Normal Modes – Quasi-Normal Modes

2 Normal Modes – Quasi-Normal Modes – Resonances

Before discussing quasi-normal modes it is useful to remember what normal modes are!

Compact classical linear oscillating systems such as finite strings, membranes, or cavities filled with electromagnetic radiation have preferred time harmonic states of motion ( $ω$ is real):

if dissipation is neglected. (We assume $χ$ to be some complex valued field.) There is generally an infinite collection of such periodic solutions, and the “general solution” can be expressed as a superposition,

of such normal modes. The simplest example is a string of length $L$ which is fixed at its ends. All such systems can be described by systems of partial differential equations of the type ( $χ$ may be a vector)

where $A$ is a linear operator acting only on the spatial variables. Because of the finiteness of the system the time evolution is only determined if some boundary conditions are prescribed. The search for solutions periodic in time leads to a boundary value problem in the spatial variables. In simple cases it is of the Sturm–Liouville type. The treatment of such boundary value problems for differential equations played an important role in the development of Hilbert space techniques.

A Hilbert space is chosen such that the differential operator becomes symmetric. Due to the boundary conditions dictated by the physical problem, $A$ becomes a self-adjoint operator on the appropriate Hilbert space and has a pure point spectrum. The eigenfunctions and eigenvalues determine the periodic solutions (1 ).

The definition of self-adjointness is rather subtle from a physicist’s point of view since fairly complicated “domain issues” play an essential role. (See [43 ] where a mathematical exposition for physicists is given.) The wave equation modeling the finite string has solutions of various degrees of differentiability. To describe all “realistic situations”, clearly $∞ C$ functions should be sufficient. Sometimes it may, however, also be convenient to consider more general solutions.

From the mathematical point of view the collection of all smooth functions is not a natural setting to study the wave equation because sequences of solutions exist which converge to non-smooth solutions. To establish such powerful statements like (2 ) one has to study the equation on certain subsets of the Hilbert space of square integrable functions. For “nice” equations it usually happens that the eigenfunctions are in fact analytic. They can then be used to generate, for example, all smooth solutions by a pointwise converging series (2 ). The key point is that we need some mathematical sophistication to obtain the “completeness property” of the eigenfunctions.

This picture of “normal modes” changes when we consider “open systems” which can lose energy to infinity. The simplest case are waves on an infinite string. The general solution of this problem is

with “arbitrary” functions $A$ and $B$ . Which solutions should we study? Since we have all solutions, this is not a serious question. In more general cases, however, in which the general solution is not known, we have to select a certain class of solutions which we consider as relevant for the physical problem.

Let us consider for the following discussion, as an example, a wave equation with a potential on the real line,

Cauchy data $χ (0,x),∂tχ(0,x)$ which have two derivatives determine a unique twice differentiable solution. No boundary condition is needed at infinity to determine the time evolution of the data! This can be established by fairly simple PDE theory [116 ].

There exist solutions for which the support of the fields are spatially compact, or – the other extreme – solutions with infinite total energy for which the fields grow at spatial infinity in a quite arbitrary way!

From the point of view of physics smooth solutions with spatially compact support should be the relevant class – who cares what happens near infinity! Again it turns out that mathematically it is more convenient to study all solutions of finite total energy. Then the relevant operator is again self-adjoint, but now its spectrum is purely “continuous”. There are no eigenfunctions which are square integrable. Only “improper eigenfunctions” like plane waves exist. This expresses the fact that we find a solution of the form (1 ) for any real $ω$ and by forming appropriate superpositions one can construct solutions which are “almost eigenfunctions”. (In the case $V (x) ≡ 0$ these are wave packets formed from plane waves.) These solutions are the analogs of normal modes for infinite systems.

Let us now turn to the discussion of “quasi-normal modes” which are conceptually different to normal modes. To define quasi-normal modes let us consider the wave equation (5 ) for potentials with $V ≥ 0$ which vanish for $|x| > x0$ . Then in this case all solutions determined by data of compact support are bounded: $|χ(t,x)| < C$ . We can use Laplace transformation techniques to represent such solutions. The Laplace transform $ˆχ (s,x )$ ( $s > 0$ real) of a solution $χ(t,x)$ is

and satisfies the ordinary differential equation

where

is the homogeneous equation. The boundedness of $χ$ implies that $ˆχ$ is analytic for positive, real $s$ , and has an analytic continuation onto the complex half plane $Re (s) > 0$ .

Which solution $ˆχ$ of this inhomogeneous equation gives the unique solution in spacetime determined by the data? There is no arbitrariness; only one of the Green functions for the inhomogeneous equation is correct!

All Green functions can be constructed by the following well known method. Choose any two linearly independent solutions of the homogeneous equation $f− (s,x)$ and $f+(s,x)$ , and define

where $W (s)$ is the Wronskian of $f −$ and $f+$ . If we denote the inhomogeneity of (7

) by $j$ , a solution of (7

) is

We still have to select a unique pair of solutions $f ,f − +$ . Here the information that the solution in spacetime is bounded can be used. The definition of the Laplace transform implies that $ˆχ$ is bounded as a function of $x$ . Because the potential $V$ vanishes for $|x| > x0$ , the solutions of the homogeneous equation (8

) for $|x | > x0$ are

The following pair of solutions

which is linearly independent for $Re (s) > 0$ , gives the unique Green function which defines a bounded solution for $j$ of compact support. Note that for $Re (s) > 0$ the solution $f +$ is exponentially decaying for large $x$ and $f−$ is exponentially decaying for small $x$ . For small $x$ however, $f+$ will be a linear combination $−sx sx a(s)e + b(s)e$ which will in general grow exponentially. Similar behavior is found for $f−$ .

Quasi-Normal mode frequencies $sn$ can be defined as those complex numbers for which

that is the two functions become linearly dependent, the Wronskian vanishes and the Green function is singular! The corresponding solutions $f+(sn,x)$ are called quasi eigenfunctions.

Are there such numbers $sn$ ? From the boundedness of the solution in spacetime we know that the unique Green function must exist for $Re (s) > 0$ . Hence $f+,f−$ are linearly independent for those values of $s$ . However, as solutions $f+,f −$ of the homogeneous equation (8 ) they have a unique continuation to the complex $s$ plane. In [34 ] it is shown that for positive potentials with compact support there is always a countable number of zeros of the Wronskian with $Re (s) < 0$ .

What is the mathematical and physical significance of the quasi-normal frequencies $sn$ and the corresponding quasi-normal functions $f+$ ? First of all we should note that because of $Re (s) < 0$ the function $f+$ grows exponentially for small and large $x$ ! The corresponding spacetime solution $esntf+(sn,x)$ is therefore not a physically relevant solution, unlike the normal modes.

If one studies the inverse Laplace transformation and expresses $χ$ as a complex line integral ( $a > 0$ ),

one can deform the path of the complex integration and show that the late time behavior of solutions can be approximated in finite parts of the space by a finite sum of the form

Here we assume that $Re (sn+1) < Re (sn) < 0$ , $sn = αn + iβn$ . The approximation $∼$ means that if we choose $x0$ , $x1$ , $𝜖$ and $t0$ then there exists a constant $C (t0,x0,x1, 𝜖)$ such that

holds for $t > t0$ , $x0 < x < x1$ , $𝜖 > 0$ with $C (t0,x0,x1, 𝜖)$ independent of $t$ . The constants $an$ depend only on the data [34

]! This implies in particular that all solutions defined by data of compact support decay exponentially in time on spatially bounded regions. The generic leading order decay is determined by the quasi-normal mode frequency with the largest real part $s1$ , i.e. slowest damping. On finite intervals and for late times the solution is approximated by a finite sum of quasi eigenfunctions (15

It is presently unclear whether one can strengthen (16 ) to a statement like (2 ), a pointwise expansion of the late time solution in terms of quasi-normal modes. For one particular potential (Pöschl–Teller) this has been shown by Beyer [42 ].

Let us now consider the case where the potential is positive for all $x$ , but decays near infinity as happens for example for the wave equation on the static Schwarzschild spacetime. Data of compact support determine again solutions which are bounded [117 ]. Hence we can proceed as before. The first new point concerns the definitions of $f±$ . It can be shown that the homogeneous equation (8 ) has for each real positive $s$ a unique solution $f+ (s,x)$ such that $limx → ∞ (esxf+ (s,x)) = 1$ holds and correspondingly for $f −$ . These functions are uniquely determined, define the correct Green function and have analytic continuations onto the complex half plane $Re (s) > 0$ .

It is however quite complicated to get a good representation of these functions. If the point at infinity is not a regular singular point, we do not even get converging series expansions for $f±$ . (This is particularly serious for values of $s$ with negative real part because we expect exponential growth in $x$ ).

The next new feature is that the analyticity properties of $f±$ in the complex $s$ plane depend on the decay of the potential. To obtain information about analytic continuation, even use of analyticity properties of the potential in $x$ is made! Branch cuts may occur. Nevertheless in a lot of cases an infinite number of quasi-normal mode frequencies exists.

The fact that the potential never vanishes may, however, destroy the exponential decay in time of the solutions and therefore the essential properties of the quasi-normal modes. This probably happens if the potential decays slower than exponentially. There is, however, the following way out: Suppose you want to study a solution determined by data of compact support from $t = 0$ to some large finite time $t = T$ . Up to this time the solution is – because of domain of dependence properties – completely independent of the potential for sufficiently large $x$ . Hence we may see an exponential decay of the form (15 ) in a time range $t1 < t < T$ . This is the behavior seen in numerical calculations. The situation is similar in the case of $α$ -decay in quantum mechanics. A comparison of quasi-normal modes of wave equations and resonances in quantum theory can be found in the Appendix A.