Lecture 6: Scattering Theory and Eigenvalues for the Schrödinger equation
Lecture plan. We will continue the discussion of the KdV equation and the Schrödinger equation, and the
amazing connection between these two equations.
The KdV equation:
(1)
ut + uux + uxxx = 0,
and the Schrödinger equation:
−6ψxx − uψ = Eψ
(2)
We will also talk about the problem of asymptotics of the Airy integral.
Scattering and inverse Scattering Theory for the Schrödinger equation
We are considering the time-independent Schrödinger equation,
−Eψ − ψxx + V ψ = 0.
(3)
We have proposed that there are fundamental normalized solutions which are defined for E real, as follows:
√
(4)
ψ1 (x) = e−i
(5)
ψ2 (x) = ei
Ex
√
Ex
1 + O x−1 , as x → +∞,
1 + O x−1 , as x → −∞.
These solutions exist for all ∞ < x < ∞, and possess asymptotics “at the other ∞” as follows:
R2 i√Ex
1 −i√Ex
1 + O(x−1 ) +
1 + O(x−1 ) , x → −∞,
e
e
T2
T2
√
1 i√Ex
R
1 −i Ex
ψ2 =
e
1 + O(x−1 ) +
e
1 + O(x−1 ) , x → −∞ .
T1
T1
(6)
ψ1 =
(7)
By multiplying the formula (6) by T2 (and (7) by T1 ), one finds
√
√
1 + O(x−1 ) + R2 e−i Ex 1 + O(x−1 ) , x → −∞,
√
√
T1 ψ2 = e−i Ex 1 + O(x−1 ) + R1 ei Ex 1 + O(x−1 ) , x → −∞ .
T2 ψ1 = ei
(8)
(9)
Ex
Summarizing from last time, we have given an interpretation of these two solutions (and their asymptotics),
as follows: The solution ψ1 represents a purely left-moving traveling wave at +∞, which behaves at −∞ as a
sum of two traveling waves, one moving left and one moving right. So one may interpret the equation (8) for
ψ1 as follows: we have a traveling wave of unit modulus seeded at −∞ which heads toward the “medium”,
and interacts with the potential V . There is some of the energy in this which is “transmitted” through the
medium and appears as a right-moving wave at +∞ (with “size” given by T2 ). The remainder of the energy
is “reflected” back toward −∞ in the form of a left-moving traveling wave whose size is given by R2 .
There is a similar interpretation for the equation (7) which you should work out.
Here are some questions we must discuss
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Does there exist solutions as described above, and how does one prove it?
What is the behavior of solutions for E < 0?
What is the spectrum of the differential operator?
What spectral information does one need to be able to reconstruct V ?
How does the “spectral data” evolve if we let the potential evolve according to the KdV equation?
But first we will discuss eigenvalues.
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Eigenvalues for the Schrödinger equation
In this section we will suppose that V < 0, and also that E < 0. In this case, when x → ∞, solutions
exhibit either exponential decay or exponential growth. It is then possible that there is a solution to the
Sch”o—dinger equation which is exponentially decaying at both +∞ and −∞. Such a solution is referred
to as an eigenfunction, and the corresponding value of E is referred to as an eigenvalue.
Fact: There exists a unique solution, ψ+ (x; E), normalized at +∞ so that it is exponentially decaying:
√
(10)
ψ+ = e− −Ex 1 + O x−1 .
Fact: There is another solution, ψ̂+ , which is exponentially growing:
√
(11)
ψ̂+ = e −Ex 1 + O x−1 ,
which is not uniquely determined by this normalization (you can add any multiple of ψ+ to it).
Similarly, at −∞, there are two solutions normalized as follows:
√
(12)
ψ− = e −Ex 1 + O |x|−1 .
√
ψ̂− = e− −Ex 1 + O |x|−1 ,
(13)
the first one being uniquely determined by the boundary condition, and the second one not.
Now since ψ− and ψ̂− are linearly independent solutions of the Schrödinger equation one may express ψ+
as a linear combination of them:
(14)
ψ+ = A(E)ψ− + B(E)ψ̂− .
For special values of E, it happens that B = 0. When this happens we see that ψ+ is decaying in both
directions, and that yields for us an L2 eigenfunction, with E being the associated eigenvalue.
It turns out that under suitable assumptions on the potential V , there is at most a finite number of L2
eigenvalues, which will be denoted by {Ej }N
j=1 . For each one, the “normalization constant” is defined to be
the coefficient A(E) appearing in (14), which is then expressed as Aj := A(Ej ).
So we can now summarize scattering theory: given a potential V , one asks to determine all of the spectrum,
which is the positive real axis, and the L2 eigenvalues. On the spectrum, one asks to determine the following
associated quantities:
• On R+ , one determines the reflection coefficient: R1 or R2 (from which, it turns out, one can
determine the remaining scattering functions).
• For each eigenvalue Ej , one determines the normalization constant, Aj .
The quest for this information is referred to as scattering theory or direct spectral theory.
If you know the scattering data, it turns out you can determine from that information the potential V .
The problem of going from the scattering data backwards and determining the potential is referred to as
inverse scattering theory or inverse spectral theory.
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