The main result is that, inside a magnetic island, there is a Continuous Spectrum very similar to that of tokamak plasmas, where a generalized safety factor q can be defined and where a wide frequency gap is formed, analogous to the ellipticity induced Alfven eigenmode gap in tokamaks. The presence of this gap is due to the strong eccentricity of the island cross section. The importance of the existence of such a gap is recognized in potentially hosting magnetic island induced Alfven eigenmodes (MiAE). Due to the frequency dependence of the shear Alfven wave continuum on the magnetic island size, the possibility of utilizing MiAE frequency scalings as a novel magnetic island diagnostic is also discussed.
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Continuous Spectrum of shear alfven waves within magnetic islands
Physical Review Letters, 2010Co-Authors: Alessandro Biancalani, Liu Chen, Francesco Pegoraro, Fulvio ZoncaAbstract:The radial structure of the shear Alfven wave Continuous Spectrum is calculated inside the separatrix of a magnetic island. We find that, within a magnetic island, there is a Continuous Spectrum very similar to that of tokamak plasmas, where a generalized safety factor q can be defined and a wide frequency gap is formed, analogous to the ellipticity induced Alfven eigenmode gap in tokamaks. This is due to the strong eccentricity of the island cross section. In this gap, a magnetic-island induced Alfven eigenmode (MIAE) can exist as a bound state, essentially free of continuum damping, which can be resonantly excited by energetic particles and interact nonlinearly with the magnetic island. Because of the frequency dependence of the shear Alfven wave continuum on the magnetic-island size, the possibility of utilizing MIAE frequency scalings as a novel magnetic-island diagnostic is also discussed.
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Shear Alfven wave Continuous Spectrum within magnetic islands
Physics of Plasmas, 2010Co-Authors: Alessandro Biancalani, Liu Chen, Francesco Pegoraro, Fulvio ZoncaAbstract:The radial structure of the Continuous Spectrum of shear Alfven waves is calculated in this paper within the separatrix of a magnetic island. Geometrical effects due to the noncircularity of the flux surface's cross section are retained to all orders. On the other hand, we keep only curvature effects responsible for the beta-induced gap in the low-frequency part of the Continuous Spectrum. Modes with different helicity from that of the magnetic island are considered. The main result is that, inside a magnetic island, there is a Continuous Spectrum very similar to that of tokamak plasmas, where a generalized safety factor q can be defined and where a wide frequency gap is formed, analogous to the ellipticity induced Alfven eigenmode gap in tokamaks. The presence of this gap is due to the strong eccentricity of the island cross section. The importance of the existence of such a gap is recognized in potentially hosting magnetic-island induced Alfven eigenmodes (MiAE). Due to the frequency dependence of the shear Alfven wave continuum on the magnetic-island size, the possibility of utilizing MiAE frequency scalings as a novel magnetic-island diagnostic is also discussed.
Alessandro Biancalani - One of the best experts on this subject based on the ideXlab platform.
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shear alfven wave Continuous Spectrum within magnetic islands
Physics of Plasmas, 2010Co-Authors: Alessandro Biancalani, Liu Chen, Francesco Pegoraro, Fulvio ZoncaAbstract:The radial structure of the Continuous Spectrum of shear Alfven waves is calculated in this paper within the separatrix of a magnetic island. Geometrical effects due to the noncircularity of the flux surface's cross section are retained to all orders. On the other hand, only curvature effects responsible for the beta-induced gap in the low-frequency part of the Continuous Spectrum are kept. Modes with different helicity from that of the magnetic island are considered. The main result is that, inside a magnetic island, there is a Continuous Spectrum very similar to that of tokamak plasmas, where a generalized safety factor q can be defined and where a wide frequency gap is formed, analogous to the ellipticity induced Alfven eigenmode gap in tokamaks. The presence of this gap is due to the strong eccentricity of the island cross section. The importance of the existence of such a gap is recognized in potentially hosting magnetic island induced Alfven eigenmodes (MiAE). Due to the frequency dependence of the shear Alfven wave continuum on the magnetic island size, the possibility of utilizing MiAE frequency scalings as a novel magnetic island diagnostic is also discussed.
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Continuous Spectrum of shear alfven waves within magnetic islands
Physical Review Letters, 2010Co-Authors: Alessandro Biancalani, Liu Chen, Francesco Pegoraro, Fulvio ZoncaAbstract:The radial structure of the shear Alfven wave Continuous Spectrum is calculated inside the separatrix of a magnetic island. We find that, within a magnetic island, there is a Continuous Spectrum very similar to that of tokamak plasmas, where a generalized safety factor q can be defined and a wide frequency gap is formed, analogous to the ellipticity induced Alfven eigenmode gap in tokamaks. This is due to the strong eccentricity of the island cross section. In this gap, a magnetic-island induced Alfven eigenmode (MIAE) can exist as a bound state, essentially free of continuum damping, which can be resonantly excited by energetic particles and interact nonlinearly with the magnetic island. Because of the frequency dependence of the shear Alfven wave continuum on the magnetic-island size, the possibility of utilizing MIAE frequency scalings as a novel magnetic-island diagnostic is also discussed.
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Shear Alfven wave Continuous Spectrum within magnetic islands
Physics of Plasmas, 2010Co-Authors: Alessandro Biancalani, Liu Chen, Francesco Pegoraro, Fulvio ZoncaAbstract:The radial structure of the Continuous Spectrum of shear Alfven waves is calculated in this paper within the separatrix of a magnetic island. Geometrical effects due to the noncircularity of the flux surface's cross section are retained to all orders. On the other hand, we keep only curvature effects responsible for the beta-induced gap in the low-frequency part of the Continuous Spectrum. Modes with different helicity from that of the magnetic island are considered. The main result is that, inside a magnetic island, there is a Continuous Spectrum very similar to that of tokamak plasmas, where a generalized safety factor q can be defined and where a wide frequency gap is formed, analogous to the ellipticity induced Alfven eigenmode gap in tokamaks. The presence of this gap is due to the strong eccentricity of the island cross section. The importance of the existence of such a gap is recognized in potentially hosting magnetic-island induced Alfven eigenmodes (MiAE). Due to the frequency dependence of the shear Alfven wave continuum on the magnetic-island size, the possibility of utilizing MiAE frequency scalings as a novel magnetic-island diagnostic is also discussed.
Alexander Kiselev - One of the best experts on this subject based on the ideXlab platform.
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Absolutely Continuous Spectrum of Stark operators
Arkiv för Matematik, 2003Co-Authors: Michael Christ, Alexander KiselevAbstract:We prove several new results on the absolutely Continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely Continuous Spectrum. Then we establish stability of the absolutely Continuous Spectrum in more general situations, where imbedded singular Spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely Continuous Spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is | q ( x )|≤ C (1+| x |)^−1/4−ε; in the smoothness direction, a sufficient condition in Hölder classes is q ∈ C ^1/2+ε( R ). On the other hand, we show that there exist potentials which both satisfy | q ( x )|≤ C (1+| x |)^−1/4 and belong to C ^1/2( R ) for which the Spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.
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Absolutely Continuous Spectrum of Stark operators
Arkiv för Matematik, 2003Co-Authors: Michael Christ, Alexander KiselevAbstract:We prove several new results on the absolutely Continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely Continuous Spectrum. Then we establish stability of the absolutely Continuous Spectrum in more general situations, where imbedded singular Spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely Continuous Spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is |q(x)|≤C(1+|x|)−1/4−e; in the smoothness direction, a sufficient condition in Holder classes isq∈C1/2+e(R). On the other hand, we show that there exist potentials which both satisfy |q(x)|≤C(1+|x|)−1/4 and belong toC1/2(R) for which the Spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.
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Imbedded Singular Continuous Spectrum for Schr\"odinger Operators
arXiv: Spectral Theory, 2001Co-Authors: Alexander KiselevAbstract:We construct examples of potentials $V(x)$ satisfying $|V(x)| \leq \frac{h(x)}{1+x},$ where the function $h(x)$ is growing arbitrarily slowly, such that the corresponding Schr\"odinger operator has imbedded singular Continuous Spectrum. This solves one of the fifteen "twenty-first century" problems for Schr\"odinger operators posed by Barry Simon. The construction also provides the first example of a Schr\"odinger operator for which M\"oller wave operators exist but are not asymptotically complete due to the presence of singular Continuous Spectrum.
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Absolutely Continuous Spectrum of Stark operators
arXiv: Spectral Theory, 2001Co-Authors: Michael Christ, Alexander KiselevAbstract:We prove several new results on the absolutely Continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely Continuous Spectrum. Then we establish stability of the absolutely Continuous Spectrum in more general situations, where imbedded singular Spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely Continuous Spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is $|q(x)| \leq C(1+|x|)^{-{1/4}-\epsilon};$ in the smoothness direction, a sufficient condition in H\"older classes is $q \in C^{{1/2}+\epsilon}(\reals)$. On the other hand, we show that there exist potentials which both satisfy $|q(x)| \leq C(1+|x|)^{-1/4}$ and belong to $C^{1/2}(\reals)$ for which the Spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.
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Absolutely Continuous Spectrum of perturbed Stark operators
Transactions of the American Mathematical Society, 1999Co-Authors: Alexander KiselevAbstract:We prove new results on the stability of the absolutely cont;inuous Spectrum for perturbed Stark operators with decaying or satisfying certain smoothness assumption perturbation. We show that the absolutely Continuous Spectrum of the Stark operator is stable if the perturbing potential decays at the rate (1 + x) 3 -e or if it is Continuously differentiable with derivative from the H6lder space C, (R), with any ae > 0. 0. INTRODUCTION In this paper, we study the stability of the absolutely Continuous spect;rum of one-dimensional Stark operators under various classes of perturbations. Stark Schr6dinger operators describe behavior of the charged particle in the constant electric field. The absolutely Continuous Spectrum is a manifestation of the fact that the particle described by the operator propagates to infinity at a rather fast rate (see, e.g. [2], [12]). It is therefore interesting to describe the classes of perturbations which preserve the absolutely Continuous Spectrum of the Stark operators. In the first part of this work, we study perturbations of Stark operators by decaying potentials. This part is inspired by the recent work of Naboko and Pushnitski [14]. The general picture that we prove is very similar to the case of perturbatiorns of free Schr6dinger operators [9]. In accordance with physical intuition, however, the absolutely Continuous Spectrum is stable under stronger perturbations than in the free case. If in the free case the short range potentials preserving purely absolutely Continuous Spectrum of the free operator are given by condition (on the power scale) lq(x)l < C(1 + xKl)'E, in the Stark operator case the corresponding condition reads lq(x)l < C(1 + IXD)-2-E. If c is allowed to be zero in the above bounds, imbedded eigenvalues may occur in both cases (see, e.g. [14], [151). Moreover, in both cases if we allow potential to decay slower by an arbitrary function growing to infinity, very rich singular Spectrum, such as a dense set of eigenvalues, -may occur (see [13] for the free case and [141 for the Stark case for precise formulation and proofs of these results). The first part of this work draws the parallel further, showing that the absolutely Continuous Spectrum of Stark operators is preserved under perturbations satisfying lq(x)l < C(1 + Jxl)-3-E, in particular even in the regimes where a dense set of eigenvalues occurs; hence in such cases these eigenvalues are genuinely imbedded. Similar results for the free case were proven in [9], Received by the editors April 14, 1997. 1991 Mathematics Subject Classification. Primary 34L40, 81Q10. ?)1999 American Mathematical Society 243 This content downloaded from 207.46.13.156 on Sat, 10 Sep 2016 04:57:39 UTC All use subject to http://about.jstor.org/terms 244 ALEXANDER KISELEV [10]. Our main strategy of the proof here is similar to that in [9] and [10]: we study the asymptotics of the generalized eigenfunctions and then apply Gilbert-Pearson theory [7] to derive spectral consequences. While the main new tool we introduce in our treatment of Stark operators is the same as in the free case, namely the a.e. convergence of the Fourier-type integral operators, there are some major differences. First of all, the spectral parameter enters the final equations that we study in a different way and this makes analysis more complicated. Secondly, we employ a different method to analyze the asymptotics. Instead of Harris-Lutz asymptotic method we study appropriate Prilfer transform variables, simplifying the overall consideration. In the second part of the work we discuss perturbations by potentials having some additional smoothness properties, but without decay. It turns out that for Stark operators the effects of decay or of additional smoothness of potential on the spectral properties are somewhat similar. It was known for a long time that if a potential perturbing Stark operator has two bounded derivatives the Spectrum remains purely absolutely Continuous (actually, certain growth of derivatives is also allowed, see Section 2 for details or Walter [21] for the original result). We note that the results similar to Walter's on the preservation of absolutely Continuous Spectrum were also obtained in [4] by applying different types of technique (Mourre method instead of studying asymptotics of solutions). On the other hand, if the perturbing potential is a sequence of derivatives of 6 functions in integer points on R with certain couplings, the Spectrum may turn pure point [3], [5], [1]. In some sense, the 6' interaction is the most singular and least "differentiable" among all available natural perturbations of one-dimensional Schrbdinger operators [11]. Hence we have very different spectral properties on the very opposite sides of the smoothness scale. This work closes part of the gap. We improve the well-known results of Walter [21] concerning the minimal smoothness required for the preservation of the absolutely Continuous Spectrum and show that in fact existence and minimal smoothness of the first derivative is sufficient to imply absolute continuity of the Spectrum. After submitting this paper, the author learned about the work of J. Sahbani [17], where, in particular, the results related to Theorem 2.1 of the present work are proven. Sahbani's results are slightly stronger than Theorem 2.1: the derivative of potential V'(x) is required to be bounded and Dini Continuous in order for the absolutely Continuous Spectrum to be preserved. In addition, he shows that the imbedded singular Spectrum in this case may only consist of isolated eigenvalues. The approach employed in [17] is an extension of conjugate operator method. 1. DECAYING PERTURBATIONS Consider a self-adjoint operator Hq defined by the differential expression Hqu =-u" -xu + q(x)u on the L2(-oo, oo). Let us introduce some notation. For the function f C L2 we denote by bf its Fourier transform:
Barry Simon - One of the best experts on this subject based on the ideXlab platform.
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Operators with singular Continuous Spectrum, V. Sparse potentials
Proceedings of the American Mathematical Society, 1996Co-Authors: Barry Simon, Günter StolzAbstract:By presenting simple theorems for the absence of positive eigenvalues for certain one-dimensional Schrodinger operators, we are able to construct explicit potentials which yield purely singular Continuous Spectrum.
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SINGULAR Continuous Spectrum FOR PALINDROMIC SCHRODINGER OPERATORS
Communications in Mathematical Physics, 1995Co-Authors: A. Hof, Oliver Knill, Barry SimonAbstract:We give new examples of discrete Schrodinger operators with potentials taking finitely many values that have purely singular Continuous Spectrum. If the hullX of the potential is strictly ergodic, then the existence of just one potentialx inX for which the operator has no eigenvalues implies that there is a generic set inX for which the operator has purely singular Continuous Spectrum. A sufficient condition for the existence of such anx is that there is az∈X that contains arbitrarily long palindromes. Thus we can define a large class of primitive substitutions for which the operators are purely singularly Continuous for a generic subset inX. The class includes well-known substitutions like Fibonacci, Thue-Morse, Period Doubling, binary non-Pisot and ternary non-Pisot. We also show that the operator has no absolutely Continuous Spectrum for allx∈X ifX derives from a primitive substitution. For potentials defined by circle maps,x n =1 J (θ0+nα), we show that the operator has purely singular Continuous Spectrum for a generic subset inX for all irrational α and every half-open intervalJ.
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Singular Continuous Spectrum is generic
arXiv: Spectral Theory, 1994Co-Authors: Rafael Del Rio, Svetlana Jitomirskaya, Nikolai Makarov, Barry SimonAbstract:In a variety of contexts, we prove that singular Continuous Spectrum is generic in the sense that for certain natural complete metric spaces of operators, those with singular Spectrum are a dense $G_\delta$.
Francesco Pegoraro - One of the best experts on this subject based on the ideXlab platform.
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shear alfven wave Continuous Spectrum within magnetic islands
Physics of Plasmas, 2010Co-Authors: Alessandro Biancalani, Liu Chen, Francesco Pegoraro, Fulvio ZoncaAbstract:The radial structure of the Continuous Spectrum of shear Alfven waves is calculated in this paper within the separatrix of a magnetic island. Geometrical effects due to the noncircularity of the flux surface's cross section are retained to all orders. On the other hand, only curvature effects responsible for the beta-induced gap in the low-frequency part of the Continuous Spectrum are kept. Modes with different helicity from that of the magnetic island are considered. The main result is that, inside a magnetic island, there is a Continuous Spectrum very similar to that of tokamak plasmas, where a generalized safety factor q can be defined and where a wide frequency gap is formed, analogous to the ellipticity induced Alfven eigenmode gap in tokamaks. The presence of this gap is due to the strong eccentricity of the island cross section. The importance of the existence of such a gap is recognized in potentially hosting magnetic island induced Alfven eigenmodes (MiAE). Due to the frequency dependence of the shear Alfven wave continuum on the magnetic island size, the possibility of utilizing MiAE frequency scalings as a novel magnetic island diagnostic is also discussed.
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Continuous Spectrum of shear alfven waves within magnetic islands
Physical Review Letters, 2010Co-Authors: Alessandro Biancalani, Liu Chen, Francesco Pegoraro, Fulvio ZoncaAbstract:The radial structure of the shear Alfven wave Continuous Spectrum is calculated inside the separatrix of a magnetic island. We find that, within a magnetic island, there is a Continuous Spectrum very similar to that of tokamak plasmas, where a generalized safety factor q can be defined and a wide frequency gap is formed, analogous to the ellipticity induced Alfven eigenmode gap in tokamaks. This is due to the strong eccentricity of the island cross section. In this gap, a magnetic-island induced Alfven eigenmode (MIAE) can exist as a bound state, essentially free of continuum damping, which can be resonantly excited by energetic particles and interact nonlinearly with the magnetic island. Because of the frequency dependence of the shear Alfven wave continuum on the magnetic-island size, the possibility of utilizing MIAE frequency scalings as a novel magnetic-island diagnostic is also discussed.
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Shear Alfven wave Continuous Spectrum within magnetic islands
Physics of Plasmas, 2010Co-Authors: Alessandro Biancalani, Liu Chen, Francesco Pegoraro, Fulvio ZoncaAbstract:The radial structure of the Continuous Spectrum of shear Alfven waves is calculated in this paper within the separatrix of a magnetic island. Geometrical effects due to the noncircularity of the flux surface's cross section are retained to all orders. On the other hand, we keep only curvature effects responsible for the beta-induced gap in the low-frequency part of the Continuous Spectrum. Modes with different helicity from that of the magnetic island are considered. The main result is that, inside a magnetic island, there is a Continuous Spectrum very similar to that of tokamak plasmas, where a generalized safety factor q can be defined and where a wide frequency gap is formed, analogous to the ellipticity induced Alfven eigenmode gap in tokamaks. The presence of this gap is due to the strong eccentricity of the island cross section. The importance of the existence of such a gap is recognized in potentially hosting magnetic-island induced Alfven eigenmodes (MiAE). Due to the frequency dependence of the shear Alfven wave continuum on the magnetic-island size, the possibility of utilizing MiAE frequency scalings as a novel magnetic-island diagnostic is also discussed.