Canonical systems whose Weyl coefficients have regularly varying - - PowerPoint PPT Presentation

Canonical systems whose Weyl coefficients have regularly varying asymptotics

Matthias Langer University of Strathclyde, Glasgow based on joint work with Raphael Pruckner and Harald Woracek (Vienna)

Canonical systems Consider the 2 × 2 canonical system y′(x) = zJH(x)y(x), x ∈ (0, ∞), (CS) where

• y . . . 2-vector function
• z ∈ C
• J =

0 −1

1

• H . . . locally integrable (on [0, ∞)) function whose values are

2 × 2 real non-negative matrices (H . . . ‘Hamiltonian’) H does not vanish on any set of positive measure

• H is in the limit point case at ∞, i.e.

∞

tr H(x) dx = ∞.

Weyl function Let W(x, z) be the (fundamental) solution of ∂ ∂xW(x, z)J = zW(x, z)H(x), x ∈ (0, ∞); W(0, z) = I. Note that the rows of W satisfy (CS) and

w11(0, z)

w12(0, z)

• =

1

• ,

w21(0, z)

w22(0, z)

• =

1

• .

Since H is in the limit point case at b, the following limit exists and is independent of τ ∈ R ∪ {∞}, qH(z) := lim

x→∞

w11(x, z)τ + w12(x, z) w21(x, z)τ + w22(x, z), z ∈ C \ R. The function qH is called Weyl function for the Hamiltonian H. It is also characterised by the property that, for z ∈ C \ R,

w11(·, z)

w12(·, z)

• − qH(z)

w21(·, z)

w22(·, z)

• ∈ L2((0, ∞), H(x)dx)

Spectral measure The Weyl function is a Nevanlinna function, i.e. Im qH(z) ≥ 0, qH(z) = qH(z) when Im z > 0. Hence it has an integral representation qH(z) = a + bz +

• R
• 1

t − z − t 1 + t2

• dµ(t),

z ∈ C \ R, where a ∈ R, b ≥ 0 and µ is a Borel measure on R such that

• R

1 1 + t2 dµ(t) < ∞. The measure µ is a spectral measure since the generalised Fourier transform f →

∞ w11(x, ·)

w12(x, ·)

• H(x)f(x) dx

establishes a unitary equivalence of the underlying operator (or relation) corresponding to (CS) with the multiplication operator in L2(R, µ).

Inverse spectral theorem (de Branges). The mapping H → qH establishes a one-to-one correspondence between all Hamiltonians (up to reparameterisation) and all Nevanlinna functions. Reparameterisation: ˜ H(x) = η′(x)H(η(x)) for a strictly increasing bijection η. The restriction of the mapping H → qH to Hamiltonians with tr H(x) = 1 a.e. is a bijection. Question: how are properties of H related to properties of qH?

Basic asymptotic properties of the Weyl function Since qH is a Nevanlinna function, ∃ c1, c2, r0 > 0: c1 r ≤ |qH(ri)| ≤ c2r, r ≥ r0. The extreme cases are well known:

• lim sup

r→∞

|qH(ri)| r > 0 ⇐ ⇒ qH(ri) ∼ ibr with b > 0 ⇐ ⇒ ∃ ε > 0. H|(0,ε) =

h1 0

0 0

• a.e.
• lim inf

r→∞ r|qH(ri)| < ∞

⇐ ⇒ qH(ri) ∼ ic1 r with c > 0 ⇐ ⇒ qH(z) =

• R

1 t − z dµ(t), µ finite ⇐ ⇒ ∃ ε > 0. H|(0,ε) =

0 0

0 h2

• a.e.

Asymptotics of the spectral function or the Titchmarsh–Weyl coefficient for Sturm–Liouville equations: Marchenko 1952 Hille 1963 Everitt 1972 Atkinson 1981 Bennewitz 1989 . . . Strings: Kac 1971, . . . Kasahara 1976 Kasahara, Watanabe 2010 Canonical systems: Eckhardt, Kostenko, Teschl 2018

Regularly varying functions A measurable function a : (0, ∞) → (0, ∞) is called regularly varying at ∞ with index α ∈ R if ∀ λ > 0. lim

r→∞

a(λr) a(r) = λα. Examples:

• a(r) = rα(log r)β1 · · · (log · · · log
• m

r)βm, α, β1, . . . , βm ∈ R

• a(r) = rαe

log r log log r,

α ∈ R,

• a(r) = rαe(log r)β cos((log r)β),

α ∈ R, β ∈ (0, 1

2).

for r large enough. When α = 0, a is called slowly varying.

Regular and rapid variation at 0 A function a : (0, ∞) → (0, ∞) is called regularly varying at 0 with index α ∈ R if r → 1/a(1

r) is regularly varying at ∞ with index α,

• r equivalently,

∀ λ > 0. lim

xց0

a(λx) a(x) = λα. A function a is called rapidly varying at 0 with index ∞ if lim

xց0

a(λx) a(x) =

      

0, λ ∈ (0, 1), 1, λ = 1, ∞, λ ∈ (1, ∞). Example: a(x) = e−1

x.

Assumptions for the main theorem Let H =

h1 h3

h3 h2

• be a Hamiltonian as above.

(i) Assume that neither H =

h1 0

0 0

• nor H =

0 0

0 h2

• n any interval
• f the form (0, ε) with ε > 0.

(ii) Set mi(t) :=

t

0 hi(s) ds,

i = 1, 2, 3. (iii) By (i), m1m2 is a strictly increasing bijection from (0, ∞)

• nto itself.

Hence, for every r > 0 there exists a unique ˚ x(r) > 0 such that (m1m2)(˚ x(r)) = 1 r2 . (iv) Assume that m1+m2 is regularly varying at 0 with positive index. This is satisfied, e.g. if tr H = 1 a.e.

• Theorem. Let the assumptions from above be satisfied. TFAE

(i) ∃ a : (0, ∞) → (0, ∞), regularly varying at ∞, ∃ ω ∈ C \ {0}: ∀ r > 0. qH(ri) = iωa(r)

• 1 + R(r)
• ;

lim

r→∞ R(r) = 0.

(ii) m1, m2 are regularly or rapidly varying at 0 with indices ρi and δ := lim

xց0

m3(x)

• m1(x)m2(x)

exists. (iii) Let ˚ x(r) be the unique solution of (m1m2)(˚ x(r)) = 1

r2 , r > 0.

The function aH(r) :=

• m1(˚

x(r)) m2(˚ x(r)) , r > 0, is regularly varying at ∞

with index α, ∀ r > 0, z ∈ C+. qH(rz) = iω

z

i

αaH(r)

• 1 + R(z, r)
• ;

∗ lim

r→∞ R(z, r) = 0 locally uniformly for z ∈ C+;

∗ α = ρ2 − ρ1 ρ2 + ρ1 ; ω ∈ C \ {0} with | arg ω| ≤ π

2(1 − |α|).

The role of ω. Assume that (i)–(iii) hold. Recall that qH(rz) ∼ iω

z

i

αaH(r),

r → ∞, with α = ρ2−ρ1

ρ2+ρ1 ∈ [−1, 1],

| arg ω| ≤ π

2(1 − |α|).

− → z → iω

z

i

α

π|α| arg ω The coefficient ω depends explicitly on α (and hence on ρ1, ρ2) and on δ = lim

x→0

m3(x)

• m1(x)m2(x)
• where |δ| ≤
• 1 − α2

. For fixed α the mapping δ → arg ω is strictly decreasing. Further, arg ω = 0 ⇐ ⇒ δ = 0.

The spectral measure Since H is not of the form

h1 h3

h3 h2

• n any interval of the form (0, ε),

b = 0 in the integral representation of qH, i.e. qH(z) = a +

• R
• 1

t − z − t 1 + t2

• dµ(t),

Assume that (i)–(iii) are satisfied and that α = ±1. With the help of a Tauberian theorem one can show that µ([0, t)) = |ω| α + 1 sin

• arg ω + π

2(1 − α)

• ta(t)
• 1 + o(1)
• ,

µ((−t, 0]) = |ω| α + 1 sin

• arg ω + π

2(1 + α)

• ta(t)
• 1 + o(1)
• ,

as t → ∞.