Noneuclidean Harmonic Analysis

Terras, Audrey

doi:10.1137/1024040

Cited by 14 publications

(4 citation statements)

References 23 publications

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“…Note, however, that the hyperbolic metric in (3.22) is written in half-space coordinates as dr 2 +|dy| 2 r 2 = dt 2 + e 2t |dy| 2 , so in order to account for a weight of the form r δ one would need to use this transform written in rectangular coordinates. This is well known and comes Kontorovich-Lebedev formulas ( [84]). Nevertheless, for our purposes it is more suitable to use this transform in geodesic polar coordinates as it is described in Section 10.2.…”

Section: Fredholm Properties -Surjectivitymentioning

confidence: 87%

On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program

Ao¹,

Chan²,

DelaTorre³

et al. 2019

Duke Math. J.

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We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, for which we establish the classical gluing method of Mazzeo and Pacard ([63]) for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of a fractional order ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. Note, however, that no traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we use complex analysis and some non-Euclidean harmonic analysis to examine a fractional Schrödinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.

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Section: Fredholm Properties -Surjectivitymentioning

confidence: 87%

On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program

Ao¹,

Chan²,

DelaTorre³

et al. 2019

Duke Math. J.

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“…Now look for a transformation z = z(y) so that A simple calculation similar as in section I1 yields z = In y, which is essentially the hyperbolic distance on the y-axis from 5' = i. z(y) has the property ( 0 ,~) H R. We have dy/y = dz, and the kinetic term in the exponential in the path integral (7) gives in a Taylor expansion :…”

Section: The Poincar6 Upper Half-planementioning

confidence: 99%

“…But be careful: A2 has negative Gaussian curvature K = -1, as well as U , D and S , i.e., they are everywhere saddle-shaped. A more convenient description for A2 reads in pseudospherical polar coordinates (t, #) [4,7, 8, 111 :…”

mentioning

confidence: 99%

The Path Integral on the Poincaré Disc, the Poincaré Upper Half-Plane and on the Hyperbolic Strip

Grosche

1990

Fortschr. Phys.

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In this paper rigorous path integral treatments are presented of free motion on the Poincaré disc, the Poincaré upper half‐plane and on the hyperbolic strip, three spaces which are analytically equivalent to each other. Whereas the path integral treatments on the disc and on the strip are new, two further path integral treatments are discussed for the Poincaré upper half‐plane to the existing one. All the calculations are mainly based on Fourier‐expansions of the Feynman kernels which can be easily performed. The remaining path integrals on D, U and S can be reduced to the path integral problems on the pseudosphere Δ2, Liouville quantum mechanics and the modified Pöschl‐Teller potential problem, respectively, where the results by means of path integrals are known. The corresponding normalized wave‐functions and the energy‐spectrum are derived. The energy‐spectra are the same in all three spaces and read E = (1/2m) (p2 + 1/4) (p ‐momentum). The “zero‐point” energy E0 = 1/8m is discussed which can be interpreted in terms of the Heisenberg uncertainty relation. The equivalences between the Feynman kernels on Δ2, D, U and S are also discussed.

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“…SeeFaraut [40] and Terras[134] for the occurrence of the Kontorovich-Lebedev transform on rank one spaces.Berezin & Karpelevi~ [13] state without proof that the spherical functions for (G,K) = (SU(n,n+k),S(U(n)xU(n+k» res tric ted to A "'" ~,J-, ... ,n I"'" n n l <, '< (ch2t.-ch2t. SeeFaraut [40] and Terras[134] for the occurrence of the Kontorovich-Lebedev transform on rank one spaces.Berezin & Karpelevi~ [13] state without proof that the spherical functions for (G,K) = (SU(n,n+k),S(U(n)xU(n+k» res tric ted to A "'" ~,J-, ... ,n I"'" n n l <, '< (ch2t.-ch2t.…”

mentioning

confidence: 99%