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Eigenvalues of the fractional Laplace operator in the interval
 J. Funct. Anal
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Spectral analysis of subordinate Brownian motions in halfline. Preliminary version
 Preprint, 2010, arXiv:1006.0524v1. SUBORDINATE BROWNIAN MOTIONS IN HALFLINE 57
"... Abstract. We study onedimensional Lévy processes with LévyKhintchine exponent ψ(ξ2), where ψ is a complete Bernstein function. These processes are subordinate Brownian motions corresponding to subordinators, whose Lévy measure has completely monotone density; or, equivalently, symmetric Lévy proce ..."
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Cited by 14 (2 self)
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Abstract. We study onedimensional Lévy processes with LévyKhintchine exponent ψ(ξ2), where ψ is a complete Bernstein function. These processes are subordinate Brownian motions corresponding to subordinators, whose Lévy measure has completely monotone density; or, equivalently, symmetric Lévy processes whose Lévy measure has completely monotone density on (0,∞). Examples include symmetric stable processes and relativistic processes. The main result is a formula for the generalized eigenfunctions of transition operators of the process killed after exiting the halfline. A generalized eigenfunction expansion of the transition operators is derived. As an application, a formula for the distribution of the first passage time (or the supremum functional) is obtained.
Eigenvalue gaps for the Cauchy process and a Poincare inequality
 J. Funct. Anal
, 2006
"... A connection between the semigroup of the Cauchy process killed upon exiting a domain D and a mixed boundary value problem for the Laplacian in one dimension higher known as the mixed Steklov problem, was established in [6]. From this, a variational characterization for the eigenvalues λn, n ≥ 1, of ..."
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Cited by 11 (4 self)
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A connection between the semigroup of the Cauchy process killed upon exiting a domain D and a mixed boundary value problem for the Laplacian in one dimension higher known as the mixed Steklov problem, was established in [6]. From this, a variational characterization for the eigenvalues λn, n ≥ 1, of the Cauchy process in D was obtained. In this paper we obtain a variational characterization of the difference between λn and λ1. We study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for λ ∗ −λ1 where λ ∗ is the eigenvalue corresponding to the “first ” antisymmetric eigenfunction for D. The proof is based on a variational characterization of λ ∗ − λ1 and on a weighted Poincaré–type inequality. The Poincaré inequality is valid for all α symmetric stable processes, 0 < α ≤ 2, and any other process obtained from Brownian motion by subordination. We also prove upper bound estimates for the spectral gap λ2 − λ1 in bounded convex domains.
Spectral gap for the Cauchy process on convex, symmetric domains
 Comm. Partial Differential Equations
"... Let D ⊂ R2 be a bounded convex domain which is symmetric relative to both coordinate axes. Assume that [−a, a] × [−b, b], a ≥ b> 0 is the smallest rectangle (with sides parallel to the coordinate axes) containing D. Let {λn}∞n=1 be the eigenvalues corresponding to the semigroup of the Cauchy pro ..."
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Cited by 9 (5 self)
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Let D ⊂ R2 be a bounded convex domain which is symmetric relative to both coordinate axes. Assume that [−a, a] × [−b, b], a ≥ b> 0 is the smallest rectangle (with sides parallel to the coordinate axes) containing D. Let {λn}∞n=1 be the eigenvalues corresponding to the semigroup of the Cauchy process killed upon exiting D. We obtain the following estimate on the spectral gap: λ2 − λ1 ≥ Cb a2 where C is an absolute constant. The estimate is obtained by proving new weighted Poincare ́ inequalities and appealing to the connection between the eigenvalue problem for the Cauchy process and a mixed boundary value problem for the Laplacian in one dimension higher
SPECTRAL PROPERTIES OF THE CAUCHY PROCESS ON HALFLINE AND INTERVAL
, 906
"... Abstract. We study the spectral properties of the transition semigroup of the killed onedimensional Cauchy process on the halfline (0, ∞) and the interval (−1, 1). This process is related to the square root of onedimensional Laplacian A = − − d2 dx2 with a Dirichlet exterior condition (on a comp ..."
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Cited by 9 (2 self)
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Abstract. We study the spectral properties of the transition semigroup of the killed onedimensional Cauchy process on the halfline (0, ∞) and the interval (−1, 1). This process is related to the square root of onedimensional Laplacian A = − − d2 dx2 with a Dirichlet exterior condition (on a complement of a domain), and to a mixed Steklov problem in the halfplane. For the halfline, an explicit formula for generalized eigenfunctions ψλ of A is derived, and then used to construct spectral representation of A. Explicit formulas for the transition density of the killed Cauchy process in the halfline (or the heat kernel of A in (0, ∞)), and for the distribution of the first exit time from the halfline follow. The formula for ψλ is also used to construct approximations to eigenfunctions of A in the interval. For the eigenvalues λn of A in the interval the asymptotic formula λn = nπ π 1 − + O ( ) is 2 8 n derived, and all eigenvalues λn are proved to be simple. Finally, efficient numerical methods of estimation of eigenvalues λn are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to 9th decimal point. 1.
Eigenvalues of the fractional Laplace operator in the unit ball
, 2015
"... We describe a highly efficient numerical scheme for finding twosided bounds for the eigenvalues of the fractional Laplace operator (−∆)α/2 in the unit ball D ⊂ Rd, with a Dirichlet condition in the complement of D. The standard Rayleigh–Ritz variational method is used for the upper bounds, while th ..."
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We describe a highly efficient numerical scheme for finding twosided bounds for the eigenvalues of the fractional Laplace operator (−∆)α/2 in the unit ball D ⊂ Rd, with a Dirichlet condition in the complement of D. The standard Rayleigh–Ritz variational method is used for the upper bounds, while the lower bounds involve the lessknown Aronszajn method of intermediate problems. Both require explicit expressions for the fractional Laplace operator applied to a linearly dense set of functions in L2(D). We use appropriate Jacobitype orthogonal polynomials, which were studied in a companion paper [15]. Our numerical scheme can be applied analytically when polynomials of degree two are involved. This is used to partially resolve the conjecture of T. Kulczycki, which claims that the second smallest eigenvalue corresponds to an antisymmetric function: we prove that this is the case when either d ≤ 2 and α ∈ (0, 2], or d ≤ 9 and α = 1, and we provide strong numerical evidence in the general case.
SPECTRAL PROPERTIES OF THE CAUCHY PROCESS
, 2009
"... This note is an announcement of the results. The full version of this paper, including full proofs, will be available soon. ..."
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Cited by 1 (0 self)
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This note is an announcement of the results. The full version of this paper, including full proofs, will be available soon.
Spectral gap for stable process on convex double symmetric domains
, 2006
"... We study the semigroup of the symmetric αstable process in bounded domains in Rd. We obtain a variational formula for the spectral gap, i.e. the difference between two first eigenvalues of the generator of this semigroup. This variational formula allows us to obtain lower bound estimates of the spe ..."
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We study the semigroup of the symmetric αstable process in bounded domains in Rd. We obtain a variational formula for the spectral gap, i.e. the difference between two first eigenvalues of the generator of this semigroup. This variational formula allows us to obtain lower bound estimates of the spectral gap for convex planar domains which are symmetric with respect to both coordinate axes. For rectangles, using ”midconcavity ” of the first eigenfunction [5], we obtain sharp upper and lower bound estimates of the spectral gap. 1