ON A THEOREM OF KAC AND GILBERT

BARRY SIMON

Abstract. We prove a general operator theoretic result that asserts that many multiplicity two selfadjoint operators have simple singular spectrum.

1. Introduction

In 1963, I.S. Kac [5] proved that whole-line Schrödinger operators, −\frac{d}{dx^{2}} + V (x), for fairly general V ’s have simple singular spectrum. It is well known (e.g., V = 0) that the absolutely continuous spectrum can have multiplicity two and that under limit point hypotheses, eigenvalues are simple. But the simplicity of the singular continuous spectrum is surprisingly subtle. Some insight into the result was obtained by Gilbert [3], who found a proof using the subordinacy theory of Gilbert-Pearson [4]. The proof is elegant but depends on the substantial machinery of subordinacy. Our purpose here is to note an abstract result that relates these things to the celebrated result of Aronszajn-Donoghue [1]:

Theorem 1. Let A be a bounded selfadjoint operator on H and φ ∈ H a cyclic vector for A. Suppose λ ∈ ℝ\{0} and

B = A + λ⟨φ, · ⟩φ  (1.1)

Then the singular spectral measures for A and B are disjoint.

We state this and the next theorem in the bounded case for simplicity; we discuss the general case later. Here’s the main result of this note:

Theorem 2. Let H = K_{1} ⊕ K_{2} and P : H → K_{1}, the canonical projection. Let A_{j} ∈ L(K_{j}) for j = 1, 2, and φ ∈ H so that φ_{1} = P φ and φ_{2} = (1 − P)φ are cyclic for A_{1} and A_{2}. Let λ ∈ ℝ\{0}. Then

C = A_{1} ⊕ A_{2} + λ(φ, ·)φ  (1.2)

has simple singular spectrum.

Date: April 16, 2004.

∗ Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125.

E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-0140592.

Remark. If A_{1} is unitarily equivalent to A_{2} and has a.c. spectrum, then C has multiplicity two a.c. spectrum. So it is interesting that the singular spectrum is simple.

In Section 2, we prove Theorem 2. In Section 3, we apply it to wholeline Jacobi matrices. In Section 4, we discuss extensions of Theorem 2 to the case of unbounded selfadjoint operators and to unitary operators. In Section 5, we apply the results of Section 4 to Schrödinger operators and to extended CMV matrices.

2. Proof of Theorem 2

Let dμ_{j} be the spectral measure for φ_{j} and A_{j} and B = A_{1} ⊕ A_{2}. Thus K_{j} ≅ L^{2}(ℝ, dμ_{j}) in such a way that A_{j} is multiplication by x and φ_{j} ≅ 1.

Pick disjoint sets X, Y, Z ⊂ ℝ whose union is ℝ so dμ_{1} ↾ X is equivalent to dμ_{2} ↾ X and μ_{1}(Y) = μ_{2}(Z) = 0. For example, if dμ_{1} = f(dμ_{1} + dμ_{2}), then one can take X = {x | 0 < f(x) < 1}, Y = {x | f(x) = 0}, Z = {x | f(x) = 1}.

Let L_{1} be the cyclic subspace generated by φ and B and L_{2} = L_{1}^{⊥}. Then ψ = (χ_{X}, −χ_{X}) is a cyclic vector for B ↾ L_{2} and its spectral measure is

dμ_{ψ}^{B} = χ_{X}(x)(dμ_{1} + dμ_{2})  (2.1)

In particular,

(dμ_{ψ}^{B})_{s} ≤ (dμ_{1} + dμ_{2})_{s}  (2.2)

By definition of L_{1}, φ is cyclic for B ↾ L_{1} and

C ↾ L_{1} = B + λ(φ, ·)φ

so, by Theorem 1,

(dμ_{φ}^{C})_{s} ⊥ (dμ_{φ}^{B})_{s} = (dμ_{1} + dμ_{2})_{s}  (2.3)

Thus, the singular parts of dμ_{φ}^{C} and dμ_{ψ}^{C} = dμ_{ψ}^{B} are disjoint, which implies that the singular spectrum of C is simple. □

The proof shows that the singular parts of B and C overlap in χ_{X}(dμ_{1} + dμ_{2})_{s} and, in particular,

Corollary 2.1. B and C have mutually singular parts if and only if A_{1} and A_{2} have mutually singular parts.

3. Application to Jacobi Matrices

A two-sided Jacobi matrix is defined by two two-sided sequences, {b_{n}}_{n=−∞}^{∞} and {a_{n}}_{n=−∞}^{∞} with b_{n} ∈ ℝ and a_{n} ∈ (0, ∞) and sup_{n}(|a_{n}| + |b_{n}|) < ∞. It defines a bounded operator J on ℓ^{2}(ℤ) by

(Ju)_{n} = a_{n−1}u_{n−1} + b_{n}u_{n} + a_{n}u_{n+1}  (3.1)

so

J = (\begin{matrix}⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ \\ … & b_{−1} & a_{−1} & 0 & 0 & … \\ … & a_{−1} & b_{0} & a_{0} & 0 & … \\ … & 0 & a_{0} & b_{1} & a_{1} & … \\ ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ \end{matrix})  (3.2)

Theorem 3.1. The singular spectrum of J is simple.

Proof. Let K_{1} = ℓ^{2}((−∞, −1]), K_{2} = ℓ^{2}([0, ∞)), and φ the vector with components

φ_{j} = \begin{cases}1 & j = −1, 0 \\ 0 & j ≠ −1, 0 \end{cases}  (3.3)

so P φ = δ_{−1}; (1 − P)φ = δ_{0}.

Then

J − a_{−1}(φ, φ) = A_{1} ⊕ A_{2}

where A_{2} is the one-sided Jacobi matrix with

A_{2} = (\begin{matrix}b_{0} − a_{−1} & a_{0} & 0 & … \\ a_{0} & b_{1} & a_{2} & … \\ … & … & … & … \end{matrix})

and A_{1} in δ_{−1}, δ_{−2}, … basis is

A_{1} = (\begin{matrix}b_{−1} − a_{−1} & a_{−2} & 0 & … \\ a_{−2} & b_{−2} & a_{−3} & … \\ … & … & … & … \end{matrix})

Thus P φ is cyclic for A_{1} and (1 − P)φ is cyclic for A_{2}. Theorem 2 applies and implies the desired result. □

4. Unitary and Unbounded Selfadjoint Operators

Let U_{1}, U_{2}, and W be unitary operators on K_{1}, K_{2}, and H = K_{1} ⊕K_{2} so W − U_{1} ⊕ U_{2} is rank one, and so if φ ∈ ker(W − U_{1} ⊕ U_{2})^{⊥}, then P φ is cyclic for U_{1} and (1 − P)φ is cyclic for U_{2}, where P is the canonical projection of H to K_{1}. Suppose W − U_{1} ⊕ U_{2} ≠ 0. Then Theorem 4.1. The singular spectrum of W is simple.

Proof. We begin by proving the unitary analog of the Aronszajn-Donoghue theorem. If W − V is rank one and nonzero, then for φ a unit vector in ker(W − V)^{⊥}, we have W φ = λV φ for some λ ∈ ∂𝔻 = {z ∈ ℂ | |z| = 1}. By a direct calculation (see, e.g., [6]),

(φ, \frac{W + z}{W − z} φ) = \frac{1 + zg(z)}{1 − zg(z)}  (4.1)
(φ, \frac{V + z}{V − z} φ) = \frac{1 + zf(z)}{1 − zf(z)}  (4.2)

and

g(z) = λ^{−1}f(z)  (4.3)

If φ cyclic for W, the singular spectrum in W is supported on those z ∈ ∂𝔻 with

lim_{r↑1} rz g(rz) = 1

and similarly, the singular spectrum of V on the set of z ∈ ∂𝔻 with

lim_{r↑1} rz f(rz) = 1

By (4.3), these sets are disjoint.

This proves the Aronszajn-Donoghue theorem in the unitary case, and that implies this theorem by mimicking the proof of Theorem 2. □

Next, let A_{1}, A_{2}, and C be potentially unbounded selfadjoint operators on K_{1}, K_{2}, and H = K_{1} ⊕ K_{2}. Suppose D ≡ (A_{1} ⊕ A_{2} − i)^{−1} − (C − i)^{−1} is rank one with φ ∈ (ker D)^{⊥} so P φ is cyclic for A_{1} and (1 − P)φ for A_{2}. Then with

U_{j} = (A_{j} + i)(A_{j} − i)^{−1} W = (C + i)(C − i)^{−1}

Theorem 4.1 applies, so

Theorem 4.2. C has simple singular spectrum.

5. Extended CMV Matrices and Schrödinger Operators

Extended CMV matrices enter in the theory of the orthogonal polynomials on the unit circle [6]. They are defined by a family of Verblunsky coefficients {α_{j}}_{j=−∞}^{∞} with α_{j} ∈ 𝔻 = {z ∈ ℂ | |z| = 1} as follows. Let Θ(α) be the 2 × 2 matrix

Θ(α) = ( \begin{matrix}ᾱ & ρ \\ ρ & −α \end{matrix} )  (5.1)

where ρ = (1 − |α|^{2})^{1/2}.

Think of ℓ^{2}(ℤ), first as a direct sum ⊕_{n=−∞}^{n}ℂ^{2} with the n-th factor spanned by (δ_{2n}, δ_{2n+1}) and let M = ⊕Θ(α_{2n}), then as a direct sum with the n-th factor spanned by (δ_{2n+1}, δ_{2n+2}) and L = ⊕Θ(α_{2n+1}). Then E = LM is the extended CMV matrix.

We claim

Theorem 5.1. E always has simple singular spectrum.

Proof. Let

x = \frac{1 + ᾱ_{−1}}{1 + α_{−1}}

Then

det(Θ(α_{−1}) − (\begin{matrix}x & 0 \\ 0 & 1 \end{matrix})) = 0

by a simple calculation. It is thus rank one, so if Ẽ is defined by replacing Θ(α_{−1}) by ( \begin{matrix}x & 0 \\ 0 & 1 \end{matrix} ), Ẽ − E is rank one. On the other hand, Ẽ is a direct sum of two half-line CMV matrices and it is easy to see P φ and (1 − P)φ are cyclic. Thus Theorem 4.1 applies. □

Finally, we turn to the Schrödinger operator case that motivated us in the first place. Suppose H = −\frac{d^{2}}{dx^{2}} + V where V ∈ L_{loc}^{1}(−∞, ∞) is limit point at both +∞ and −∞. Let H_{1} (resp. H_{2}) be H on L^{2}(0, ∞) (resp. L^{2}(−∞, 0)) with u(0) = 0 boundary conditions. Then

(H − i) − (H_{1} ⊕ H_{2} − i)^{−1}

is rank one by the explicit Green’s function formulae [2], and its kernel is spanned by a function φ with

(−\frac{d^{2}}{dx^{2}} + V )φ = iφ x ≠ 0  (5.2)

with φ L^{2} at +∞ and −∞ and φ(0_{+}) = φ(0_{−}). (5.2) implies φ ↾ [0, ∞) (resp. φ ↾ (−∞, 0]) is cyclic for H_{1} (resp. H_{2}). Theorem 4.2 applies and yields the Kac-Gilbert theorem:

Theorem 5.2. −\frac{d^{2}}{dx^{2}} + V has simple singular spectrum.

References

[1] N. Aronszajn and W.F. Donoghue, On exponential representations of analytic functions in the upper half-plane with positive imaginary part, J. Analyse Math. 5 (1957), 321–388.

[2] F. Gesztesy and B. Simon, The xi function, Acta Math. 176 (1996), no. 1, 49–71.

[3] D.J. Gilbert, On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints, Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), 213–229.

[4] D.J. Gilbert and D.B. Pearson, On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Anal. Appl. 128 (1987), 30–56.

[5] I.S. Kac, Spectral multiplicity of a second-order differential operator and expansion in eigenfunction, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 1081– 1112. [Russian]

[6] B. Simon, Orthogonal Polynomials on the Unit Circle, AMS Colloquium Series, American Mathematical Society, Providence, RI, expected 2004.