Alexei Iantchenko | Malmö University (original) (raw)
Papers by Alexei Iantchenko
Quarterly Journal of Mechanics and Applied Mathematics, Feb 28, 2019
A first-order asymptotic model for describing waves propagating along the surface of a functional... more 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 ...
arXiv (Cornell University), Sep 20, 2022
We present a comprehensive analysis of wavenumber resonances or leaky modes associated with the R... more We present a comprehensive analysis of wavenumber resonances or leaky 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 analyze 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 mentioned Riemann surface as expected.
arXiv (Cornell University), Mar 1, 2015
We give an expansion of the solution of the evolution equation for the massless Dirac fields in t... more We give an expansion of the solution of the evolution equation for the massless Dirac fields in the outer region of de Sitter-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 cutoff resolvent estimates for the semiclassical Schrödinger operators from [4]. The method extends to the Dirac operators on spherically symmetric asymptotically hyperbolic manifolds.
arXiv (Cornell University), May 17, 2011
We consider a periodic Jacobi operator H with finitely supported perturbations on Z. We solve the... more 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 − (λ) + 1, where R − is the reflection coefficient.
arXiv (Cornell University), May 16, 2010
We consider the "weighted" operator P k = −∂xa(x)∂x on the line with a steplike coefficient which... more We consider the "weighted" operator P k = −∂xa(x)∂x on the line with a steplike 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 → ∞. 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 k → ∞ the resonance spectrum converges to the real axis.
Springer eBooks, Feb 8, 2006
arXiv (Cornell University), Nov 30, 2015
We provide the full asymptotic description of the quasi-normal modes (resonances) in any strip of... more 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 [17], [18] to the (uncharged) Kerr-de Sitter black holes. We show that the mass of the Dirac field does not have effect on the two leading terms in the expansions of resonances.
We consider a periodic Jacobi operator J with finitely supported perturbations on the half-lattic... more We consider a 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 all eigenvalues, resonances and from the set of zeros of S(λ) − 1, where S(λ) is the scattering matrix.
arXiv (Cornell University), Jul 14, 2014
The quasi-normal modes for black holes are the resonances for the scattering of incoming waves by... more 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-Nordström 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 Schrödinger operator with exponentially decreasing positive potential. We give exact relation between the sets of Dirac and Schrödinger 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 Sá Barreto and Zworski in [2].
Inverse Problems, Oct 17, 2011
We consider a periodic Jacobi operator J with finitely supported perturbations on the half-lattic... more We consider a 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 all eigenvalues, resonances and from the set of zeros of S(λ) − 1, where S(λ) is the scattering matrix.
Journal of Differential Equations, Feb 1, 2012
We consider a periodic Jacobi operator H with finitely supported perturbations on Z. We solve the... more 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 − (λ) + 1, where R − is the reflection coefficient.
HAL (Le Centre pour la Communication Scientifique Directe), Feb 16, 2023
In this paper we study Love waves in a layered, elastic half-space. We first address the direct p... more 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 frequencywavenumber couples of the Love waves. Contents 1. Introduction 1 2. Characterization of Love waves: dispersion relation and first results 5 3. Regularity and monotonicity of branches of wavenumbers 13 4. The simple square well: direct computations 20 5. The double square well 25 Appendix 31 References 36
Journal of Mathematical Physics, 2022
arXiv: Analysis of PDEs, 2017
In this paper, we present a semiclassical description of surface waves or modes in an elastic med... more 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 Verdiere 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.
Proceedings of the Royal Society A, 2023
We present a comprehensive analysis of wavenumber resonances or leaking modes associated with the... more 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.
Journal of Mathematical Analysis and Applications, 2014
We consider the 1D Dirac operator on the half-line with compactly supported potentials. We study ... more 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.
Asymptotic Analysis, 2015
We consider the radial Dirac operator with compactly supported potentials. We study resonances as... more 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.
Quarterly Journal of Mechanics and Applied Mathematics, Feb 28, 2019
A first-order asymptotic model for describing waves propagating along the surface of a functional... more 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 ...
arXiv (Cornell University), Sep 20, 2022
We present a comprehensive analysis of wavenumber resonances or leaky modes associated with the R... more We present a comprehensive analysis of wavenumber resonances or leaky 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 analyze 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 mentioned Riemann surface as expected.
arXiv (Cornell University), Mar 1, 2015
We give an expansion of the solution of the evolution equation for the massless Dirac fields in t... more We give an expansion of the solution of the evolution equation for the massless Dirac fields in the outer region of de Sitter-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 cutoff resolvent estimates for the semiclassical Schrödinger operators from [4]. The method extends to the Dirac operators on spherically symmetric asymptotically hyperbolic manifolds.
arXiv (Cornell University), May 17, 2011
We consider a periodic Jacobi operator H with finitely supported perturbations on Z. We solve the... more 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 − (λ) + 1, where R − is the reflection coefficient.
arXiv (Cornell University), May 16, 2010
We consider the "weighted" operator P k = −∂xa(x)∂x on the line with a steplike coefficient which... more We consider the "weighted" operator P k = −∂xa(x)∂x on the line with a steplike 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 → ∞. 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 k → ∞ the resonance spectrum converges to the real axis.
Springer eBooks, Feb 8, 2006
arXiv (Cornell University), Nov 30, 2015
We provide the full asymptotic description of the quasi-normal modes (resonances) in any strip of... more 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 [17], [18] to the (uncharged) Kerr-de Sitter black holes. We show that the mass of the Dirac field does not have effect on the two leading terms in the expansions of resonances.
We consider a periodic Jacobi operator J with finitely supported perturbations on the half-lattic... more We consider a 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 all eigenvalues, resonances and from the set of zeros of S(λ) − 1, where S(λ) is the scattering matrix.
arXiv (Cornell University), Jul 14, 2014
The quasi-normal modes for black holes are the resonances for the scattering of incoming waves by... more 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-Nordström 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 Schrödinger operator with exponentially decreasing positive potential. We give exact relation between the sets of Dirac and Schrödinger 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 Sá Barreto and Zworski in [2].
Inverse Problems, Oct 17, 2011
We consider a periodic Jacobi operator J with finitely supported perturbations on the half-lattic... more We consider a 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 all eigenvalues, resonances and from the set of zeros of S(λ) − 1, where S(λ) is the scattering matrix.
Journal of Differential Equations, Feb 1, 2012
We consider a periodic Jacobi operator H with finitely supported perturbations on Z. We solve the... more 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 − (λ) + 1, where R − is the reflection coefficient.
HAL (Le Centre pour la Communication Scientifique Directe), Feb 16, 2023
In this paper we study Love waves in a layered, elastic half-space. We first address the direct p... more 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 frequencywavenumber couples of the Love waves. Contents 1. Introduction 1 2. Characterization of Love waves: dispersion relation and first results 5 3. Regularity and monotonicity of branches of wavenumbers 13 4. The simple square well: direct computations 20 5. The double square well 25 Appendix 31 References 36
Journal of Mathematical Physics, 2022
arXiv: Analysis of PDEs, 2017
In this paper, we present a semiclassical description of surface waves or modes in an elastic med... more 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 Verdiere 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.
Proceedings of the Royal Society A, 2023
We present a comprehensive analysis of wavenumber resonances or leaking modes associated with the... more 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.
Journal of Mathematical Analysis and Applications, 2014
We consider the 1D Dirac operator on the half-line with compactly supported potentials. We study ... more 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.
Asymptotic Analysis, 2015
We consider the radial Dirac operator with compactly supported potentials. We study resonances as... more 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.