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Fractional Laplace operator in the unit ball Kwaśnicki, Mateusz
Description
The eigenvalues $\lambda_n$ of the fractional Laplace operator $(-\Delta)^{\alpha/2}$ in the unit ball are not known explicitly, and many apparently simple questions concerning $\lambda_n$ remain unanswered. In my recent joint work with Bartłomiej Dyda and Alexey Kuznetsov we address two examples of such questions. Until recently, evaluating $\lambda_n$ was difficult. We provide two efficient numerical methods for finding lower and upper numerical estimates for $\lambda_n$. For the upper bounds, we use standard Rayleigh–Ritz variational method, while lower bounds involve Weinstein–Aronszajn method of intermediate problems. Both require closed-form expressions for the fractional Laplace operator. We use explicit formulae for the eigenvalues and eigenvectors of the operator $(1 - |x|^2)^{\alpha/2}_+ (-\Delta)^{\alpha/2}$, a topic that will be discussed in detail by Alexey Kuznetsov. The second problem that we address is a conjecture due to Tadeusz Kulczycki, which asserts that all eigenfunctions corresponding to $\lambda_2$ are antisymmetric. This was known to be true only in dimension $1$ when $\alpha \ge 1$. We are able to extend this result to arbitrary $\alpha$ in dimensions $1$ and $2$, and to $\alpha = 1$ in dimensions up to $9$. We prove this result by applying our estimates analytically. This is practically doable only when $2 \times 2$ matrices are involved. Larger matrices can be treated numerically, and such experiments strongly support the conjecture in full generality. At the end of my talk I will present an intriguing open problem, which originates in the following observation: the operators $A = -(1 - |x|^2)^{\alpha/2}_+ (-\Delta)^{\alpha/2}$ and $B = (1 - |x|^2) \Delta - (2 + \alpha) x \cdot \nabla$ have identical eigenfunctions! In dimension $1$ this can be used to prove that the time-changed isotropic $\alpha$-stable Lévy process generated by $A$ can be constructed by subordinating the Jacobi diffusion generated by $B$ using an appropriate subordinator. This result, however, does not extend to higher dimensions!
Item Metadata
Title |
Fractional Laplace operator in the unit ball
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2016-11-10T15:37
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Description |
The eigenvalues $\lambda_n$ of the fractional Laplace operator $(-\Delta)^{\alpha/2}$ in the unit ball are not known explicitly, and many apparently simple questions concerning $\lambda_n$ remain unanswered. In my recent joint work with Bartłomiej Dyda and Alexey Kuznetsov we address two examples of such questions.
Until recently, evaluating $\lambda_n$ was difficult. We provide two efficient numerical methods for finding lower and upper numerical estimates for $\lambda_n$. For the upper bounds, we use standard Rayleigh–Ritz variational method, while lower bounds involve Weinstein–Aronszajn method of intermediate problems. Both require closed-form expressions for the fractional Laplace operator. We use explicit formulae for the eigenvalues and eigenvectors of the operator $(1 - |x|^2)^{\alpha/2}_+ (-\Delta)^{\alpha/2}$, a topic that will be discussed in detail by Alexey Kuznetsov.
The second problem that we address is a conjecture due to Tadeusz Kulczycki, which asserts that all eigenfunctions corresponding to $\lambda_2$ are antisymmetric. This was known to be true only in dimension $1$ when $\alpha \ge 1$. We are able to extend this result to arbitrary $\alpha$ in dimensions $1$ and $2$, and to $\alpha = 1$ in dimensions up to $9$. We prove this result by applying our estimates analytically. This is practically doable only when $2 \times 2$ matrices are involved. Larger matrices can be treated numerically, and such experiments strongly support the conjecture in full generality.
At the end of my talk I will present an intriguing open problem, which originates in the following observation: the operators $A = -(1 - |x|^2)^{\alpha/2}_+ (-\Delta)^{\alpha/2}$ and $B = (1 - |x|^2) \Delta - (2 + \alpha) x \cdot \nabla$ have identical eigenfunctions! In dimension $1$ this can be used to prove that the time-changed isotropic $\alpha$-stable Lévy process generated by $A$ can be constructed by subordinating the Jacobi diffusion generated by $B$ using an appropriate subordinator. This result, however, does not extend to higher dimensions!
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Extent |
39 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Wrocław University of Technology
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Series | |
Date Available |
2017-06-18
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0348340
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Postdoctoral
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International