#### Date of Submission

4-28-1984

#### Date of Award

4-28-1985

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Delhi)

#### Supervisor

Sinha, Kalyan Bidhan (TSMU-Delhi; ISI)

#### Abstract (Summary of the Work)

In the present work we shall deal with the asymptotic completeness problem in three and four (Quantum Mechanical) particle scattering. This thesis is divided into three chapters. In the first chapter we give an introduction to Scattering Phencmenon and give a description of the N-particle completeness problem.Then we collect some results preliminary to the later chapters and some results that would complete a discussion of the problem.The second chapter consists of some technical results and a reduction of the asymptotic completeness problem in N-particlerscattering via time dependent methods. The last chapter has two sections. The first section deals with verifying the conditions laid dowm in chapter II, for completeness to follow, in the case of three particles. This we do under explicit conditions on the potentials. The potentials decay at the rate of x|2-e in the pair directions and have some local singularity.The second section of the last chapter has completeness for the four particle case. Here apart from some smoothness conditions on the pair potentials we impose implicit restrictions on the pair Hamiltonians.We deal with only separable Hilbert spaces with inner products linear in the second variable and assume all the operator theory basic to our discussion. This background is well covered in (AJS, K1, RSI-II,W].For a self-adjoint operator L on a Hilbert space H we use the following o (L), , (L)P, . C(L), ac (T.) and o SC(L) denote the spectrum of L, the point, continuocus, absolutely continuous and singularly continuous spectra of L while Hp, (L), Hc. and Hsc (L) the corresponding SC spectral subspaces of L in H. By o (L) we denote the essential spectrum ess which is the union of continuous spectrum, eigenvalues of infinite multiplicity and the accumulation points of eigenvalues of L. By EL we denote the projection onto Hp (L). The range of a closed operator A will be denoted by R(A). An equation like a - b* + c* wil1 mean two separate equations one for each sign. Similarly a statement s means two independent statements s* and s. An operator on H and on HeK are denoted by the same letter C. For any tWo operators A,B, by aa(B) we mean the n-fold commutator (A,... (A,B]...1.In the discussion of many particles Greek letters index rairs of particles while Roman letters index the particles themselves. The summaticn I in chapters I and II is over the set of y's given by {y = (ij):1sisjs N}. For any real valued function V: IR + IR we define, whenever it exists, E lim v(t) = t+t00 lim S ds (s). t+t0 We indicate in f.1, any reference to other works. All the absolute constants are denoted by the letter K.$ 1. Scattering cross-sections Scattering is an essential part of Physics by means of which a Physicist tests his theories, discovers interactions among fundamental particles and studies the structure of matter in general. The scattering experiment consists, usually, of an incident beam of particles, the target (or the scattering centre) which scatters these particles and a detector to detect (count) the particles going out after scattering. The observable quantity in a scattering experiment is the cross-section which has an expression

#### Control Number

ISILib-TH168

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Krishna, M. Dr., "Some Spectral Properties of Three and Four Body Scrodinger Operators by the Methods of Time Dependent Scattering Theory." (1985). *Doctoral Theses*. 209.

https://digitalcommons.isical.ac.in/doctoral-theses/209

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842986