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Siarhei Finski - One of the best experts on this subject based on the ideXlab platform.
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analytic torsion for surfaces with cusps i Compact Perturbation theorem and anomaly formula
Communications in Mathematical Physics, 2020Co-Authors: Siarhei FinskiAbstract:We define the analytic torsion associated with a Riemann surface endowed with a metric having Poincare-type singularities in the neighborhood of a finite number of points and a Hermitian vector bundle with at most logarithmic singularities at those points, coming from the metric on the negative power of the canonical line bundle twisted by the divisor of the points. Then we provide a relation between this analytic torsion and the Ray–Singer analytic torsion of the Compactified surface. From this relation we then establish the anomaly formula, which describes how the analytic torsion changes under the change of the metric on the surface and on the vector bundle.
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Quillen metric for singular families of Riemann surfaces with cusps and Compact Perturbation theorem
arXiv: Differential Geometry, 2019Co-Authors: Siarhei FinskiAbstract:We study the behavior of the Quillen metric for the family of Riemann surfaces with cusps when the additional cusps are created by degeneration. More precisely, in our previous paper, we've seen that the renormalization of the Quillen metric associated with a family of Riemann surfaces with cusps extends continuously over the locus of singular curves. The main result of this article shows that, modulo some explicit universal constant, this continuous extension coincides with the Quillen metric of the normalization of singular curves. This result shows that the Quillen metric is compatible with the adjunction of cusps. When this theorem is applied directly to the moduli space of curves, we obtain the compatibility of the Quillen metric with clutching morphisms in the moduli space of pointed stable curves. As one application, we obtain the compatibility between our definition of the analytic torsion and the definition of Takhtajan-Zograf using lengths of closed geodesics. As a consequence of the proof of the main theorem, we get an explicit relation in terms of Bott-Chern forms between the Quillen metric associated with a cusped metric and the Quillen metric associated with a metric on the Compactified Riemann surface. This refines relative Compact Perturbation theorem we obtained before by pinning down the universal constant.
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analytic torsion for surfaces with cusps i Compact Perturbation theorem and anomaly formula
arXiv: Differential Geometry, 2018Co-Authors: Siarhei FinskiAbstract:Let $\overline{M}$ be a Compact Riemann surface and let $g^{TM}$ be a metric over $\overline{M} \setminus D_M$, where $D_M \subset \overline{M}$ is a finite set of points. We suppose that $g^{TM}$ is equal to the Poincare metric over a punctured disks around the points of $D_M$. The metric $g^{TM}$ endows the twisted canonical line bundle $\omega_M(D)$ with the induced Hermitian norm $\|\cdot\|_M$ over $\overline{M} \setminus D_M$. Let $(\xi, h^{\xi})$ be a holomorphic Hermitian vector bundle over $\overline{M}$. In this article we define the analytic torsion $T(g^{TM}, h^{\xi} \otimes \|\cdot\|_M^{2n})$ associated with $(M, g^{TM})$ and $(\xi \otimes \omega_M(D)^n, h^{\xi} \otimes \|\cdot\|_M^{2n})$ for $n \leq 0$. We prove that $T(g^{TM}, h^{\xi} \otimes \|\cdot\|_M^{2n})$ is related to the analytic torsion of non-cusped surfaces. Then we prove the anomaly formula for the associated Quillen norm. The results of this paper will be used in the sequel to study the regularity of the Quillen norm and its asymptotics in a degenerating family of Riemann surfaces with cusps and to prove the curvature theorem. We also prove that our definition of the analytic torsion for hyperbolic surfaces is compatible with the one obtained through Selberg trace formula by Takhtajan-Zograf.
Kui Du - One of the best experts on this subject based on the ideXlab platform.
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On Well-Conditioned Spectral Collocation and Spectral Methods by the Integral Reformulation
SIAM Journal on Scientific Computing, 2016Co-Authors: Kui DuAbstract:Well-conditioned spectral collocation and spectral methods have recently been proposed to solve differential equations. In this paper, we revisit the well-conditioned spectral collocation methods proposed in [T. A. Driscoll, J. Comput. Phys., 229 (2010), pp. 5980-5998] and [L.-L. Wang, M. D. Samson, and X. Zhao, SIAM J. Sci. Comput., 36 (2014), pp. A907--A929], and the ultraspherical spectral method proposed in [S. Olver and A. Townsend, SIAM Rev., 55 (2013), pp. 462--489] for an $m$th-order ordinary differential equation from the viewpoint of the integral reformulation. Moreover, we propose a Chebyshev spectral method for the integral reformulation. The well-conditioning of these methods is obvious by noting that the resulting linear operator is a Compact Perturbation of the identity. Numerical examples are given to confirm the well-conditioning of the Chebyshev spectral method.
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On well-conditioned spectral collocation and spectral methods by the integral reformulation
arXiv: Numerical Analysis, 2015Co-Authors: Kui DuAbstract:Well-conditioned spectral collocation and spectral methods have recently been proposed to solve differential equations. In this paper, we revisit the well-conditioned spectral collocation methods proposed in [T.~A. Driscoll, {\it J. Comput. Phys.}, 229 (2010), pp.~5980-5998] and [L.-L. Wang, M.~D. Samson, and X.~Zhao, {\it SIAM J. Sci. Comput.}, 36 (2014), pp.~A907--A929], and the ultraspherical spectral method proposed in [S.~Olver and A.~Townsend, {\it SIAM Rev.}, 55 (2013), pp.~462--489] for an $m$th-order ordinary differential equation from the viewpoint of the integral reformulation. Moreover, we propose a Chebyshev spectral method for the integral reformulation. The well-conditioning of these methods is obvious by noting that the resulting linear operator is a Compact Perturbation of the identity. The adaptive QR approach for the ultraspherical spectral method still applies to the almost-banded infinite-dimensional system arising in the Chebyshev spectral method for the integral reformulation. Numerical examples are given to confirm the well-conditioning of the Chebyshev spectral method.
Diomba Sambou - One of the best experts on this subject based on the ideXlab platform.
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Lieb-Thirring type inequalities for non self-adjoint Perturbations of magnetic Schrödinger operators
2013Co-Authors: Diomba SambouAbstract:Let $H := H_{0} + V$ and $H_{\perp} := H_{0,\perp} + V$ be respectively Perturbations of the free Schrödinger operators $H_{0}$ on $L^{2}\big(\mathbb{R}^{2d+1}\big)$ and $H_{0,\perp}$ on $L^{2}\big(\mathbb{R}^{2d}\big)$, $d \geq 1$ with constant magnetic field of strength $b>0$, and $V$ is a complex relatively Compact Perturbation. We prove Lieb-Thirring type inequalities for the discrete spectrum of $H$ and $H_{\perp}$. In particular, these estimates give $a\, priori$ information on the distribution of the discrete eigenvalues around the Landau levels of the operator, and describe how fast sequences of eigenvalues converge.
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Lieb-Thirring type inequalities for non-selfadjoint Perturbations of magnetic Schrödinger operators
2013Co-Authors: Diomba SambouAbstract:Let $H := H_{0} + V$ and $H_{\perp} := H_{0,\perp} + V$ be respectively Perturbations of the free Schrödinger operators $H_{0}$ on $L^{2}\big(\mathbb{R}^{2d+1}\big)$ and $H_{0,\perp}$ on $L^{2}\big(\mathbb{R}^{2d}\big)$, $d \geq 1$ with constant magnetic field of strength $b>0$, and $V$ is a complex relatively Compact Perturbation. We prove Lieb-Thirring type inequalities for the discrete spectrum of $H$ and $H_{\perp}$. In particular, these estimates give $a\, priori$ information on the distribution of eigenvalues around the Landau levels of the operator, and describe how fast sequences of eigenvalues converge.
V. H. Cortés - One of the best experts on this subject based on the ideXlab platform.
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Spectral stability for Compact Perturbations of Toeplitz matrices
Journal of Mathematical Analysis and Applications, 2016Co-Authors: M. A. Astaburuaga, Olivier Bourget, V. H. CortésAbstract:Abstract Let f be a regular real-valued non-constant symbol defined on the one dimensional torus T . Denote respectively by κ and T, its set of critical points and the associated Toeplitz matrix on l 2 ( N ) . If V is a suitable Compact Perturbation, we prove that the operator T + V has no singular continuous spectrum and only finite point spectrum away from the set of thresholds f ( κ ) . We also obtain some propagation estimates and apply these results to concrete examples.
Zhu Sen - One of the best experts on this subject based on the ideXlab platform.
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Small Compact Perturbation of Operators in β_∞(Ω)
2008Co-Authors: Zhu Sen, Wang Li-feiAbstract:We consider an approximation problem related to strongly irreducible operators,that is,does the direct sum of a strongly irreducible operator inβ_∞(Ω) and certain operator have a small Compact Perturbation which is a strongly irreducible operator inβ_∞(Ω)? In this paper,we prove that the direct sum of any strongly irreducible operator inβ_∞(Ω) and certain biquasitriangular operator have small Compact Perturbations which are strongly irreducible operators inβ_∞(Ω).
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small Compact Perturbation of operators in β_ ω
东北数学(英文版), 2008Co-Authors: Zhu Sen, Wang LifeiAbstract:We consider an approximation problem related to strongly irreducible operators,that is,does the direct sum of a strongly irreducible operator inβ_∞(Ω) and certain operator have a small Compact Perturbation which is a strongly irreducible operator inβ_∞(Ω)? In this paper,we prove that the direct sum of any strongly irreducible operator inβ_∞(Ω) and certain biquasitriangular operator have small Compact Perturbations which are strongly irreducible operators inβ_∞(Ω).