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Additional resources for Spectral Properties of Hamiltonian Operators
Example text
0') P. , CtEA j,c~ j where I . and J need the I are H. A, Neumark E A we ~ ~ define ~ then similarly By H [21], closed P P~=O and J,~ definition of H J,~ P. 6 all ~ striction U(y) A ]H (or IH. 1 in We also this for basis. IN) is all its H. ~ ~ and of J H~ H. ,m. y E ~. for is called_D N in such all N for that represented dimension It in every is the re- similarly -generating, ~ .......... all y 6 F and the D that and y E F. for by follows H. every matrix y to if DR(y) the Uj(y) H. (i and H Every D -generating respectively.
By A. 2. ,~ N be positive numbers. jn n - ~ for u E C ~ (R3n), where 5. E 1 =T Ji'''Jn construction ms given u E C~ o (R3N). Theorem. j n and 5-3 u l of theorem J l .. " J n TJl'''jnU for Ji 1 R 3. A p p l y t h e A. , jnUPf 2 j=l we T. jn T-bound not instead larger than n have F denote = - ~n ( u _1 A i=l 2~ji) Ji J 1=1 ~ [pTufl for all transformation - lxjl2(Fu)(x) 2~iji 2! ~u)(x)lt2dx ~. Ji I u ~ C~o (R3N) in L2(R3N). d. 4 Theorem. T Jl"''Jn Suppose that A is an operator in L 2 ( R J n ) w h i c h -bounded with T J1" is ' "'Jn-bound not exceeding b.
M_l T' as in part part of theorem such that the assertion internal the two internal n-1 it T' n u E C~ o follows are essentially have for now are subsystems _ ~ A. °n_lU[} The operators " selfadjoint; u;I (R3N). 5). jn. n-1 t by - ~ ~. 1 i=I that I l T ~ ! . n - 1 u]] --< I I T ' l . . k . kmU! I for all u ¢ C 000 ( ~ N ) equivalence. every W J n ul] ~ subsystem in terms j l , . . %) W. n_ I = U W j i'" "jnU* a c t s and is is generated h-small This Let at is replaced i = 1 , . . ~n-1; with domain C°°(~ 3N) such that o only o n the internal coordinates Yl ~ ' ' ' ~ Y n - 1 operator W I"" "n-1 in L i ( ~ 3 n - 3 ) by a symmetric b x : ( z l , .
