ABSTRACT We propose an efficient and reliable technique to calculate highly localized Whispering ... more ABSTRACT We propose an efficient and reliable technique to calculate highly localized Whispering Gallery Modes (WGMs) inside an oblate spheroidal cavity. The idea is to first separate variables in spheroidal coordinates and then to deal with two ODEs, related to the angular and radial coordinates solved using high order finite difference schemes. It turns out that, due to solution structure, the efficiency of the calculation is greatly enhanced by using variable stepsizes to better reflect the behaviour of the evaluated functions. We illustrate the approach by numerical experiments.
International Journal of Theoretical Physics, 2000
In the present contribution we compare the new Multitaper Filtering technique with the very popul... more In the present contribution we compare the new Multitaper Filtering technique with the very popular Filter Diagonalization Method. The substitution of a time-independent problem, like the standard Schrödinger equation, by a time-dependent one from the Filter Diagonalization Method allows the employment of and comparison with standard signal processing filtration machinery. The use of zero-order prolate spheroidal tapers as filtering functions
International Journal of Computer Mathematics, 2008
ABSTRACT Eigenfunctions of the Finite Fourier Transform, often referred to as ‘prolates’, are ban... more ABSTRACT Eigenfunctions of the Finite Fourier Transform, often referred to as ‘prolates’, are band-limited and highly concentrated at a finite time-interval. Both features are acquired by the convolution of a band-limited function with a prolate. This permits interpolation of such a convolution by the Walter and Shen sampling formula in terms of prolates, although the Fourier transform of the convolution is not necessarily even continuous and the concentration interval is twice as large as that of a prolate. Rigorous error estimates are given as dependent on the truncation limits. The accuracy achieved is tested by numerical examples.
ABSTRACT In this paper, we discuss the progress in the numerical simulation of the so-called ‘whi... more ABSTRACT In this paper, we discuss the progress in the numerical simulation of the so-called ‘whispering gallery’ modes (WGMs) occurring inside a prolate spheroidal cavity. These modes are mainly concentrated in a narrow domain along the equatorial line of a spheroid and they are famous because of their extremely high quality factor. The scalar Helmholtz equation provides a sufficient accuracy for WGM simulation and (in a contrary to its vector version) is separable in spheroidal coordinates. However, the numerical simulation of ‘whispering gallery’ phenomena is not straightforward. The separation of variables yields two spheroidal wave ordinary differential equations (ODEs), first only depending on the angular, second on the radial coordinate. Though separated, these equations remain coupled through the separation constant and the eigenfrequency, so that together with the boundary conditions they form a singular self-adjoint two-parameter Sturm–Liouville problem. We discuss an efficient and reliable technique for the numerical solution of this problem which enables calculation of highly localized WGMs inside a spheroid. The presented approach is also applicable to other separable geometries. We illustrate the performance of the method by means of numerical experiments.
A detailed account is given of a recent modification of the Filter Diagonalization technique that... more A detailed account is given of a recent modification of the Filter Diagonalization technique that serves to analyze a signal spectrum within a selected energy range. Our approach employs for filtering the eigenfunctions of the Finite Fourier Transform, or prolates, which ...
ABSTRACT We propose an efficient and reliable technique to calculate highly localized Whispering ... more ABSTRACT We propose an efficient and reliable technique to calculate highly localized Whispering Gallery Modes (WGMs) inside an oblate spheroidal cavity. The idea is to first separate variables in spheroidal coordinates and then to deal with two ODEs, related to the angular and radial coordinates solved using high order finite difference schemes. It turns out that, due to solution structure, the efficiency of the calculation is greatly enhanced by using variable stepsizes to better reflect the behaviour of the evaluated functions. We illustrate the approach by numerical experiments.
International Journal of Theoretical Physics, 2000
In the present contribution we compare the new Multitaper Filtering technique with the very popul... more In the present contribution we compare the new Multitaper Filtering technique with the very popular Filter Diagonalization Method. The substitution of a time-independent problem, like the standard Schrödinger equation, by a time-dependent one from the Filter Diagonalization Method allows the employment of and comparison with standard signal processing filtration machinery. The use of zero-order prolate spheroidal tapers as filtering functions
International Journal of Computer Mathematics, 2008
ABSTRACT Eigenfunctions of the Finite Fourier Transform, often referred to as ‘prolates’, are ban... more ABSTRACT Eigenfunctions of the Finite Fourier Transform, often referred to as ‘prolates’, are band-limited and highly concentrated at a finite time-interval. Both features are acquired by the convolution of a band-limited function with a prolate. This permits interpolation of such a convolution by the Walter and Shen sampling formula in terms of prolates, although the Fourier transform of the convolution is not necessarily even continuous and the concentration interval is twice as large as that of a prolate. Rigorous error estimates are given as dependent on the truncation limits. The accuracy achieved is tested by numerical examples.
ABSTRACT In this paper, we discuss the progress in the numerical simulation of the so-called ‘whi... more ABSTRACT In this paper, we discuss the progress in the numerical simulation of the so-called ‘whispering gallery’ modes (WGMs) occurring inside a prolate spheroidal cavity. These modes are mainly concentrated in a narrow domain along the equatorial line of a spheroid and they are famous because of their extremely high quality factor. The scalar Helmholtz equation provides a sufficient accuracy for WGM simulation and (in a contrary to its vector version) is separable in spheroidal coordinates. However, the numerical simulation of ‘whispering gallery’ phenomena is not straightforward. The separation of variables yields two spheroidal wave ordinary differential equations (ODEs), first only depending on the angular, second on the radial coordinate. Though separated, these equations remain coupled through the separation constant and the eigenfrequency, so that together with the boundary conditions they form a singular self-adjoint two-parameter Sturm–Liouville problem. We discuss an efficient and reliable technique for the numerical solution of this problem which enables calculation of highly localized WGMs inside a spheroid. The presented approach is also applicable to other separable geometries. We illustrate the performance of the method by means of numerical experiments.
A detailed account is given of a recent modification of the Filter Diagonalization technique that... more A detailed account is given of a recent modification of the Filter Diagonalization technique that serves to analyze a signal spectrum within a selected energy range. Our approach employs for filtering the eigenfunctions of the Finite Fourier Transform, or prolates, which ...
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Papers by T. Levitina