On the Eigenfunctions of Many Particle Systems in Quantum Mechanics
On the Eigenfunctions of Many Particle Systems in Quantum Mechanics
Tosio Kato
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The operators G V. Are somewhat more complicated. According to the definition of the product of two operators in a Hilbert space, G7. Is defined - 22 - only for functions u such that V. U and G(7. U) are both defined. But the V. Are in general unbounded and not defined everywhere, and hence the GV. Are also not defined everjrwhere. It can be shown, however, that the G7, are never- theless bounded. In what follows let V stand for any one of the V. For simplicity. Then the operator OV is obviousl...y a restriction of the integral operator A defined by (U. L) (Au) (x) - I g(x-y) V(y) u(y) dy, and, as we shall show, A is a bounded linear operator defined everywhere in ^ ♦ More generally, we can regard A as a linear operator which transforms an element from a Banach space L^ ■» L^(E ) into another Banach space L^, Here m L^ is, as usual, the space of all (equivalent classes of) complex-valued functions u(x) with the finite p-nonn 1/p (U. 2) Hull r!u(x)|Pdx We shall consider the values 2 § p $ oo and define as usual Iju ||^" sup|u(x)|, Ap a linear operator on L^ to L*', Au is defined for u e.
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