Properties of a Many-Particle Hamiltonian under Complex Scaling

Properties of a Many-Particle Hamiltonian under Complex Scaling. 

Here f= f(z) is an analytic function originally defined on the real axis x, which is assumed to be analytic also in the point z = t x. An essential problem in the method of complex scaling in quantum mechanics is hence to study whether a wave function y = y(x) defined on the real axis may be continued analytically out in the complex plane to the point z = rjx. Since many analytic functions have natural boundaries, it is from the very beginning evident that there may be considerable restrictions on the parameter q itself. More generally, one may define the operator u through the relation 

There are many ways to show that the dilatation operator u satisfies the condition (3.60), which is characteristic for the restricted similarity transformations. By putting t] = exp(0), it was proven in eq. (A.3.14) that u may be written in the form 

Per-Olov Lowdin, Piotr Froelich, and Manoj Mishra 

Let us assume that f = f(x) is analytic around the point x=0 within a radius p, so that - for I x I p - one has the Taylor expansion 

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