Anderson localization
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1: Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate
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►Anderson, P.
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2: 19.35 Other Applications
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►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute to high precision (Borwein and Borwein (1987, p. 26)).
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3: Bibliography Q
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Sharp estimates for complete elliptic integrals.
SIAM J. Math. Anal. 27 (3), pp. 823–834.
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4: Bibliography
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Generalized elliptic integrals and modular equations.
Pacific J. Math. 192 (1), pp. 1–37.
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Functional inequalities for complete elliptic integrals and their ratios.
SIAM J. Math. Anal. 21 (2), pp. 536–549.
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Functional inequalities for hypergeometric functions and complete elliptic integrals.
SIAM J. Math. Anal. 23 (2), pp. 512–524.
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Hypergeometric Functions and Elliptic Integrals.
In Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa (Eds.),
pp. 48–85.
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Inequalities for elliptic integrals.
Publ. Inst. Math. (Beograd) (N.S.) 37(51), pp. 61–63.
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5: 31.3 Basic Solutions
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denotes the solution of (31.2.1) that corresponds to the exponent at and assumes the value there.
If the other exponent is not a positive integer, that is, if , then from §2.7(i) it follows that exists, is analytic in the disk , and has the Maclaurin expansion
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►Solutions (31.3.1) and (31.3.5)–(31.3.11) comprise a set of 8 local solutions of (31.2.1): 2 per singular point.
…For example, is equal to
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►The full set of 192 local solutions of (31.2.1), equivalent in 8 sets of 24, resembles Kummer’s set of 24 local solutions of the hypergeometric equation, which are equivalent in 4 sets of 6 solutions (§15.10(ii)); see Maier (2007).
6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►Should be bounded but random, leading to Anderson localization, the spectrum could range from being a dense point spectrum to being singular continuous, see Simon (1995), Avron and Simon (1982); a good general reference being Cycon et al. (2008, Ch. 9 and 10).
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►Thus, and this is a case where is not continuous, if , , there will be an eigenfunction localized in the vicinity of , with a negative eigenvalue, thus disjoint from the continuous spectrum on .
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7: 15.17 Mathematical Applications
8: 4.42 Solution of Triangles
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9: 31.7 Relations to Other Functions
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31.7.1
►Other reductions of to a , with at least one free parameter, exist iff the pair takes one of a finite number of values, where .
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31.7.2
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31.7.3
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31.7.4
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