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1: 33.22 Particle Scattering and Atomic and Molecular Spectra

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§33.22(vii) Complex Variables and Parameters

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Searches for resonances as poles of the $S$-matrix in the complex half-plane . See for example Csótó and Hale (1997).

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Eigenstates using complex-rotated coordinates $r\to r{\mathrm{e}}^{\mathrm{i}\theta }$, so that resonances have square-integrable eigenfunctions. See for example Halley et al. (1993).

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2: Bibliography G

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Stochastic resonance. Rev. Modern Phys. 70 (1), pp. 223–287.

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Referenced by:

§18.39(iii).

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3: Bibliography C

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$S$-matrix and $R$-matrix determination of the low-energy ${}_{}{}^5\mathrm{He}$ and ${}_{}{}^5\mathrm{Li}$ resonance parameters. Phys. Rev. C 55 (1), pp. 536–539.

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External Links:

Document

Referenced by:

2nd item.

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4: Bibliography M

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Resonance Radiation and Excited Atoms. 2nd edition, Cambridge Univerity Press, Cambridge, England.

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Notes:

Reprinted by Cambridge University Press in 1971.

External Links:

FullText, ZentralBlatt

Referenced by:

§7.21.

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5: Bibliography S

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Resonances in $n$-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory. Ann. of Math. (2) 97, pp. 247–274.

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External Links:

ISSN 0003-486X, Document, Link, MathReview (K. Kumar)

Referenced by:

§1.18(viii).

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6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Note that the integral in (1.18.66) 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 ${\lambda }_{\mathrm{res}}-\mathrm{i}{\mathrm{\Gamma }}_{\mathrm{res}}/2$ corresponding to quantum resonances, or decaying quantum states with lifetimes proportional to $1/{\mathrm{\Gamma }}_{\mathrm{res}}$. …This is accomplished by the variable change $x\to x{\mathrm{e}}^{\mathrm{i}\theta }$, in $ℒ$, which rotates the continuous spectrum ${𝝈}_c\to {𝝈}_c{\mathrm{e}}^{-2\mathrm{i}\theta }$ and the branch cut of (1.18.66) into the lower half complex plain by the angle $-2\theta$, with respect to the unmoved branch point at $\lambda =0$; thus, providing access to resonances on the higher Riemann sheet should $\theta$ be large enough to expose them. …