On the Principle of Limiting Absorptions
On the Principle of Limiting Absorptions
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We shall °(2) Introduce the class of functions D ', where - /c < z < z, as ^1^2 ^ ^ ' the completion in W^ ^ (O ) of the set of functions defined in 2 z^z^ O and vanishing in boundary strip along P . Note that ^1^2 Q ^12 for all functions *, such that * e D^^; O W^^MO^ ^ ), ^1^2 ^ ^l"'2 45 A4) e L (Q ), there holds the following inequality, analogous to inequality (9) k=l dx c^^ f [h|2 + |A*|^]dx, (57) ■1^2 '1^2 where z^ < z^ < z^ < Zg, c^^^ = c^^(z^, Z2, z-^, zp Let now ^ e L^(0 ^ ), |^ e L„(Q^ ^ ). Then, as it is 2 z^z^ 6x^ 2 z^z^ easily seen. /l*l^dxi^-|^ / |*|^d. + 2iz, -. J / dx, (58) '1^2 where i = 1, 2, - K < z-^ < z^, dx = dx-j^dx^, . . . , dx^_^ "(2) We shall denote by D^ ' the totality of all functions °(2) which are defined in O'' and for which 4) e D' ' for any z-, z„, Z -| Zp X c.
where - i< < z-, ■< Zp.
Let M, and *, (x) be the eigenvalues and eigenfunctions of the equation a Sx' ix* = (59) 44 In the region t] with the initial condition 0| = 0, regarded o ^ in the sense * e D(ri).
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To quickly assess the difficulty of the text, read a short excerpt:
We shall °(2) Introduce the class of functions D ', where - /c < z < z, as ^1^2 ^ ^ ' the completion in W^ ^ (O ) of the set of functions defined in 2 z^z^ O and vanishing in boundary strip along P . Note that ^1^2 Q ^12 for all functions *, such that * e D^^; O W^^MO^ ^ ), ^1^2 ^ ^l"'2 45 A4) e L (Q ), there holds the following inequality, analogous to inequality (9) k=l dx c^^ f [h|2 + |A*|^]dx, (57) ■1^2 '1^2 where z^ < z^ < z^ < Zg, c^^^ = c^^(z^, Z2, z-^, zp Let now ^ e L^(0 ^ ), |^ e L„(Q^ ^ ). Then, as it is 2 z^z^ 6x^ 2 z^z^ easily seen. /l*l^dxi^-|^ / |*|^d. + 2iz, -. J / dx, (58) '1^2 where i = 1, 2, - K < z-^ < z^, dx = dx-j^dx^, . . . , dx^_^ "(2) We shall denote by D^ ' the totality of all functions
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