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Additional info for Communications in Mathematical Physics - Volume 302
7. Weak Limit of Zeros of Heine-Stieltjes Polynomials . . . . . 8. Heun’s Differential Equation ( p = 2) . . . . . . . . . 9. General Families of A-Critical Measures . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 56 59 64 67 81 87 90 101 108 54 A. Martínez-Finkelshtein, E. A. Rakhmanov 1. Generalized Lamé Equation Let us start with a classical problem more than 125 years old.
55). Deforming the contour using analyticity of Gλ across oscillatory eigenvalues λ± (ε) we obtain G = G˜ + O, where ε ε O(x, t; y) := eλ+ (ε)t ϕ+ε (x)ϕ˜+ε (y)tr + eλ− (ε)t ϕ− (x)ϕ˜− (y)tr 42 B. Texier, K. Zumbrun ε is the sum of the residues of the integrand at λ± (the right- and left-eigenfunctions ϕ± ε are defined in Sect. 2). The Green function G˜ is the kernel of the integral operand ϕ˜± ator S(ε, t) defined in Sect. 3. 23 applies to G. 21) connecting −ν − iν0 and −ν + iν0 . 8) −ν−iν0 −ν+i∞ 1 GII (ε, x, t; y) := eλt Gλ (ε, x, y) dλ.
18, (Id − S(ε, T ))−1 belongs to L(X 2 , X 1 ), for all (ε, T ) ∈ . Proof. 23)(iii). 5) are all bounded in L(X 1 ). 12). 27 to handle the contribution of the reactive term. 66), there are two terms in the upper bound for (S + R) +4 . 23) is small enough. 16) the growth assumption on ω. 9. 8). 66)(ii), and θ0 to be small ε ∈ H 2. 2. Reduction. 10. 17) is defined in Sect. 3, is equivalent to b = (Id − S(ε, T ))−1 N˜ (a, b, ε, T ) + ω for ω ∈ Ker(Id − S(ε, T )) ∩ X 1 . Proof. A simple consequence of the definition in the above section of the right inverse (Id − S(ε, T ))−1 .