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Communications in Mathematical Physics - Volume 223 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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6, p. 40]. Let (Pn ) be a sequence of monic polynomials, orthogonal with respect to a positive measure σ supported by the interval [0, ∞[, and let (Kn ) be the monic polynomials Generalized q-Hermite Polynomials 41 orthogonal with respect to the measure x dσ (x). They are called the kernel polynomials for the parameter value 0. 6) are orthogonal with respect to the symmetric measure µ on the real line determined by the equations g(x 2 ) dµ(x) = g(x) dσ (x), where g is an arbitrary continuous function on [0, ∞[ of at most polynomial growth.

Math. 137, 82–203 (1998) 21. : Orthogonal Polynomials. Fourth Edition, Providence, RI: American Mathematical Society, 1975 Literature for Further Reading 1. : A realization of the q-Harmonic Osciallator. Theoretical and Mathematical Physics, Vol. 87, No. 1, 1991, pp. 442–444 2. : Difference Analogs of the Harmonic Oscillator. Theoretical and Mathematical Physics, Vol. 85, No. 1, 1991, pp. 1055–1062 3. : A Relativistic Model of the isotropic Oscillator. Annalen der Physik, 7. Folge, 42, 1, 25–30 (1985) 4.

Pick one such x1 . For any z2 ∈ D(0, 2) let gt (z2 ) = 1 0 (0) (0) u(x1 + te2πiθ , z2 ) dθ − u(x1 , z2 ). 15) By Jensen’s formula, see Theorem 2 in Sect. 16) (0) where n(s, z2 ) = µ(D(x1 , s), z2 ) with the Riesz measure µ(·, z2 ) of u(·, z2 ). Clearly, (0) µv (D(x1 , s)) = 1 −1 n(s, x2 ) dx2 . 16), and our choice of x1 , 1 −1 gt (x2 ) dx2 = t 0 (0) µv (D(x1 , s)) t ds ≤ C . s ρ Now fix some r ∈ (0, 1/2) and define 2−j g2j r . 17) Integrated Density of States for Schrödinger Operators 63 The subharmonicity of z1 → u(z1 , z2 ) implies that gt ≥ 0 so that G is the sum of nonnegative terms.

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