Expansion in Eigenfunctions of the Automorphic Laplacian on the Lobachevsky Plane

1990 ◽  
pp. 13-27
Author(s):  
Alexei B. Venkov
Keyword(s):  
Lobachevsky Plane ◽  
Expansion In Eigenfunctions

On a singular eigenvalue problem

1968 ◽  
Vol 64 (2) ◽  
pp. 439-446 ◽  
Author(s):  
D. Naylor ◽  
S. C. R. Dennis
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Bessel Functions ◽  
Asymptotic Condition ◽  
Real Constant ◽  
Limit Circle ◽  
Expansion In Eigenfunctions ◽  
Image Position

Sears and Titchmarsh (1) have formulated an expansion in eigenfunctions which requires a knowledge of the s-zeros of the equationHere ka > 0 is supposed given and β is a real constant such that 0 ≤ β < π. The above equation is encountered when one seeks the eigenfunctions of the differential equationon the interval 0 < α ≤ r < ∞ subject to the condition of vanishing at r = α. Solutions of (2) are the Bessel functions J±is(kr) and every solution w of (2) is such that r−½w(r) belongs to L2 (α, ∞). Since the problem is of the limit circle type at infinity it is necessary to prescribe a suitable asymptotic condition there to make the eigenfunctions determinate. In the present instance this condition is

Decomposition of boundary representations on the Lobachevsky plane associated with linear bundles

2019 ◽  
pp. 368-375
Author(s):  
Larisa I. Grosheva
Keyword(s):  
Exceptional Case ◽  
Lobachevsky Plane ◽  
Canonical Representations ◽  
Boundary Representations

Earlier we described canonical (labelled by λ ∈C) and accompanying boundary representations of the group G = SU(1,1) on the Lobachevsky plane D in sections of linear bundles and decomposed canonical representations into irreducible ones. Now we decompose representations acting on distributions concentrated at the boundary of D. In the generic case 2λ ∉N they are diagonalizable, in the exceptional case Jordan blocks appear.

scholarly journals Ising models on the Lobachevsky plane

1990 ◽  
Vol 128 (1) ◽  
pp. 63-76 ◽  
Author(s):  
C. M. Series ◽  
Ya. G. Sinai
Keyword(s):  
Ising Models ◽  
Lobachevsky Plane
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