Linear Operators: Spectral theory |
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Page 1305
... boundary condition . A set of boundary conditions is said to be separated if it ( or , more generally , some set of boundary conditions equivalent to it ) contains no mixed boundary conditions . In all other cases the set is said to be ...
... boundary condition . A set of boundary conditions is said to be separated if it ( or , more generally , some set of boundary conditions equivalent to it ) contains no mixed boundary conditions . In all other cases the set is said to be ...
Page 1310
... conditions . Let 20. Then the boundary conditions are real , and there is exactly one solution p ( t , λ ) of ( t − 2 ) p = 0 square - integrable at a and satisfying the boundary conditions at a , and exactly one solution y ( t , λ ) ...
... conditions . Let 20. Then the boundary conditions are real , and there is exactly one solution p ( t , λ ) of ( t − 2 ) p = 0 square - integrable at a and satisfying the boundary conditions at a , and exactly one solution y ( t , λ ) ...
Page 1321
... boundary conditions defining T. PROOF . We have seen in the derivation of Theorem 8 that the functions a , ( t ) and ẞ , ( t ) are uniquely determined by the jump equa- tions and by the boundary conditions B ( K ) = 0 , i = 1 ...
... boundary conditions defining T. PROOF . We have seen in the derivation of Theorem 8 that the functions a , ( t ) and ẞ , ( t ) are uniquely determined by the jump equa- tions and by the boundary conditions B ( K ) = 0 , i = 1 ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero