Spectral Theory in Hilbert Space

2. UNIQUENESS THEOREMS 87
hand, the equation with q = 0 has a Weyl solution on [0, ∞), so that
in fact this value of ℓ gives a point which is in all K(c). This may of
course also be verified directly (do it!). The proof is now complete.
2. Uniqueness theorems
Given q, b, and the boundary conditions, one may in principle determine
m and thus dρ. We will take as our basic inverse problem to
determine q (and possibly b and the boundary conditions) when dρ is
given. Around 1950 Gelfand and Levitan [9] gave a rather complete
solution to this problem. Their solution includes uniqueness, i.e., a
proof that different boundary value problems can not yield the same
spectral measure, reconstruction, i.e., a method (an integral equation)
whereby one, at least in principle, can determine q from the spectral
measure, and characterization, i.e., a description of those measures
that are spectral measures for some equation.
To discuss the full Gelfand-Levitan theory here would take us too
far afield. Instead we will confine ourselves to the problem of uniqueness,
i.e., to show that two different operators can not have the same
spectral measure. This problem was solved independently by Borg [8]
and Marčenko [10] just before the Gelfand-Levitan theory appeared.
To state the theorem we introduce, in addition to the operator T , another
similar operator ˜ T , corresponding to a boundary condition of the
form (12.2), but with an angle ˜α ∈ [0, π), an interval [0, ˜ b), a potential
˜q and, if needed, a boundary condition at ˜ b. Let the corresponding
spectral measure be d˜ρ.
Theorem 12.4 (Borg-Marčenko). If dρ = d˜ρ, then ˜ T = T , i.e.,
˜α = α, ˜ b = b and ˜q = q.
A few years ago Barry Simon [11] proved a ‘local’ version of this
uniqueness theorem. This was a product of a new strategy developed by
Simon for obtaining the results of Gelfand and Levitan. I will give my
own proof [6], which is quite elementary and does not use the machinery
of Simon. We will use the same idea to prove Theorem 12.4.
In order to state Simon’s theorem, one should first note that knowing
m is essentially equivalent to knowing dρ, at least if the boundary
condition (12.2) is known. Knowing m one can in fact find dρ
via the Stieltjes inversion formula, and knowing dρ one may calculate
the integral in the representation of m. By Theorem 12.2 we always
have B = 0, and A may be determined (if α = 0) since we also have
m(iν) → cot α as ν → ±∞. We denote the m-functions associated
with T and ˜ T by m and ˜m respectively. Then Simon’s theorem is the
following.
Theorem 12.5 (Simon). Suppose that 0 < a ≤ min(b, ˜ b). Then
α = ˜α and q = ˜q a.e. on (0, a) if (m(λ) − ˜m(λ))e 2(a−ε) Re √ −λ → 0 for