| 1. |
- Achieng, Pauline, 1990-
(författare)
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Reconstruction of solutions of Cauchy problems for elliptic equations in bounded and unbounded domains using iterative regularization methods
- 2023
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Cauchy problems for elliptic equations arise in applications in science and engineering. These problems often involve finding important information about an elliptical system from indirect or incomplete measurements. Cauchy problems for elliptic equations are known to be disadvantaged in the sense that a small pertubation in the input can result in a large error in the output. Regularization methods are usually required in order to be able to find stable solutions. In this thesis we study the Cauchy problem for elliptic equations in both bounded and unbounded domains using iterative regularization methods. In Paper I and II, we focus on an iterative regularization technique which involves solving a sequence of mixed boundary value well-posed problems for the same elliptic equation. The original version of the alternating iterative technique is based on iterations alternating between Dirichlet-Neumann and Neumann-Dirichlet boundary value problems. This iterative method is known to possibly work for Helmholtz equation. Instead we study a modified version based on alternating between Dirichlet-Robin and Robin-Dirichlet boundary value problems. First, we study the Cauchy problem for general elliptic equations of second order with variable coefficients in a limited domain. Then we extend to the case of unbounded domains for the Cauchy problem for Helmholtz equation. For the Cauchy problem, in the case of general elliptic equations, we show that the iterative method, based on Dirichlet-Robin, is convergent provided that parameters in the Robin condition are chosen appropriately. In the case of an unbounded domain, we derive necessary, and sufficient, conditions for convergence of the Robin-Dirichlet iterations based on an analysis of the spectrum of the Laplacian operator, with boundary conditions of Dirichlet and Robin types.In the numerical tests, we investigate the precise behaviour of the Dirichlet-Robin iterations, for different values of the wave number in the Helmholtz equation, and the results show that the convergence rate depends on the choice of the Robin parameter in the Robin condition. In the case of unbounded domain, the numerical experiments show that an appropriate truncation of the domain and an appropriate choice of Robin parameter in the Robin condition lead to convergence of the Robin-Dirichlet iterations.In the presence of noise, additional regularization techniques have to implemented for the alternating iterative procedure to converge. Therefore, in Paper III and IV we focus on iterative regularization methods for solving the Cauchy problem for the Helmholtz equation in a semi-infinite strip, assuming that the data contains measurement noise. In addition, we also reconstruct a radiation condition at infinity from the given Cauchy data. For the reconstruction of the radiation condition, we solve a well-posed problem for the Helmholtz equation in a semi-infinite strip. The remaining solution is obtained by solving an ill-posed problem. In Paper III, we consider the ordinary Helmholtz equation and use seperation of variables to analyze the problem. We show that the radiation condition is described by a non-linear well-posed problem that provides a stable oscillatory solution to the Cauchy problem. Furthermore, we show that the ill–posed problem can be regularized using the Landweber’s iterative method and the discrepancy principle. Numerical tests shows that the approach works well.Paper IV is an extension of the theory from Paper III to the case of variable coefficients. Theoretical analysis of this Cauchy problem shows that, with suitable bounds on the coefficients, can iterative regularization methods be used to stabilize the ill-posed Cauchy problem.
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| 2. |
- Argatov, Ivan, et al.
(författare)
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A simple mathematical model for the resonance frequency analysis of dental implant stability : Implant clamping quotient.
- 2019
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Ingår i: Mechanics research communications. - : Elsevier. - 0093-6413 .- 1873-3972. ; 95, s. 67-70
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Tidskriftsartikel (refereegranskat)abstract
- A simple mathematical model for free vibrations of an elastically clamped beam is suggested to interpret the results of the resonance frequency analysis developed for implant stability measurements in terms of the Implant Stability Quotient (ISQ) units. It is shown that the resonance frequency substantially depends on the lateral compliance of the implant/bone system. Based on the notion of the lateral stiffness of the implant/bone system, a new measure of the implant stability is introduced in the form similar to the ISQ scale and is called the Implant Clamping Quotient (ICQ), because it characterizes the jawbone’s clamp of the implant. By definition, the ICQ unit is equal to a percentage of the original scale for the lateral stiffness of the implant/bone system.
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| 3. |
- Argatov, Ivan, et al.
(författare)
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How to define the storage and loss moduli for a rheologically nonlinear material?
- 2017
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Ingår i: Continuum Mechanics and Thermodynamics. - : Springer. - 0935-1175 .- 1432-0959. ; 29:6, s. 1375-1387
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Tidskriftsartikel (refereegranskat)abstract
- A large amplitude oscillatory shear (LAOS) is considered in the strain-controlled regime, and the interrelation between the Fourier transform and the stress decomposition approaches is established. Several definitions of the generalized storage and loss moduli are examined in a unified conceptual scheme based on the Lissajous–Bowditch plots. An illustrative example of evaluating the generalized moduli from a LAOS flow is given.
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| 4. |
- Argatov, Ivan, et al.
(författare)
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Rayleigh surface waves in functionally graded materials : long-wave limit
- 2019
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Ingår i: Quarterly Journal of Mechanics and Applied Mathematics. - : Oxford Univeristy Press. - 0033-5614 .- 1464-3855. ; 72:2, s. 197-211
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Tidskriftsartikel (refereegranskat)abstract
- A first-order asymptotic model for describing waves propagating along the surface of a functionally graded isotropic elastic half-space is constructed in the long-wave limit under the assumption of a finitely supported perturbation of the half-space properties. Explicit approximations for the Rayleigh waves are derived under the assumption that the semi-infinite elastic medium is slightly inhomogeneous
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| 5. |
- Argatov, Ivan, et al.
(författare)
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Resonance spectrum for a continuously stratified layer : application to ultrasonic testing
- 2013
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Ingår i: Waves in Random and Complex Media. - : Taylor & Francis. - 1745-5030 .- 1745-5049. ; 23:1, s. 24-42
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Tidskriftsartikel (refereegranskat)abstract
- Ultrasound wave propagation in a nonhomogeneous linearly elastic layer of constant thickness immersed between homogeneous fluid and solid media is considered. The resonances (scattering poles) for the corresponding acoustic propagator are studied. It is shown that the distribution of the resonances depends on the smoothness of the coefficients that characterize physical properties of the layer and the ambient media. Namely, if the coefficients have jump discontinuities at the boundaries, then the resonances are asymptotically distributed along a straight line parallel to the real axis on the unphysical sheet of the complex frequency plane. On the contrary, if the coefficients are continuous, then it is shown that the resonances are asymptotically distributed along a logarithmic curve. The developed mathematical model is applied to the ultrasonic testing of the articular cartilage (AC) layer attached to the subchondral bone from one side and being in contact with a solution on the other side. It is conjectured that the spacing between two successive resonances may be sensitive to AC degeneration. The application of the obtained results to the development of ultrasonic testing for quantitative evaluation of AC is discussed.
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| 6. |
- Chepkorir, Jennifer, 1989-
(författare)
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Regularization methods for solving Cauchy problems for elliptic and degenerate elliptic equations
- 2024
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis, we study Cauchy problems for the elliptic and degenerate elliptic equations. These problems are ill-posed. We split the boundary of the domain into two parts. On one of them, say Γ0, we have available Cauchy data and on remaining part Γ1 we introduce unknown Robin data. To construct the operator equation which replaces our Cauchy problem we use two boundary value problems (BVP). The first one is the mixed BVP with Robin condition on Γ1 and with Dirichlet condition on Γ0 and the second BVP with Dirichlet Data on Γ1 and with Robin data on Γ0. The well–posedness of these problems is achieved by an appropriate choice of parameters in Robin boundary conditions. The first Dirichlet–Robin BVP is used to construct the operator equation replacing the Cauchy problem and the second Robin–Dirichlet problem for adjoint operator. Using these problems we can apply various regularization methods for stable reconstruction of the solution. In Paper I, the Cauchy problem for the elliptic equation with variable coefficients, which includes Helmholtz type equations, is analyzed. A proof showing that the Dirichlet–Robin alternating algorithm is convergent is given, provided that the parameters in the Robin conditions are chosen appropriately. Numerical experiments that shows the behaviour of the algorithm are given. In particular, we show how the speed of convergence depends on the choice of Robin parameters. In Paper II, the Cauchy problem for the Helmholtz equation, for moderate wave numbers k2, is considered. The Cauchy problem is reformulated as an operator equation and iterative method based on Krylov subspaces are implemented. The aim is to achieve faster convergence in comparison to the Alternating algorithm from the previous paper. Methods such as the Landweber iteration, the Conjugate gradient method and the generalized minimal residual method are considered. We also discuss how the algorithms can be adapted to also cover the case of non–symmetric differential operators. In Paper III, we look at a steady state heat conduction problem in a thin plate. The plate connects two cylindrical containers and fix their relative positions. A two dimensional mathematical model of heat conduction in the plate is derived. Since the plate has sharp edges on the sides we obtained a degenerate elliptic equation. We seek to find the temperature on the interior cylinder by using data on the exterior cylinder. We reformulate the Cauchy problem as an operator equation, with a compact operator. The operator equation is solved using the Landweber method and the convergence is investigated. In Paper IV, the Cauchy problem for a more general degenerate elliptic equation is considered. We stabilize the computations using Tikhonov regularization. The normal equation, in the Tikhonov algorithm, is solved using the Conjugate gradient method. The regularization parameter is picked using either the L–curve or the Discrepancy principle. In all papers, numerical examples are given where we solve the various boundary value problems using a finite difference scheme. The results show that the suggested methods work quite well.
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| 7. |
- de Hoop, Maarten, et al.
(författare)
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Semiclassical analysis of elastic surface waves
- 2017
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- In this paper, we present a semiclassical description of surface waves or modes in an elastic medium near a boundary, in spatial dimension three. The medium is assumed to be essentially stratified near the boundary at some scale comparable to the wave length. Such a medium can also be thought of as a surficial layer (which can be thick) overlying a half space. The analysis is based on the work of Colin de Verdi\`ere on acoustic surface waves. The description is geometric in the boundary and locally spectral "beneath" it. Effective Hamiltonians of surface waves correspond with eigenvalues of ordinary differential operators, which, to leading order, define their phase velocities. Using these Hamiltonians, we obtain pseudodifferential surface wave equations. We then construct a parametrix. Finally, we discuss Weyl's formulas for counting surface modes, and the decoupling into two classes of surface waves, that is, Rayleigh and Love waves, under appropriate symmetry conditions.
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| 8. |
- de Hoop, Maarten V., et al.
(författare)
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Analysis of wavenumber resonances for the Rayleigh system in a half space
- 2023
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Ingår i: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences. - : Royal Society. - 1364-5021 .- 1471-2946. ; 479:2277
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Tidskriftsartikel (refereegranskat)abstract
- We present a comprehensive analysis of wavenumber resonances or leaking modes associated with the Rayleigh operator in a half space containing a heterogeneous slab, being motivated by seismology. To this end, we introduce Jost solutions on an appropriate Riemann surface, a boundary matrix and a reflection matrix in analogy to the studies of scattering resonances associated with the Schrödinger operator. We analyse their analytic properties and characterize the distribution of these wavenumber resonances. Furthermore, we show that the resonances appear as poles of the meromorphic continuation of the resolvent to the nonphysical sheets of the Riemann surface as expected.
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| 9. |
- De Hoop, Maarten V, et al.
(författare)
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Inverse problem for Love waves in a layered, elastic half-space
- 2024
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Ingår i: Inverse Problems. - : Institute of Physics Publishing (IOPP). - 0266-5611 .- 1361-6420. ; :045013, s. 1-44
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study Love waves in a layered, elastic half-space. We first address the direct problem and we characterize the existence of Love waves through the dispersion relation. We then address the inverse problem and we show how to recover the parameters of the elastic medium from the empirical knowledge of the frequency–wavenumber couples of the Love waves.
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| 10. |
- de Hoop, Maarten V., et al.
(författare)
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Inverse problem for the Rayleigh system with spectral data
- 2022
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Ingår i: Journal of Mathematical Physics. - : American Institute of Physics (AIP). - 0022-2488 .- 1089-7658. ; 63:3, s. 1-33
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Tidskriftsartikel (refereegranskat)abstract
- We analyze an inverse problem associated with the time-harmonic Rayleigh system on a flat elastic half-space concerning the recovery of Lamé parameters in a slab beneath a traction-free surface. We employ the Markushevich substitution, while the data are captured in a Jost function, and we point out parallels with a corresponding problem for the Schrödinger equation. The Jost function can be identified with spectral data. We derive a Gel’fand-Levitan type equation and obtain uniqueness with two distinct frequencies.
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| 11. |
- de Hoop, Maarten, V, et al.
(författare)
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Semiclassical inverse spectral problem for seismic surface waves in isotropic media : part I. Love waves
- 2020
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Ingår i: Inverse Problems. - : Institute of Physics Publishing (IOPP). - 0266-5611 .- 1361-6420. ; 36:7
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Tidskriftsartikel (refereegranskat)abstract
- We analyze the inverse spectral problem on the half line associated with elastic surface waves. Here, we focus on Love waves. Under certain generic conditions, we establish uniqueness and present a reconstruction scheme for theS-wavespeed with multiple wells from the semiclassical spectrum of these waves.
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| 12. |
- de Hoop, Maarten, V, et al.
(författare)
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Semiclassical inverse spectral problem for seismic surface waves in isotropic media : part II. Rayleigh waves
- 2020
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Ingår i: Inverse Problems. - : Institute of Physics Publishing (IOPP). - 0266-5611 .- 1361-6420. ; 36:7
-
Tidskriftsartikel (refereegranskat)abstract
- We analyze the inverse spectral problem on the half line associated with elastic surface waves. Here, we extend the treatment of Love waves [5] to Rayleigh waves. Under certain conditions, and assuming that the Poisson ratio is constant, we establish uniqueness and present a reconstruction scheme for the S-wave speed with multiple wells from the semiclassical spectrum of these waves.
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| 13. |
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| 14. |
- Iantchenko, Alexei
(författare)
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An Inverse Problem for Trapping Point Resonances
- 2008
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Ingår i: Letters in Mathematical Physics. - : Springer. - 0377-9017 .- 1573-0530. ; 86:2-3, s. 151-157
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Tidskriftsartikel (refereegranskat)abstract
- We consider semi-classical Schr¨odinger operator P(h)=−h2+V (x) in Rn such that the analytic potential V has a non-degenerate critical point x0 = 0 with critical value E0 and we can define resonances in some fixed neighborhood of E0 when h > 0 is small enough. If the eigenvalues of the Hessian are Z-independent the resonances in hδ-neighborhood of E0 (δ>0) can be calculated explicitly as the eigenvalues of the semiclassical Birkhoff normal form. Assuming that potential is symmetric with respect to reflections about the coordinate axes we show that the classical Birkhoff normal form determines the Taylor series of the potential at x0. As a consequence, the resonances in a hδ-neighborhood of E0 determine the first N terms in the Taylor series of V at x0. The proof uses the recent inverse spectral results of V. Guillemin and A. Uribe.
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| 15. |
- Iantchenko, Alexei, et al.
(författare)
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Inverse problems in Seismology with spectral and resonance data
- 2021
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Semiclassical analysis can be employed to describe surface waves in an elastic half space which is quasi-stratified near its boundary. The propagation of such waves is governed by effective Hamiltonians on the boundary with a space-adiabatic behavior. Effective Hamiltonians of surface waves correspond to eigenvalues of ordinary differential operators, which, to leading order, define their phase velocities. In case of isotropic medium, the surface wave decouples up to principal parts, into Love and Rayleigh waves.We present the conditional recovery of the Lamé parameters from spectral data, in two inverse problems approaches:- semiclassical techniques using the semiclassical spectra as the data;- exact methods for Sturm-Liouville operators, using the discrete and continuous spectra, or the Weyl function, as the data based on the solution of the Gel’fand-Levitan-Marchenko equation.We conclude with discussion using resonances (leaking modes) as data.
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| 16. |
- Iantchenko, Alexei, Professor
(författare)
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Inverse problems in surface-wave tomography with spectral and resonance data
- 2023
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- Semiclassical analysis can be employed to describe surface waves in an elastic half space which is quasi-stratified near its boundary. In the case of an isotropic medium, the surface wave decouples up to principal parts into Love and Rayleigh waves associated to scalar and matrix spectral problems, respectively. Since the mathematical features (such as spectrum, resonances) of these problems can be extracted from the seismograms, we are interested in recovering the Lam e parameters from these data. We generalize spectral methods for Schrodinger operators to the Rayleigh problem, which is essentially not of Schrodinger type; and give comprehensive analysis of the wavenumber resonances, known in seismology as leaking modes.This is joint work with Maarten V. de Hoop.
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| 17. |
- Iantchenko, Alexei
(författare)
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Olika kulturers bidrag till matematik
- 2005
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Annan publikation (populärvet., debatt m.m.)abstract
- Matematiken är internationell och det matematiska språket, symbolerna, är samma överallt i världen. Många olika kulturer har bidragit till matematikens utveckling. I denna föreläsningspresentation diskuteras matematikutveckling i olika länder och matematiker av olika ursprung. Begränsning tidsperiod:1800-1900.
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| 18. |
- Iantchenko, Alexei, et al.
(författare)
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On the positivity of the Jansen-Heß operator for arbitrary mass
- 2003
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Ingår i: Annales de l'Institute Henri Poincare. Physique theorique. - : Springer Science and Business Media LLC. - 1424-0637 .- 1424-0661. ; 4:6, s. 1083-1099
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Tidskriftsartikel (refereegranskat)abstract
- A fully consistent interpretation of the Dirac equation is only possible in the framework of quantum field theory. This is due to the possibility of negative-energy solutions, or, in mathematical terms, to the fact that the Dirac operator is not bounded from below. Far from the pair-production threshold, however, one can recover an approximate single-particle picture for the relativistic electron by restricting the Dirac Hamiltonian to its positive spectral subspace. Alternatively, one can employ a Foldy-Wouthuysen-type transformation to derive an approximate operator in the positive spectral subspace of the free Hamiltonian, which contains the effects of the Coulomb field up to the second order [G. Jansen and B. Hess, Phys. Rev. A 39 (1989), no. 11, 6016--6017]. The positivity of the Jansen-Hess (JH) operator has been proved recently for nuclear charges up to the critical value $Z_c = 25$Zc=25 [R. Brummelhuis, H. K. H. Siedentop and E. Stockmeyer, Doc. Math. 7 (2002), 167--182 (electronic); MR1911215 (2004b:81046)]. In this paper, the proof of positivity is extended up to $Z_c = 114$Zc=114 by considering the JH Hamiltonian in a momentum-space, partial wave representation. One first proves positivity of the kernels of the partial-wave components of the JH operator. As a second step, one then establishes the monotonicity properties of these kernels with respect to the orbital quantum number. This is enough to prove positivity analytically for $Z\leq 81$Z≤81, and through the numerical solution of a transcendental equation for $Z \leq 114$Z≤114.
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| 19. |
- Iantchenko, Alexei
(författare)
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Periodic Jacobi operator with finitely supported perturbation on the half-lattice
- 2010
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- We consider the periodic Jacobi operator $J$ with finitely supported perturbations on $\ell^2(\N)$ subject to Dirichlet boundary condition at $n=0$. We classify all states of $J$ and give their properties. We solve the inverse resonance problem (including characterization): we prove that mapping from real perturbations to the associated regularized Jost functions is one-to-one and onto.
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| 20. |
- Iantchenko, Alexei, et al.
(författare)
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Periodic Jacobi operator with finitely supported perturbation on the half-lattice
- 2011
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Ingår i: Inverse Problems. - : Institute of Physics Publishing (IOPP). - 0266-5611 .- 1361-6420. ; 27:11
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Tidskriftsartikel (refereegranskat)abstract
- We consider the periodic Jacobi operator $J$ with finitely supported perturbations on the half-lattice. We describe all eigenvalues and resonances of $J$ and give their properties. We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the Jost functions is one-to-one and onto, we show how the Jost functions can be reconstructed from the eigenvalues, resonances and the set of zeros of $S(\l)-1,$ where $S(\l)$ is the scattering matrix.
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| 21. |
- Iantchenko, Alexei, et al.
(författare)
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Periodic Jacobi operator with finitely supported perturbations : the inverse resonance problem
- 2011
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We consider a periodic Jacobi operator $H$ with finitely supported perturbations on ${\Bbb Z}.$ We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the scattering data: the inverse of the transmission coefficient and the Jost function on the right half-axis, is one-to-one and onto. We consider the problem of reconstruction of the scattering data from all eigenvalues, resonances and the set of zeros of $R_-(\lambda)+1,$ where $R_-$ is the reflection coefficient.
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| 22. |
- Iantchenko, Alexei, et al.
(författare)
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Periodic Jacobi operator with finitely supported perturbations
- 2010
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We describe the spectral properties of the Jacobi operator $(Hy)_n= a_{n-1} y_{n-1}+a_{n}y_{n+1}+b_ny_n,$ $n\in\Z,$ with $a_n=a_n^0+ u_n,$ $b_n= b_n^0+ v_n,$ where sequences $a_n^0>0,$ $b_n^0\in\R$ are periodic with period $q$, and sequences $ u_n,$ $ v_n$ have compact support. In the case $ u_n\equiv 0$ we obtain the asymptotics of the spectrum in the limit of small perturbations $ v_n.$
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| 23. |
- Iantchenko, Alexei, et al.
(författare)
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Periodic Jacobi operator with finitely supported perturbations : The inverse resonance problem
- 2012
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Ingår i: Journal of Differential Equations. - : Elsevier. - 0022-0396 .- 1090-2732. ; 252:3, s. 2823-2844
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Tidskriftsartikel (refereegranskat)abstract
- We consider a periodic Jacobi operator H with finitely supported perturbations on Z. We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the scattering data, the inverse of the transmission coefficient and the Jost function on the right half-axis, is one-to-one and onto. We consider the problem of reconstruction of the scattering data from all eigenvalues, resonances and the set of zeros of R_(lambda) + 1, where R_ is the reflection coefficient. (C) 2011 Elsevier Inc. All rights reserved.
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| 24. |
- Iantchenko, Alexei, et al.
(författare)
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Quadratic equations : Preliminary notes
- 2010
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Annan publikation (populärvet., debatt m.m.)abstract
- Explanatory materials about quadratic equations to be better prepared for studying mathematics at the University. Used at Aberystwyth Mathematical Club.
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| 25. |
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| 26. |
- Iantchenko, Alexei
(författare)
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Quasi-normal modes for de Sitter–Reissner–Nordström black holes
- 2017
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Ingår i: Mathematical Research Letters. - : International Association of Computer Science and Information Technology Press. - 1073-2780 .- 1945-001X. ; 24:1, s. 83-117
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Tidskriftsartikel (refereegranskat)abstract
- The quasi-normal modes for black holes are the resonances for the scattering of incoming waves by black holes. Here we consider scattering of massless uncharged Dirac fields propagating in the outer region of de Sitter-Reissner-Nordstr¨om black hole, which is spherically symmetric charged exact solution of the Einstein-Maxwell equations. Using the spherical symmetry of the equation and restricting to a fixed harmonic the problem is reduced to a scattering problem for the 1D massless Dirac operator on the line. The resonances for the problem are related to the resonances for a certain semiclassical Schr¨odinger operator with exponentially decreasing positive potential. We give exact relation between the sets of Dirac and Schr¨odinger resonances. The asymptotic distribution of the resonances is close to the lattice of pseudopoles associated to the non-degenerate maxima of the potentials. Using the techniques of quantum Birkhoff normal form we give the complete asymptotic formulas for the resonances. In particular, we calculate the first three leading terms in the expansion. Moreover, similar results are obtained for the de Sitter-Schwarzschild quasi-normal modes, thus improving the result of S´a Barreto and Zworski in [2].
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| 27. |
- Iantchenko, Alexei
(författare)
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Quasi-normal modes for Dirac fields in Kerr-Newman-de Sitter black holes
- 2018
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Ingår i: Analysis and Applications. - : World Scientific. - 0219-5305 .- 1793-6861. ; 16:4, s. 449-524
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Tidskriftsartikel (refereegranskat)abstract
- We provide the full asymptotic description of the quasi-normal modes (resonances) in any strip of fixed width for Dirac fields in slowly rotating Kerr-Newman-de Sitter black holes. The resonances split in a way similar to the Zeeman effect. The method is based on the extension to Dirac operators of techniques applied by Dyatlov in [Quasi-normal modes and exponential energy decay for the Kerr-de Sitter black hole, Commun. Math. Phys. 306(1) (2011) 119-163; Asymptotic distribution of quasi-normal modes for Kerr-de Sitter black holes. Ann. Henri Poincare 13(5) (2012) 1101-1166] to the (uncharged) Kerr-de Sitter black holes. We show that the mass of the Dirac field does not have an effect on the two leading terms in the expansions of resonances. We give an expansion of the solution of the evolution equation for the Dirac fields in the outer region of the slowly rotating Kerr-Newman-de Sitter black hole which implies the exponential decay of the local energy. Moreover, using the r-normal hyperbolicity of the trapped set and applying the techniques from [Asymptotics of linear waves and resonances with applications to black holes, Commun. Math. Phys. 335 (2015) 1445-1485: Resonance projectors and asymptotics for r-normally hyperbolic trapped sets. J. Amer. Math. Soc. 28 (2015) 311-381]. we give location of the resonance free band and the Weyl-type formula for the resonances in the band near the real axis.
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| 28. |
- Iantchenko, Alexei
(författare)
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Resonance expansions of massless Dirac fields propagating in the exterior of a de Sitter–Reissner–Nordström black hole
- 2017
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Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier. - 0022-247X .- 1096-0813. ; 454:2, s. 639-658
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Tidskriftsartikel (refereegranskat)abstract
- We give an expansion of the solution of the evolution equation for the massless Dirac fields in the outer region of deSitter–Reissner–Nordström black hole in terms of resonances. By means of this method we describe the decay of local energy for compactly supported data. The proof uses the cut-off resolvent estimates for the semi-classical Schrödinger operators from [4]. The method extends to the Dirac operators on spherically symmetric asymptotically hyperbolic manifolds.
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| 29. |
- Iantchenko, Alexei, et al.
(författare)
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Resonance Spectrum for a Continuously Stratified Layer with Application to Ultrasound Testing of Articular Cartilage
- 2011
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Ingår i: Numerical Analysis and Applied Mathematics ICNAAM 2011. - : American Institute of Physics (AIP). ; , s. 451-454
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Konferensbidrag (refereegranskat)abstract
- Ultrasound wave propagation in the articular cartilage layer is considered. The cartilage is modeled by a nonhomogeneous linearly elastic layer of constant thickness. The resonances for the corresponding acoustic propagator are studied. It is shown that the resonances are asymptotically distributed along a straight line parallel to the real axis on the unphysical sheet of the complex frequency plane. The spacing between two successive resonances turns out to be sensitive to articular cartilage degeneration.
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| 30. |
- Iantchenko, Alexei
(författare)
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Resonance Spectrum For One-Dimensional Layered Media
- 2006
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- We consider the "weighted" operator $P_k=-\partial_x a(x)\partial_x$ on the line with a step-like coefficient which appears when propagation of waves thorough a finite slab of a periodic medium is studied. The medium is transparent at certain resonant frequencies which are related to the complex resonance spectrum of $P_k.$ If the coefficient is periodic on a finite interval (locally periodic) with $k$ identical cells then the resonance spectrum of $P_k$ has band structure. In the present paper we study a transition to semi-infinite medium by taking the limit $k\to \infty.$ The bands of resonances in the complex lower half plane are localized below the band spectrum of the corresponding periodic problem ($k=\infty$) with $k-1$ or $k$ resonances in each band. We prove that as $k\to \infty$ the resonance spectrum converges to the real axis.
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| 31. |
- Iantchenko, Alexei
(författare)
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Resonance spectrum for one-dimensional layered media
- 2006
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Ingår i: Applicable Analysis. - : Informa UK Limited. - 0003-6811 .- 1563-504X. ; 85:11, s. 1383-1410
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Tidskriftsartikel (refereegranskat)abstract
- We consider the “weighted” operator Pk= - ∂x a(x)∂ x on the real line with a step-like coefficient which appears when propagation of waves through a finite slab of a periodic medium is studied. The medium is transparent at certain resonant frequencies which are related to the complex resonance spectrum of Pk. If the coefficient is periodic on a finite interval (locally periodic) with k identical cells, then the resonance spectrum of Pk has band structure. In the article, we study a transition to semi-infinite medium by taking the limit ∞ . The bands of resonances in the complex lower half plane are localized below the band spectrum of the corresponding periodic problem (k=∞) with k - 1 or k resonances in each band. We prove that as ∞ , the resonance spectrum converges to the real axis.
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| 32. |
- Iantchenko, Alexei
(författare)
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Resonance spectrum for one-dimensional truncated periodic media
- 2006
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- We consider the "weighted" operator $P_k=-\partial_x a(x)\partial_x$ on the line with a step-like coefficient which appears when propagation of waves thorough a finite slab of a periodic medium is studied. The medium is transparent at certain resonant frequencies which are related to the complex resonance spectrum of $P_k.$ If the coefficient is periodic on a finite interval (locally periodic) with $k$ identical cells then the resonance spectrum of $P_k$ has band structure. In the present paper we study a transition to semi-infinite medium by taking the limit $k\to \infty.$ The bands of resonances in the complex lower half plane are localized below the band spectrum of the corresponding periodic problem ($k=\infty$) with $k-1$ or $k$ resonances in each band. We prove that as $k\to \infty$ the resonance spectrum converges to the real axis.
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| 33. |
- Iantchenko, Alexei, et al.
(författare)
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Resonances for 1D massless Dirac operators
- 2013
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We consider the 1D massless Dirac operator on the real line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: 1) asymptotics of counting function, 2) estimates on the resonances and the for- bidden domain, 3) the trace formula in terms of resonances
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| 34. |
- Iantchenko, Alexei, et al.
(författare)
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Resonances for 1D massless Dirac operators
- 2014
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Ingår i: Journal of Differential Equations. - : Elsevier. - 0022-0396 .- 1090-2732. ; 256:8, s. 3038-3066
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Tidskriftsartikel (refereegranskat)abstract
- We consider the ID massless Dirac operator on the real line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: (1) asymptotics of counting function, (2) estimates on the resonances and the forbidden domain, (3) the trace formula in terms of resonances. (C) 2014 Elsevier Inc: All rights reserved.
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| 35. |
- Iantchenko, Alexei, et al.
(författare)
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Resonances for Dirac operators on the half-line
- 2014
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Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier. - 0022-247X .- 1096-0813. ; 420:1, s. 279-313
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Tidskriftsartikel (refereegranskat)abstract
- We consider the 1D Dirac operator on the half-line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: (1) asymptotics of counting function, (2) estimates on the resonances and the forbidden domain.
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| 36. |
- Iantchenko, Alexei, et al.
(författare)
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Resonances for periodic Jacobi operators with finitely supported perturbations
- 2012
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Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier. - 0022-247X .- 1096-0813. ; 388:2, s. 1239-1253
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Tidskriftsartikel (refereegranskat)abstract
- We describe the resonances and the eigenvalues of a periodic Jacobi operator with finitely supported perturbations. In the case of small diagonal perturbations we determine their asymptotics.
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| 37. |
- Iantchenko, Alexei, et al.
(författare)
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Resonances for the radial Dirac operators
- 2015
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Ingår i: Asymptotic Analysis. - : BIOS Scientific Publishers. - 0921-7134 .- 1875-8576. ; 93:4, s. 327-370
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Tidskriftsartikel (refereegranskat)abstract
- We consider the radial Dirac operator with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: (1) asymptotics of counting function, (2) in the massless case we get the trace formula in terms of resonances.
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| 38. |
- Iantchenko, Alexei
(författare)
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Resonansfenomen i kvantmekaniken: Vad har resonanser för betydelse för vår världsbild
- 2005
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Annan publikation (populärvet., debatt m.m.)abstract
- Syfte med denna föreläsning är att introducera begreppet “resonans” och att övertyga er om att resonanser är en del av vår verklighet. Jag ska också ge exempel som visar hur resonanser kan användas för att studera vår värld. Föreläsning ska vara begriplig för alla. I föreläsningen introduceras begreppet “resonans” på grundläggande nivå. Exempel ges på hur renosanser kan användas för att studera vår värld. Renosanser är en del av vår verklighet - genom att mäta resonanser kan man skapa en bild av det verkliga objektet. Presentationen användes vid Alexei Iantchenkos docentföreläsning den 1 april 2005.
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| 39. |
- Iantchenko, Alexei
(författare)
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Scattering poles near the real axis for strictly convex obstacles
- 2004
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- To study the location of poles for the acoustic scattering matrix for two strictly convex obstacles with smooth boundaries, one uses an approximation of the quantized billiard operator M along the trapped ray between the two obstacles. Assuming that the boundaries are analytic and the eigenvalues of Poincar´e map are non-resonant we use the Birkhoff normal form for M to get the complete asymptotic expansions for the poles in any logarithmic neighborhood of real axis.
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| 40. |
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| 41. |
- Iantchenko, Alexei, et al.
(författare)
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Schrödinger Operator on the Zigzag Half-Nanotube in Magnetic Field
- 2010
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Ingår i: Mathematical Modelling of Natural Phenomena. - : EDP Sciences. - 0973-5348 .- 1760-6101. ; 5:4 Spectral problems. Issue dedicated to the memory of M. Birman, s. 175-197
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Tidskriftsartikel (refereegranskat)abstract
- We consider the zigzag half-nanotubes (tight-binding approximation) in a uniform magnetic field which is described by the magnetic Schrödinger operator with a periodic potential plus a finitely supported perturbation. We describe all eigenvalues and resonances of this operator, and theirs dependence on the magnetic field. The proof is reduced to the analysis of the periodic Jacobi operators on the half-line with finitely supported perturbations.
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| 42. |
- Iantchenko, Alexei
(författare)
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Semiclassical inverse spectral and resonance problems in semiclassical surface-wave tomography : Lectures at l'IHP, Paris, 20-21 june 2022
- 2022
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Semiclassical analysis can be employed to describe surface waves in an elastic half space which is quasi-stratified near its boundary. The propagation of such waves is governed by effective Hamiltonians on the boundary with a space-adiabatic behavior.Effective Hamiltonians of surface waves correspond to eigenvalues of ordinary differential operators, which, to leading order, define their phase velocities.Using these Hamiltonians, we obtain pseudodifferential surface wave equations. Then we carry out the semiclassical construction of general surface waveparametrices. In the process, we introduce locally Schrödinger-like operators in the boundary normal coordinate and their eigenvalues signifying effective Hamiltonians in the boundary(tangential) coordinates describing surface-wave propagation.In case of isotropic medium the surface wave decouples up to principal parts into Love and Rayleigh waves associated to scalar and matrix spectral problems, respectively.Since the mathematical features (such as spectrum, resonances) of these problems can be extracted from the seismograms, we are interested in recovering the Lamé parameters from these data. Plan of lectures:1) Semiclassical description of surface waves.2) Semiclassical inverse spectral problems for Love and Rayleigh waves: conditional recovery of S-wave speed using the semiclassical spectra as the data.3) Recovery of S-and P-wave speeds from the discrete and continuous spectra using the exact methods for Sturm-Liouville operators: Gel'fand-Levitan-Marchenko approach.4) Leaking modes as analogues of scattering resonances.The material of these lectures is result of collaboration with Maarten V. de Hoop at Rice University and his Geo-Mathematical Imaging Group.
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| 43. |
- van de Hoop, Marten, et al.
(författare)
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Semiclassical Surface Wave Tomography of Isotropic Media
- 2020
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Ingår i: Spectral Theory and Mathematical Physics. - Cham : Springer. - 9783030555566 - 9783030555559 ; , s. 105-123
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Konferensbidrag (refereegranskat)abstract
- We carry out a semiclassical analysis of surface waves in Earth which is stratified near its boundary at some scale comparable to the wave length.Propagation of such waves is governed by effective Hamiltonians which are non-homogeneous principal symbols of some pseudodifferential operators. Each Hamiltonian is identified with an eigenvalue in the discreet spectrum of a locally one-dimensional Schrödinger-like operator on the one hand, and generates a flow identified with surface wave bicharacteristics in the two-dimensional boundary on the other hand.The eigenvalues exist under certain assumptions reflecting that wave speeds near the boundary are smaller than in the deep interior. This assumption is naturally satisfied in Earth’s crust and upper mantle.Using the mentioned Hamiltonians, we obtain pseudodifferential surface wave equations. In the case of isotropic elasticity, the equations decouple into equations for Rayleigh and Love waves. In both cases, we perform a comprehensive analysis of the recovery of the S-wave speed from the semiclassical spectrum.Our approach follows the ideas of Colin de Verdière pertaining to acoustic surface waves.
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