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1: 18.1 Notation
or the dilated Chebyshev polynomials of the first and second kinds:
C n ( x ) = 2 T n ( 1 2 x ) ,
S n ( x ) = U n ( 1 2 x ) .
2: 1.18 Linear 2nd Order Differential Operators and Eigenfunction Expansions
Note that the integral in (1.18.67) is not singular if approached separately from above, or below, the real axis: in fact analytic continuation from the upper half of the complex plane, across the cut, and onto higher Riemann Sheets can access complex poles with singularities at discrete energies λ r e s - i Γ r e s / 2 corresponding to quantum resonances, or decaying quantum states with lifetimes 1 / Γ r e s . For this latter see Simon (1973), and Reinhardt (1982); wherein advantage is taken of the fact that although branch points are actual singularities of an analytic function, the location of the branch cuts are often at our disposal, as they are not singularities of the function, but simply arbitrary lines to keep a function single valued, and thus only singularities of a specific representation of that analytic function. This is accomplished by the variable change x x e i θ , in , which rotates the continuous spectrum σ c σ c e - 2 i θ and the branch cut of (1.18.67) into the lower half complex plain by the angle - 2 θ , with respect to the unmoved branch point at λ = 0 ; thus, providing access to resonances on the higher Riemann sheet should θ be large enough to expose them. This dilatation transformation, which does require analyticity of q ( x ) , or an analytic approximation thereto, leaves the poles, corresponding to the discrete spectrum, invariant, as they are, as is the branch point, actual singularities of ( z - T ) - 1 f , f .