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-the operators h and H; construction of the resolvent of the operator h for lm z ≠ 0; construction of the resolvent of the operator H for complex z; investigation of the kernel t(k,k’,z); general treatement of the kernels W αβ (k,p;k’,p’;z); estimates of the kernels Q n γ 1… γ n+1 (k,p; k’p’; z); the operator A(z); expansion theorems for the eigenfunctions of h and H; foundation of the time-dependent formulation of the scattering problem for a system described by the operators h and H; properties of h ölder functions; estimates for some integrals; proof of lemma 6.2-

The book consists of eleven sections and four appendices. In § 1 the operators in question are rigorously defined in Hilbert space, i. e. , their domain of definition is specified and their self-adjointness proved. Here are also formulated the basic conditions imposed on the potentials which in the rest of the book are tacitly assumed to be fulfilled. §§ 2 and 4, and §§ 3, 5, 6, 7, respectively, deal with the resolvents of energy operators for two and three bodies; the resolvents are expressed in term s of integral operators, and integral equations are set up and investigated in order to derive estimates for the kernels and examine their singularities in the variable z near the real axis. The obtained results are applied in §§ 8 and 9 to the proof of eigenfunction expansion theorems, and in §§10 and 11 to the time- dependent formulation of the scattering problem and the construction of the scattering operator. Appendix I gives the derivation of some properties of functions which satisfy the Holder condition, and of singular integrals containing these functions. Appendices II and III give proofs of estimates of some integrals applied in the text. Appendix IV contains remarks and references to the literature which are not mentioned in the main text. This book is a translation of MATEMATICHESKIE VOPROSY KVANTOVOI TEORII RASSEYANIYA DLYA SISTEMY TREKH CHASTITS In: Trudy Matematicheskogo Instituta imeni V. A. Steklova. LXIX Izdatel’stvo Akademii Nauk SSSR Moskva-Leningrad 1963