Quasi-Normal Modes of Stars and 8 Acknowledgments

9 Appendix: Schrödinger Equation Versus Wave Equation 

The purpose of this appendix is a brief comparison of properties of the Schrödinger equation

equation1014

and the wave equation

equation1017

for the same potential V (x).

The ansatz of ``stationary states''

equation1020

leads to the time independent Schrödinger equation

  equation1023

The ansatz (corresponding to Laplace transformation)

equation1026

gives for the wave equation

  equation1029

The equations (75Popup Equation) and (77Popup Equation) are the same if we set tex2html_wrap_inline3624 . Hence time independent scattering theory - the theory of the operator tex2html_wrap_inline3626 - does not know whether we deal with the wave equation or the Schrödinger equation.

Consider first a positive potential of compact support. From quantum mechanics we know that there are no bound states. This is easy to understand: outside the support of the potential the solutions are tex2html_wrap_inline3628, hence only negative values of E are possible eigenvalues. However, if we multiply (75Popup Equation) by tex2html_wrap_inline3632 and integrate over all x we obtain a contradiction because of the positivity of V . Hence the operator tex2html_wrap_inline3626 has no eigenfunctions and there are only scattering states, the continuous spectrum. The operator is selfadjoint on the Hilbert space of square integrable functions on the real line. Its resolvent, tex2html_wrap_inline3640 is defined for all complex E outside the continuous spectrum, which consists of the non negative real numbers. tex2html_wrap_inline3640 is an integral operator whose kernel is the Green function constructed in section  1 . There G (s, x, x ') was considered as a function of s . As a function of E, G is analytic on the whole complex plane with the exception of real tex2html_wrap_inline3654, the continuous spectrum. In terms of s, the Green function - and the resolvent - is defined on the ``physical half space'' Re (s)>0.

Resonances of Schrödinger operators can be defined as the poles of the analytic extension of the Green function or the resolvent. There is a huge amount of literature on the subject, in particular mathematical papers. A convenient starting point may be the proceedings of a conference on resonances in 1984 [4]. Existence of resonances, asymptotic distribution and also their interpretation is treated. There is in particular the recent development of ``Geometric scattering Theory'', where the use of ``pseudo differential operators'' gives strong and interesting results [149]. We describe one such result on the asymptotic distribution of quasi-normal mode frequencies of the Schwarzschild spacetime in section  2 . It is amusing to note that in the field of quantum mechanics the same difficulties in defining the notion of a resonance occurred as in relativity in the context of ``normal modes of black holes''! [189] is a good reference explaining this point.

Historically, quasi-normal modes appeared the first time in Gamow's treatment of the tex2html_wrap_inline2814 -decay. The model he studies is a potential with two positive square potentials. Radioactive decay is exponential in experiments. However, even the decay in time of solutions of the free Schrödinger equation is a power law decay (1 / t for 1-dimensional systems), similarly for potentials with compact support. So we face the difficulty to characterize that part of the time evolution, in which exponential decay is a good approximation, knowing that the final decay is polynomial! In [190] such estimates are derived.

Let us finally consider general potentials of compact support. If the potential well is deep enough a finite number of negative eigenvalues tex2html_wrap_inline3664 may exist describing bound states of the quantum system. What are the properties of the corresponding solutions of the wave equation? Let tex2html_wrap_inline3666 be an eigenfunction of tex2html_wrap_inline3626 with eigenvalue tex2html_wrap_inline3670 . Then (tex2html_wrap_inline3624)

equation1044

is a solution of the wave equation. For large positive x we have

equation1047

as the only square integrable solution of the Schrödinger equation with vanishing potential. Thus the solution of the wave equation for large x is

equation1050

This solution grows exponentially in time and falls off exponentially in space for tex2html_wrap_inline3678 . This apparently strange behaviour is possible because the energy density of the conserved energy of the wave equation

equation1055

is not positive definite, if the potential is somewhere negative. In this situation the solution can grow in time and nevertheless have conserved finite energy.

In the complex s plane the eigenvalues appear as poles of the Green function with values tex2html_wrap_inline3682 . To the right of the largest eigenvalue we have analyticity of the Green function.

Let us close this section by remarking that functional analysis techniques can also be used to develop existence theory: In the case discussed above, tex2html_wrap_inline3626 is selfadjoint on the space of square-integrable function. ``Functional calculus'' can be used to show the existence and uniqueness of solutions of the time dependent Schrödinger and wave equation, given appropriate initial data. From this point of view the time independent Green function is the primary object, which is unique because there is a unique selfadjoint operator tex2html_wrap_inline3626 on the real line.



Quasi-Normal Modes of Stars and 8 Acknowledgments

image Quasi-Normal Modes of Stars and Black Holes
Kostas D. Kokkotas and Bernd G. Schmidt
http://www.livingreviews.org/lrr-1999-2
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