BIRS Workshop Lecture Videos

Banff International Research Station Logo

BIRS Workshop Lecture Videos

Fractional Laplace operator in the unit ball Kwaśnicki, Mateusz


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 Media

Item Citations and Data


Attribution-NonCommercial-NoDerivatives 4.0 International