Topological Levinson’s theorem with two Hilbert
spaces: from finite graphs to insulators
Hideki Inoue (University of Lyon)
in collaboration withJohannes Kellendonk & Hermann Schulz-Baldes

MarcheFebruary 2, 2022

Introduction

Original version of Levinson’s theorem
Levinson’s theorem is a fundamental relation in QM originally establishedby N. Levinson in 1949.

Theorem
Consider the radial Schrödinger operator（ ） with Dirichlet b.c.

H = −∂r2 + v(r) ≡ H!=0

acting on H = L2 (R+ ) with a potential v decaying faster than r−2 asr → ∞. Then, one has the relation

scattering phase shift = dim Hpp (H) + δ,
where the correction term δ is given by!
1/2 if H has zero-energy resonance,
δ=0otherwise.

Topological version of Levinson’s theorem
Topological approach has been widely popularized.

Description with the previous model cf. [Kellendonk& Richard, 2007]
Let H0 := −∂r2 with Dirichlet boundary condition.
1. The wave operators exist and are complete:

W± := s-lim eitH e−itH0t→±∞

2. One constructs a C ∗ -subalgebra E ⊂ B(H) satisfyingπ

0 → K(H) → E −→ C(T) → 0(exact),W− ∈ E and π(W− ) ∈ C(T) is unitary.

3. It follows from K-theory that the following index theorem holds:
Wind(π(W− )) = − Index(W− ).
4. One observes that the above index theorem is equivalent to
Levinson’s theorem.

Scattering theory with two Hilbert spaces
In practice, free and perturbed systems may act on different spaces, i.e.
H0 ! H 0 ,H!HandH0 '= H.

Then one may introduce an identification operator J : H0 → H toconstruct the wave operator as
W± ≡ W± (H, H0 ; J) := s-lim eitH Je−itH0t→±∞

Motivating Question Can we still give a topologicalinterpretation for Levinson’s theorem in two Hilbert spacessetting?
ProblemC ∗ -algebras are defined as a subalgebra of B(G) for a Hilbert space G.
However, W± ∈ B(H0 , H) whenever they exist.
−→ There is no C ∗ -algebra E containing the two-space WO W− .
Let us discuss this problem with a simple example!

Introduction

Generalized Childs’ Graph Model

Mathematical Scattering Theory for Generalized Childs’ Graph Model

Topological Levinson’s Theorem

Work in progress: Bulk-edge correspondence via two spaces scattering

Generalized Childs’ Graph ModelChilds’ graph modelChilds’ graph model [Childs et. al. 2011 & 2012] consists of a bundle ofwires attached to a finite graph G1 .
! H0 : the Laplacian on the bundle of wires Z+ × V with V finite,! H: the Laplacian on the graph G obtained by attaching wires to G1「が𦥯※※※ がた ". _ '
32、o.

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iiig2+×｛33Newlyaddededges

Levinson’s theorem for Childs’ graph modelTheorem (Childs et. al. 2011 & 2012)
Set s(λ) := det(S(λ)) with S(λ) ∈ B(%2 (V)) being the scattering matrix
for the pair (H0 , H). Then, one has"
1s% (λ)s(λ)∗ dλ = #{bound states} + δ − #{vertices of G1 }2πi σ(H0 )New Question Why does #{vertices of G1 } appear?

Generalized Childs’ graph model
The free system H0 (wires)
Let Z+ := {0, 1, 2, . . .} and V be a countable set.
! A (free) particle in a bundle of wires Z+ × V are described by!

H0 := %2 (Z+ × V) ≡ %2 (Z+ ) ⊗ %2 (V),H0 := h0 ⊗ 1V ,Hilbert space
Hamiltonian

where h0 := (T + T ∗ )/2 with T being the shift operator.
! σ(H0 ) = σac (H0 ) = [−1, 1].
! We fix a spectral representation for H0
F0 H0 F0∗ =2with hλ ≡ h = % (V)"λdλ,[−1,1]F0 : H0 →" ⊕hλ dλ,[−1,1]

T Fourier transform
sine

Remark Discrete symmetries can be included by considering H0 ⊗ σ withσ = σ ∗ = σ −1 , σ ∈ ML (C).

Generalized Childs’ graph model
The perturbed system (wires + “graph”)
Let H1 be a self-adjoint operator on a separable Hilbert space H1 .
! The perturbed system is described by
H := H0 ⊕ H1 , Hilbert space'(HQ0
H :=.

 HamiltonianQ ∗ H1! The interaction term Q : H1 → H0 is a bounded operator of theform)
Q=|δ0 ⊗ δn +,Qn |,Qn ∈ H1 , Qn '= 0.

？".....○n∈Vs eが. . 二 . s i. . . Q 2.

：

Mathematical Scattering Theory for GeneralizedChilds’ Graph ModelWave and scattering operators
Set J : H0 '→ H to be the inclusion operator. Then HJ − JH0 = Q∗
and the existence of the WO can be justified with Cook’s criterion.

Lemma
For any H1 , the WO W± ≡ W± (H, H0 ; J) exist and are isometries.
! W± H0 = HW± (intertwining property) ,! Ran(W± ) ⊂ Hac (H).

Remark! By Kato-Rosenblum theorem, W± are complete for the Childs’
graph model, i.e. #V, dim H1 < ∞.

The scattering operator S and the scattering matrix S(·) are defined byS := W+∗ W− ,F0 SF0∗ =" ⊕[−1,1]S(λ)dλ.

H o y t l t l t o e

u n dStationary expressions?

We use the stationary approach to compute an explicit formula for S.

Resolvents

For z ∈ C with -(z) '= 0, we set
R0 (z) := (H0 − z)−1 ,R1 (z) = (H1 − z)−1 ,R(z) = (H − z)−1 .

Lemma
The following stationary formulas hold:" 1+ε*,f, W± f0 +H =lim R(λ ± iε)f, JR0 (λ ± iε)f0 H dλ−1 ε↓0 πfor any f ∈ Hac (H) and f0 in a dense subset of H0 .

Reduction to one Hilbert space components
Let us consider the components
W± := J ∗ W± ,which belong to B(H0 ).

1. One can compute S from the component W− as
S = s-lim eitH0 W− e−itH0 .
t→∞2. It follows from the stationary formula that
" 1+ε*,f0% , W− f0 +H0 =lim J ∗ R(λ − iε)Jf0% , R0 (λ − iε)f0 H dλ0−1 ε↓0 π
3. By Schur complement formula,J ∗ R(z)J = (H0 − QR1 (z)Q∗ − z)−1 .
4. The term −QR1 (z)Q∗ can be considered as an energy dependent
potential localized on the boundary {0} × V ⊂ Z+ × V.

Explicit formula for the one Hilbert space component W−Theorem
Suppose that σ(H0 ) ∩ σ(H1 ) has Lebsegue measure 0. Then, one has

W− = 1 +1,tanh(A) − i tanh(B) cosh(A)−1 (S − 1) + K,2

where! B := tanh−1 (H0 ),! A is a self-adjoint operator satisfying [B, A] = i1,! the remainder term K is generated by S anda(A)b(B)with a, b ∈ S(R)

Reamrk A corresponds to the generator of dilation groups in thecontinuous setting.
Remark Surprisingly, a completely same formula for the wave operator
holds for Schrödinger operators on Z+ [I-Tsuzu, 2019].

Explicit formula for the scattering matrix S(λ)
CorollarySuppose that σ(H0 ) ∩ σ(H1 ) has Lebsegue measure 0. Then, one hasS(λ) =1 + 2e+iθ M (λ),
1 + 2e−iθ M (λ)

a.e. λ ∈ [−1, 1],where! λ = cos(θ) with θ ∈ [0, π],! M (z) is “the resolvent of H1 restricted to the boundary”, that is,M (z) := ΠQR1 (z)Q∗ Π∗
with Π : H0 → %2 (V) defined by∞)x=0

δx ⊗ ψx /→ ψ0 ,

ψx ∈ %2 (V).

As a consequence, the scattering operator S is unitary.

Remarks
Corollary
If σ(H1 ) ∩ σ(H0 ) is a finite set, then the operator-valued function.
/[−1, 1] 0 λ /→ S(λ) ∈ B %2 (V)is smooth in norm.

Remark Our assumption is not optimal.

Example (attach wires everywhere)
Let H1 = %2 (V) and Q = Π∗ . Then, Q∗ Π∗ = ΠQ = 1V and therefore
S(λ) =

H1 − λ + 2e+iθ.
H1 − λ + 2e−iθ

−→ S is unitary even if H1 has ac spectrum on [−1, 1].

Topological Levinson’s TheoremC ∗ -algebraic framework cf. [Kellendonk-Richard,2007]
Assumption
Throughout this section we suppose that! σ(H1 ) ∩ σ(H0 ) is a finite set,
! N := #V < ∞.

As a consequence, (−1, 1) 0 λ /→ S(λ) ∈ MN (C) is smooth.
Let R := [−∞, ∞] be the two-point compactification of R.
! We define a C ∗ -algebra E on H0 by
102. /
1E := C ∗ a(A)b(B)c 1 a, b ∈ C R , c ∈ MN (C) ,! The set of compact operators on H0 is also expressed as1021K(H0 ) = C ∗ a(A)b(B)c 1 a, b ∈ C0 (R), c ∈ MN (C) .
! W− ∈ E and K ∈ K.
m

Quotient algebra.
/! E/K(H0 ) ∼= C(") ⊗ MN (C) with " = ∂ R × R ∼= T.
.
/! The quotient map π : E → C "; MN (C) is given by.
/ .
/π a(A)b(B) = a(−∞)b, b(+∞)a, a(+∞)b, b(−∞)a .

YA)

Tf"t o" i s Sotann"→o(B)! π(W− ) =: Γ ≡ (S ◦ tanh, Γ+ , 1, Γ− ),! The functions Γ± are given by.
/Γ± (t) = 1 − E± − tanh(t) ∓ i cosh(t)−1 E± ,t ∈ R,

where E± are projections (onto the set of resonances at ±1).

Relative topological Levinson’s theorem
We get the following index theorem not for the wave operator W− butfor the one Hilbert space component W− :Theorem
Under Assumption (A), Γ is a unitary element in C(") ⊗ MN (C).
Moreover, one has.
/Wind det(Γ) = − Index(W− )Remark With the notation s(λ) = det(S(λ)), the l.h.s. is given by.

/1Wind det(Γ) =2πi" 1−1s% (λ)s(λ)∗ dλ − δTopological interpretation of Childs-Levinson theoremConsider Childs’ graph model (i.e. #V, dim H1 < ∞).
! J : H0 '→ H is Fredholm with Index(J) = − dim H1 .

! Since W− = JW− + (finite-rank), W− is also Fredholm.
! Combining this with the completenessHsc (H) = 0andcoker(W− ) = Hpp (H).

Then, the r.h.s. of topological Levinson’s theorem is given by− Index(W− ) = − Index(J ∗ W− )= − Index(W− ) + Index(J)= dim coker(W− ) − dim H1= dim Hpp (H) − #{vertices of G1 }Conclusion Levinson’s theorem for Childs’ graph model is also arealization of the index theorem!

Work in progress: Bulk-edge correspondence via two
spaces scatteringeoccoThe model (revisited)

The model (Wires + (1 + d)-dim insulator)! Hbulk ! %2 (Z1+d ), periodic with a gap on σ(H0 ),! H1 := Hedge ! %2 (Z+ × Zd ), Zd -periodic with gapless edge states,! Wires are also attached periodically to the boundary V = Zd .

M tdue 前c owires
ned！
insulatingbulk（matingUnitarity of S for edge Hamiltonians
By the Floquet-Bloch decomposition w. r. t. the Zd -action,"""∼∼∼
H0 =H0 (k)dk, H1 =H1 (k)dk, H =H(k)dk,
W± ∼
="

Td

Td

Td

W± (k)dk,! H0 (k) = h0 .

W± ∼="Td

Td

W± (k)dk,

S∼
="

Td

S(k)dk.

! For each k ∈ Td , σ(H1 (k)) ∩ σ(H0 ) is finite.

! The scattering operator S(k) for the pair (H0 (k), H(k)) is unitary.

−→ S is unitary for edge states.

(Edge states are absorbed into the conducting wires)
Remark For H1 = Hedge , H0 can be replaced with (1 + d)-dim
half-space Laplacian ∆Z+ ×Zd (metal-insulator junction). Indeed, one
reduces the system to the generalised Childs’ graph model with #V = 1
for each fixed quasi-momentum k ∈ Td .

Work in progress: Levinson and bulk-edge correspondence
Levinson’s theorem in disordered system
Topological Levinson’s theorem for periodic edge Hamiltonians can beformulated by using trace per unit volume along the boundary Zd .
−→ Is the scattering operator S still unitary for disordered systems?

Bulk-edge correspondence via scattering theory
The Chern number of the reflection coefficient Rλ+iε , which also related
to the scattering part of H, is equal to the bulk and edge invariants[Schulz-Baldes & Toniolo ‘21].
−→ (CONJECTURE) For λ ∈ σ(H0 ), at least in K-theory levelS(λ) = Rλ+i0?

Conclusion! For fairly general class of Hamiltonians H1 we obtained(H1 , Q) /→ W± , W± , S.

! Topological Levinson’s theorem by considering the one Hilbert spacecomponent W− = J ∗ W− .
! Potential applications to the bulk-edge correspondence.
References
A. M. Childs, D. J. Strouse, Levinson’s theorem for graphs,
J. Math. Phys. 52 (2011).
A. M. Childs, D. Gosset, Levinson’s theorem for graphs II,
J. Math. Phys. 52 (2012).
H. Inoue, N. Tsuzu, Schrödinger Wave Operators on the Discrete
Half-Line, Integr. Equ. Oper. Theory 91, 1-12 (2019).
J. Kellendonk, S. Richard, The topological meaning of Levinson’s theorem,
half-bound states included, J. Phys. A: Math. Theo. 41, (2008).
H. Schulz-Baldes, D. Toniolo, Dimensional reduction and scatteringformulation for even topological invariants, Commun. Math. Phys. 381,
(2021).