Eigenvalues of the fractional Laplace operator in the interval

Abstract

Two-term Weyl-type asymptotic law for the eigenvalues of one-dimensional fractional Laplace operator (−∆) α/2 (α ∈ (0, 2)) in the interval (−1, 1) is given: the n-th eigenvalue is equal to (nπ/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.

Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well

“…Currently, the status of undoubtful relevance have approximate statements (various estimates) pertaining to the asymptotic behavior of eigenfunctions at the well boundaries and estimates, of varied degree of accuracy, of the eigenvalues, c.f. [11] and [12]- [16].…”

“…That will set a connection with the infinite well problem, considered so far in the fractional QM literature with rather limited success, c.f. [11]- [16]). A consistent spectral solution of the Cauchy well problem is our major task.…”

“…Since approximate spectral formulas and eigenfunction estimates are available for the the "fractional Laplace operator in the interval" (mathematicians' transcript of the infinite well in fractional QM), [16], the latter publication will be our reference point as far as an approximation issue of an infinite well in terms of a sequence of deepening finite wells, will become our concern.…”

“…It is worthwhile to mention that the only eigenvalue formulas for an infinite Cauchy well, that can be termed rigorous, refer to the large n limit. More explicitly [16] we actually have an approximate expression for E n , provided n is sufficiently large:…”

“…As the initial trial function in L 2 (R) we take Φ [16]; Our algorithm appears to be more reliable, since 6 and 7 refer to wells whose depths are respectively 500 and 5000. Left panel shows an enlarged vicinity of the maxima.…”