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Extra info for Communications in Mathematical Physics - Volume 210
Math. 171 Issue 2, 1–121 (1987) Reed, M. : Methods of modern mathematical physics. IV. Analysis of operators. : Acad. : Main notions of measure theory. Math. Sb. : Semibounded local Hamiltonians for perturbation of the Laplacian supported by curves with angle points in R4 (Russian). Teoret. i Mat. Fiz. 106, 179–199 (1996)                 Communicated by B. Simon Commun. Math. Phys. 210, 49 – 83 (2000) Communications in Mathematical Physics © Springer-Verlag 2000 Spectral Transformation Chains and Some New Biorthogonal Rational Functions Vyacheslav Spiridonov1 , Alexei Zhedanov2 1 Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Moscow region 141980, Russia.
3. Companion Polynomials . . . . . . . . . . . . . . 4. Symmetries of the RI I -Chain and a Similarity Reduction . . . . . 5. Rational and Elementary Functions Solutions . . . . . . . . 6. Elliptic Solutions of the Basic Equation . . . . . . . . . . 7. Elliptic Analogues of Hypergeometric Functions . . . . . . . . 8. Finite-Dimensional Biorthogonality . . . . . . . . . . . 9. Duality Property and the Normalization Constants . .
Therefore, (n) ⊕ (n) Hγ where Hγ is a multiplication by some real-analytic function defined Hγ = n∈N (n) on an open set Gγ ⊂ B. Thus, it remains to refer to the following lemma. Lemma 17. Let G be an open subset of Rd and H be an operator in L2 (G) which acts by multiplication by some nonconstant real-analytic function on G. Then the spectrum of H is purely absolutely continuous. 86 from . 4. Final Remarks The key role in the proof of Proposition 4 is played by condition (A), which implies the analyticity of the dispersion law for HT .