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
What to read after On the Principle of Limiting Absorptions?
You can find similar books in the "Read Also" column, or choose other free books by D M Eidus to read online
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