Eigenvalues of the fractional Laplace operator in the interval

By Mateusz Kwaśnicki. 2010

157 citations recorded by the source corpus.

Abstract Two-term Weyl-type asymptotic law for the eigenvalues of the one-dimensional fractional Laplace operator ( − Δ ) α / 2 ( α ∈ ( 0 , 2 ) ) in the interval ( − 1 , 1 ) is given: the n -th eigenvalue is equal to ( n π / 2 − ( 2 − α ) π / 8 ) α + O ( 1 / n ) . Simplicity of eigenvalues is proved for α ∈ [ 1 , 2 ) . L 2 and L ∞ properties of eigenfunctions are studied. We also give precise numerical bounds for the first few eigenvalues.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7d75e5a3561c2e8d400a2eb1b5bce479ebddc703

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In Computational Theory and Mathematics

1. Suleyman Ulusoy
2. E. Lenzmann
3. Rupert L. Frank
4. Bartłlomiej Dyda
5. Luis Silvestre
6. Ariel Martin Salort
7. Enrico Valdinoci
8. C. Bucur

In Mathematical Physics

1. Rupert L. Frank
2. Suleyman Ulusoy
3. E. Lenzmann
4. Bartłlomiej Dyda
5. Luis Silvestre
6. Ariel Martin Salort
7. Enrico Valdinoci
8. C. Bucur

Citations in the corpus

• Fractional calculus for power functions and eigenvalues of the fractional Laplacian (2012, methodology): “A number of methods to study this one-dimensional case, and more general cases, were developed by several authors [1], [2], [3], [4], [5], [7], [13], [14], [24], [25], [26], [27], [28], [29], [32], [36].”
• Fractional calculus for power functions and eigenvalues of the fractional Laplacian (2012, methodology): “A number of methods to study this one-dimensional case, and more general cases, were developed by several authors [1], [2], [3], [4], [5], [7], [13], [14], [24], [25], [26], [27], [28], [29], [32], [36].”
• Eigenvalues and minimizers for a non-standard growth non-local operator (2018, background): “In [22] Servadai and Valdonoci as well as Kwaśnicki in [16] study the spectrum of different non-local linear operators.”
• A UNIQUENESS RESULT FOR AN INVERSE PROBLEM IN A SPACE-TIME FRACTIONAL DIFFUSION EQUATION (2013, background): “4) We recall the following result from [8] about the asymptotic behavior of the eigenvalues of the fractional Laplacian in an interval, which is used for further estimates and proofs.”
• Uniqueness of Radial Solutions for the Fractional Laplacian (2013, methodology): “1; 1/ (with exterior Dirichlet conditions on the complement I c) obtained by Bañuelos and Kulczycki [6] ; see also [25, 26] for improvements to .”
• A tutorial on inverse problems for anomalous diffusion processes (2015, unknown): “Last, we note that for other choices of the fractional derivative, e.g., the Riemann-Liouville fractional derivative and the fractional Laplacian [6, 55], the magnitude of eigenvalues of the operator also tends to infinity, and the growth rate increases with the fractional order β .”