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Adiabatic theorems with and without spectral gap condition for
non-semisimple spectral values
J. Schmid
Fachbereich Mathematik, Universität Stuttgart,
D-70569 Stuttgart, Germany
[email protected]
We present adiabatic theorems with and without spectral gap condition for operators
A(t) : D(A(t)) ⊂ X → X in a Banach space X. In the case with spectral gap, the
considered spectral values λ(t) ∈ σ(A(t)) are not required to be semisimple and, in
particular, need not be eigenvalues. In the case without spectral gap, the considered
eigenvalues λ(t) ∈ ∂σ(A(t)) are not required to be weakly semisimple. We also discuss
adiabatic theorems for operators A(t) with time-dependent domains and an application
to operators defined by symmetric sesquilinear forms.
Keywords: Adiabatic theorems with and without spectral gap condition, not necessarily
(weakly) semisimple spectral values, time-dependent domains.
1. Introduction
Adiabatic theory for general – as opposed to skew self-adjoint [1–3] – operators is a
quite young area of research which emerged in [4] and was developed further in [5–7].
(In the complex time method [8, 9] in the spirit of [10] non-self-adjoint operators
appear as well, yet as auxiliary objects.) In general terms, adiabatic theory is
concerned with linear operators A(t) : D(A(t)) ⊂ X → X in a Banach space X
over , spectral values λ(t) of A(t) and projections P (t) in X (for t ∈ I := [0, 1])
such that
• A(t) is densely defined closed for every t ∈ I and the initial value problems
x = A(εs)x
1
1
(s ∈ [0, ]) or x = A(t)x
ε
ε
(t ∈ [0, 1])
with initial conditions x(s0 ) = y or x(t0 ) = y are well-posed on the
spaces D(A(t)) for every value of the slowness parameter ε ∈ (0, ∞)
(Condition 1.1).
• P (t) is spectrally related with λ(t) in some natural sense (specified below)
for every t ∈ I except possibly for some few t (Condition 1.2).
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What one basically wants to know in adiabatic theory is the following: when –
under which additional conditions on A, λ and P – does the evolution Uε generated
by the operators 1ε A(t) (Condition 1.1) approximately follow the spectral subspaces
P (t)X related to the spectral values λ(t) of A(t) (Condition 1.2) as the slowness
parameter ε tends to 0? Shorter and more precisely: when is it true that
(1 − P (t))Uε (t)P (0) −→ 0
(ε 0)
(with respect to a certain operator topology) for all t ∈ I? Adiabatic theorems are,
by definition, theorems that give such conditions. We distinguish adiabatic theorems with spectral gap condition (uniform or non-uniform) and adiabatic theorems
without spectral gap condition depending on whether λ(t) is isolated in σ(A(t)) for
every t ∈ I (uniformly or non-uniformly) or not.
We present, in this paper, some new adiabatic theorems for general operators A(t)
carrying further the adiabatic theory from [5, 7] mainly in two respects:
• first, by no longer requiring the considered spectral values λ(t) of A(t) to
be (weakly) semisimple
• second, by no longer requiring the domain of A(t) to be time-independent.
We do not strive for the utmost generality and do not give complete proofs here
(which can be found in [11]), but instead try to convey the central ideas.
2. Spectral-theoretic preliminaries
In this preliminary section, we identify a natural notion of spectral relatedness –
namely, associatedness in the case with spectral gap and weak associatedness in
the case without spectral gap – thereby specifying the vague Condition 1.2 above.
Suppose that A : D(A) ⊂ X → X is densely defined with ρ(A) = ∅ and λ ∈ σ(A).
If λ is isolated in σ(A), then a projection P is called associated with A and λ iff P
commutes with A and AP := A|P D(A) is bounded such that
σ(AP ) = {λ} whereas σ(A1−P ) = σ(A) \ {λ}.
Additionally, λ is called semisimple iff λ is a pole of ( . − A)−1 of order 1. If λ is
not necessarily isolated in σ(A), then a projection P is called weakly associated (of
order m) with A and λ iff P commutes with A and AP is bounded such that
AP − λ is nilpotent (of order m) whereas A1−P − λ is injective with dense range.
Additionally, λ is called weakly semisimple iff there is a projection P weakly associated with A and λ of order 1, that is, (AP − λ)1 = 0. We have the following
central properties of associatedness (which are completely well-known) and of weak
associatedness (which seem to be new).
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• If λ is isolated in σ(A), then there
exists a unique projection P associated
with A and λ, and P = (2πi)−1 γ (z − A)−1 dz (Riesz projection). If P is
associated with A and λ, and λ is a pole of ( . − A)−1 of order m, then
P X = ker(A − λ)k
and (1 − P )X = rg(A − λ)k
(k ≥ m).
• In general, there exists no projection P weakly associated with A and λ,
but if it exists it is unique. If P is weakly associated with A and λ of order
m, then
P X = ker(A − λ)k
and (1 − P )X = rg(A − λ)k
(k ≥ m).
Clearly, associatedness is a completely natural notion of spectral relatedness in the
case with spectral gap. In order to see that weak associatedness is equally natural in
the case without spectral gap, notice that it is certainly natural to require that AP
be bounded with σ(AP ) = {λ} (in other words, that AP − λ be quasinilpotent) and
that A1−P − λ be injective – and from this one naturally arrives at weak associatedness by further requiring that AP − λ be nilpotent (instead of only quasinilpotent)
and that λ belong to the continuous spectrum of A1−P (instead of the residual
spectrum). We conclude this section with the following remarks.
• If A is a spectral operator [12] (with spectral measure P A ) and λ ∈ σ(A)
such that for some bounded neighborhood σ of λ the bounded spectral
operator A|P A (σ)X is of finite type, then the projection weakly associated
with A and λ exists and is given by P A ({λ}). In particular, this is true if
A is spectral of scalar type.
• If P is associated with A and λ, then the dual operator P ∗ is associated
with A∗ and λ. An analogous implication holds for weak associatedness,
provided that A is a semigroup generator and X is reflexive.
3. Adiabatic theorems . . .
3.1. . . . with spectral gap condition
We begin with an adiabatic theorem with spectral gap condition, which treats uniform and non-uniform spectral gaps alike and which uses the convenient terminology
of λ( . ) falling into σ(A( . )) \ {λ( . )} at a point t0 (which means there is a sequence
(tn ) converging to t0 such that dist(λ(tn ), σ(A(tn )) \ {λ(tn )}) −→ 0 as n → ∞). It
is a quite simple generalization of the adiabatic theorem of Abou Salem [5] covering
the case of semisimple spectral values. Its proof rests upon solving an appropriate
commutator equation.
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Theorem 3.1. Suppose that
• A(t) : D ⊂ X → X for every t ∈ I generates a contraction semigroup and
t → A(t)x is continuously differentiable for every x ∈ D
• λ(t) for every t ∈ I is a spectral value of A(t), λ( . ) falls into σ(A( . )) \
{λ( . )} at only countably many points accumulating at only finitely many
points, and t → λ(t) is continuous
• P (t) for every t ∈ I \ N is associated with A(t) and λ(t) and I \ N t → P (t) extends to a twice strongly continuously differentiable map on the
whole of I, where N denotes the set of those points where λ( . ) falls into
σ(A( . )) \ {λ( . )}.
If Vε denotes the (adiabatic) evolution system for
sup Uε (t) − Vε (t) = O(ε)
or
1
εA
+ [P , P ] on D, then
o(1)
(ε 0),
t∈I
depending on whether N = ∅ (uniform spectral gap) or N = ∅ (non-uniform gap).
Sketch of proof. Since the case with non-uniform gap can be reduced by a standard argument to the case of uniform gap, we have only to consider this latter case.
Setting
B(t) :=
1
2πi
(z − A(t))−1 P (t)(z − A(t))−1 dz
γt
with cycles γt in ρ(A(t)) encircling λ(t) but not σ(A(t)) \ {λ(t)}, we easily obtain
(central properties of associatedness!) the commutator equation BA−AB ⊂ [P , P ].
With this commutator equation, in turn, we can rewrite the difference Vε − Uε as
τ =t
Vε (t) − Uε (t) = ε Uε (t, τ )B(τ )Vε (τ )τ =0
−ε
0
t
Uε (t, τ ) B (τ ) + B(τ )[P (τ ), P (τ )] Vε (τ ) dτ,
from which the desired conclusion follows by the boundedness of Uε and Vε in ε.
We refer to [11] for simple examples of the above theorem where the previously
known adiabatic theorems [5–7] cannot be applied. In particular, these examples
show that the considered spectral values λ(t) may well be essential singularities of
the resolvent of A(t) and need not even be eigenvalues. A similar – but weaker –
theorem is a corollary of an adiabatic theorem of higher order by Joye [6] where the
operators A(t) have to be required to analytically depend on t and the considered
spectral values are required to be poles of the resolvent.
3.2. . . . without spectral gap condition
We now present the main result of this paper: an adiabatic theorem without spectral
gap condition for not necessarily weakly semisimple eigenvalues. It generalizes an
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adiabatic theorem independently obtained by Avron, Fraas, Graf, Grech [7] and by
Schmid [13], which covers the case of weakly semisimple eigenvalues. Its proof rests
upon solving an appropriate approximate commutator equation.
Theorem 3.2. Suppose that
• A(t) : D ⊂ X → X for every t ∈ I generates a contraction semigroup and
t → A(t)x is continuously differentiable for every x ∈ D.
• λ(t) for every t ∈ I is an eigenvalue of A(t) such that λ(t)+δeiϑ(t) ∈ ρ(A(t))
for every δ ∈ (0, δ0 ] and t → λ(t), eiϑ(t) are continuously differentiable
• P (t) is weakly associated with A(t) and λ(t) for almost every t ∈ I,
M
−1
0
(δ ∈ (0, δ0 ]),
(1 − P (t)) ≤
λ(t) + δeiϑ(t) − A(t)
δ
rk P (0) < ∞ and t → P (t) is continuously differentiable.
If X is reflexive, then
sup Uε (t) − Vε (t) −→ 0
(ε 0),
t∈I
whenever the (adiabatic) evolution system Vε for 1ε A + [P , P ] exists on D. If X is
arbitrary, then supt∈I (1 − P (t))Uε (t)P (0) −→ 0 as ε 0.
Sketch of proof. We may assume by a standard approximation argument [3] using
mollification that t → P (t) is twice continuously differentiable (in which case the
existence of Vε is guaranteed). Setting m0 := rk P (0) = rk P (t) (indepent of t!) and
Bδ :=
m
0 −1 k+1
i=1
k=0
m
0 −1
k+1
Rδi P P (λ − A)k +
(λ − A)k P P Rδi
k=0
i=1
where δ := (δ1 , . . . , δm0 ) and Rδ := (λ + δ − A)−1 (1 − P ), we obtain from
P X = ker(A − λ)m0 (central properties of weak associatedness!) the approximate
commutator equation Bδ A − ABδ ⊂ [P , P ] − Cδ , where Cδ := Cδ+ − Cδ− with
Cδ+ :=
m
0 −1
k=0
δk+1
k+1
i=1
m
0 −1
k+1
Rδi P P (λ − A)k , Cδ− :=
(λ − A)k P P δk+1
Rδi .
k=0
i=1
At first glance, this solution Bδ of the approximate commutator equation might
seem a bit mysterious, but actually, it can be guessed from the solution B of the
commutator equation in the case with spectral gap: indeed, if λ is a pole of ( . −A)−1
of order at most m0 , then B can be seen to be equal to Bδ=0 with the help of
Cauchy’s theorem. With the above approximate commutator equation, we can now
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rewrite the difference Vε − Uε as
τ =t
Vε (t) − Uε (t) = ε Uε (t, τ )Bδ (τ )Vε (τ )τ =0 − ε
· [P (τ ), P (τ )] Vε (τ ) dτ +
0
t
0
t
Uε (t, τ ) Bδ (τ ) + Bδ (τ )·
Uε (t, τ )Cδ (τ )Vε (τ ) dτ
and with the assumed resolvent estimate we obtain the estimates
Bδ (t) ≤ c (δ1 · · · δm0 )−1 ,
1
0
Bδ (t) ≤ c (δ1 · · · δm0 )−(m0 +1) ,
m
0 −1
± C (τ ) dτ ≤ c
(δ1 · · · δk )−1 η ± (δk+1 )
δ
k=0
1 1 where η + (δ) := 0 δ Rδ (τ )P (τ )P (τ ) dτ and η − (δ) := 0 δ P (τ )P (τ )Rδ (τ ) dτ
for δ ∈ (0, δ0 ]. With (1 − P )X = rg(A − λ)m0 and (1 − P ∗ )X ∗ = rg(A∗ − λ)m0
(weak associatedness for semigroup generators in reflexive spaces carries over to the
dual operators!) and with the assumed resolvent estimate one can show that
δ Rδ (t),
δ Rδ (t)∗ −→ 0 (δ 0)
w.r.t. the strong operator topology for almost every t ∈ I, and since rk P (t)∗ =
rk P (t) < ∞, one even gets η ± (δ) −→ 0. Choosing now δ = δ ε = (δ1 ε , . . . , δm0 ε ) in
a suitable way, we arrive at the desired conclusion.
We refer to [11] for a wealth of simple examples of the above theorem where the
previously known adiabatic theorems [7, 13] cannot be applied (finding interesting
applied examples will be the subject of future research). In these examples the
operators A(t) need not be spectral, but for spectral operators the construction of
examples is particularly simple: one can ensure the assumptions – in particular, the
reduced resolvent estimate – of the above theorem by choosing A(t) for every t as a
spectral operator (with spectral measure P A(t) ) generating a contraction semigroup
and λ(t) as an eigenvalue of A(t) such that, apart from regularity conditions,
• A(t)|P A(t) (σ(t))D is spectral of scalar type for some punctured neighborhood
σ(t) := σ(A(t)) ∩ U r0 (λ(t)) \ {λ(t)}
of λ(t) in σ(A(t)) (r0 ∈ (0, ∞) ∪ {∞}) and rk P A(t) ({λ(t)}) < ∞ for a. e. t,
• the open sector {λ(t) + δeiϑ : δ ∈ (0, δ0 ), ϑ ∈ (ϑ(t) − ϑ0 , ϑ(t) + ϑ0 )} of radius
δ0 ∈ (0, ∞) and angle 2ϑ0 ∈ (0, π) is contained in ρ(A(t)).
In the special case where the eigenvalues λ(t) from the above theorem are purely
imaginary for every t ∈ I, these eigenvalues are automatically weakly semisimple
by the contraction semigroup assumption, and so, in this special case, the adiabatic
theorem above reduces to the theorem from [7, 13]. Simple examples [6, 11] show
that the contraction semigroup assumption cannot be essentially weakened.
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3.3. . . . for time-dependent domains
So far, we have always considered operators A(t) with time-independent domains.
It is possible to quite easily extend all the presented adiabatic theorems to operators
with time-dependent domains, which in essence is due to the following facts:
• An evolution system W = (W (t, s)) for operators with time-dependent domains is right-differentiable w.r.t. the second variable s in some appropriate
sense
• A continuous, right-differentiable map f : [a, b] → X with bounded right
derivative ∂+ f belongs to the Sobolev class W 1,∞ and therefore the fundamental theorem of calculus is available
We do not state these extended adiabatic theorems with their precise assumptions
(which can be found in [11]), but instead give an application to the special case of
skew self-adjoint operators A(t) = iAa(t) defined by symmetric sesquilinear forms
a(t) (Schrödinger operators with Rollnik potentials V (t), for instance). Consider
the situation, where H + is continuously and densely embedded in H (both Hilbert
spaces), where a(t) is a symmetric sesquilinear form on H + such that, for some
m ∈ (0, ∞),
. , .. +
t := a(t)( . , .. ) + m . , .. +
is a scalar product on H + and . +
for every t ∈ I, and
t is equivalent to . where t → a(t)(x, y) is twice continuously differentiable for all x, y ∈ H + . We then
have the following adiabatic theorem without spectral gap condition, which is a
generalization of a theorem of Bornemann [2] where the considered eigenvalues are
required to belong to the discrete spectrum. It can be proved in a considerably
simpler way than in [2], namely by applying the general adiabatic theorems for
time-dependent domains mentioned above.
Theorem 3.3. Suppose that
• A(t) = iAa(t) : D(A(t)) ⊂ H → H with a(t) as above
• λ(t) for every t ∈ I is an eigenvalue of A(t) such that t → λ(t) continuous
• P (t) is weakly associated with A(t) and λ(t) for almost every t ∈ I,
rk P (0) < ∞ and t → P (t) is continuously differentiable.
Then
sup (1 − P (t))Uε (t)P (0) ,
t∈I
sup P (t)Uε (t)(1 − P (0)) −→ 0
t∈I
(ε 0).
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