Non-commutative normal forms and inverse
spectral problems.
Anatoly Anikin
RAS Ishlinsky Institute for Problems in Mechanics, Moscow,Russia
2016, Dynamics in Siberia, Novosibirsk
A. Anikin
Normal forms and spectral problems.
Schrödinger operator with potential well
The main object is the one-dimensional Schrödinger operator:
Ĥ =
p̂2
+ V (x),
2
p̂ = −ih
d
dx
where the potential V (x) is an even degree polynomial s.t.
V (x) > 0 for x > 0, V (0) = V ′ (0) = 0 and V ′′ (0) = 1. Its
spectrum is discrete, its ’low lying eigenvalues’ (i.e. lying in
[0, Ch]) are
Ĥψν = Eν ψν ,
∞
X
Eν ∼ h ν + 1/2 +
Eνk hk
k=2
as h → 0. Number ν ∈ Z+ is fixed.
A. Anikin
Normal forms and spectral problems.
The goal
Ĥψν = Eν ψν ,
∞
X
Eν ∼ h ν + 1/2 +
Ekν hk
k=2
1. New method for calculating Ekν via Taylor coefficients of V
based on an analog of Birkhoff normal forms in a
non-commutative algebra.
2. New results about the growth of coefficients Ekν .
3. New inverse results: how to recover V from the knowledge of
Ekν ?
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Normal forms and spectral problems.
Formal graded Heisenberg algebra
Definition
The formal graded Heisenberg algebra
H is an algebra of formal
P
fz z, zj ∈ {p, x}, fz ∈ C,
non-commutative series F =
z=z1 ...zk
with identities rp = pr, rx = xr, where r = px − xp.
Remark
The grading: H =
∞
Q
Hk (or informally H =
∞
L
Hk ). Here Hk is
k=0
k=0
the space of degree k homogeneous forms. E.g.
deg p = deg x = 1, deg r = 2.
Remark
d
, r = −ih.
In quantum mechanics p = −ih dx
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Normal forms and spectral problems.
Lie algebra structure
Proposition
For any F, G there is a unique W, s.t. [F, G] ≡ FG − GF = rW.
Definition
(Quantum) Poisson bracket: {F, G} := W (or informally
.)
{F, G} = FG−GF
r
Remark
deg F = n, deg G = m ⇒ deg {F, G} = n + m − 2.
Definition
Involution:
F = F∗ .
P
fz z1 . . . zk
∗
=
P
fz zk . . . z1 . F is Hermitian, if
Remark
There is a ’homomorphism into classics’ π : H → O. Here O is an
algebra of formal Taylor series with standard product and formal
Poisson bracket.
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Normal forms and spectral problems.
Normal form in H
Definition P
eadW H :=
∞ adW k
k=0 k! H
≡ H + {W, H} + 12 {W, {W, H}} + . . .,
Proposition (D. V. Treschev, 2005)
If W ∈
∞
Q
Hk , then eadW : H → H is an automorphism of
k=3
non-commutative algebra and Lie algebra.
Proposition (A., 2009)
Assume that
∞
P
Hk , H2 =
H = H∗ = H2 +
k=3
2N
P+1
a Hermitian W =
1
2
p2 + x2 , deg Hk = k. There is
Wk s.t.
k=3
adW
e
H = H2 + G + R,
{G, H2 } = 0,
R∈
∞
Y
k=2N +2
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Normal forms and spectral problems.
Hk .
Spectrum via normal form (idea)
Let us think that we deal with operators.
i
eadW H = U ∗ NU, N = H2 + G + R, U = e− h W
U is unitary ⇒ N is self-adjoint with the same spectrum as H.
e = H2 + G. Since [N,
e H2 ] = 0, the
Let us truncate: N
e
eigenfunctions ψν of H2 are eigenfunctions of N.
Lemma (Easy exercise)
If Rk ∈ Hk , then kRψn ukL2 (R) = O(hk/2 ).
Let G =
N
P
G2k , then G2k ψν = ak (ν)hk ψν , and we arrive at
k=2
N
P
Eν = h ν + 1/2 +
ak (ν)hk + O(hN +1 ).
k=2
A. Anikin
Normal forms and spectral problems.
Spectrum via normal form
Theorem
2
b = p̂ + V (x̂), where V (x) ≥ 0 is an even degree polynomial
Let H
2
b x̂) be the
s.t. V (0) = V ′ (0) = 0, and V ′′ (0) = 1. Let G(p̂,
differential operator of degree 2N that corresponds to the normal
b has a series of eigenvalues
form of Ĥ. Then H
Eν = λν + O(hN +1 ), where Ĝψν = λν ψn u, and ψν are
2
2
eigenfunctions of Ĥ2 = p̂2 + x̂2 .
The idea of proof (suggested by V.E. Nazaikinskii) is based on
lifting a formal normal form into a pseudo-differential algebra of
ordered symbols.
Remark
In a different language: S. vu Ngoc, L. Charles, 2008.
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Normal forms and spectral problems.
Explicit formulas
Let us expand Hk =
k
L
s=0
H(s,k−s) (H(s,k−s) is generated by words,
where p appears s times, and x appears k − s times).
Proposition
The following system is a basis in H(s,r):
ps xr , ps−1 xr p, . . . , pxr ps−1 , xr ps , if s ≤ r.
xr ps , xr−1 ps x, . . . , xps xr−1 , ps xr if r ≤ s.
Definition
This basis is called primitive.
Remark
A traditional approach of hk expansions and symbol maps means
using the basis: ps xr , rps−1 xr−1 , . . . , rs−1 pxr−s+1 , rs xr−s .
A. Anikin
Normal forms and spectral problems.
Birkhoff normal form: reminder
To reduce a classical Hamiltonian H =
a normal form, we use steps as follows:
(1) Linear change of variables: p =
P∞
k=2 Hk ,
a−√
+ia+
,
2
adW3
H2 =
x=
p2 +x2
2
to
ia−√+a+
.
2
(2) A sequence of near-identity maps: e
, eadW4 , . . .:
adWk
k−1
e
: a− = ã− + O(ã− , ã+ ) , a+ = ã+ + O(ã− , ã+ )k−1
(3) Inverse change of variables to step (1): (a− , a+ ) → (p, x).
∞
N
k=3
k=3
X
X
p 2 + x2 X
(1)
ek +
e k (2)
H(p, x) =
+
G
H
→ ia− a+ +
Hk → ia− a+ +
2
∞
k=3
(3)
O(a− , a+ )N +1 →
G2k
p 2 + x2
+
2
N
X
k=3
Gk + O(a− , a+ )N +1 ,
e2k = g̃2k (a− a+ )k , G2k+1 = G
e2k+1 = 0.
= g2k (p + x ) , G
2
2 k
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Normal forms and spectral problems.
Non-commutative analog
After the changes of coordinates reducing H ∈ H to a normal
form: (p, x) → (a− , a+ ) → (ã− , a+ ) H maps to the form:
N
X
e = i (a− a+ + a+ a− ) +
G2k (a− , a+ ) + O(a− , a+ )2N +2
H
2
k=2
, where G2k =
Theorem
2k
P
s=0
s 2k 2k−s .
α2k
s a+ a− a+
2
The low lying eigenvalues of Ĥ = p̂2 + V (x) are
N
P
Ekν hk + O(hN +1 ), where
Eν = h ν + 1/2 +
k=2
Ekν
k
= k!(−i)
k X
ν+k−s
s=0
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k
α2k
s .
Normal forms and spectral problems.
Application 1: Growth of coefficients
Proposition (D. V. Treschev, 2005)
For any monomial z of type (r, k) the linear combination
k
k
P
P
z=
αs xs pr xk−s is convex, i.e. αs ≥ 0 and
αs = 1.
s=0
s=0
Remark
The expansions in hk are much worse. E.g. in the sum
k
k
P
P
xk pk =
βs rs pr−s xk−s also
βs = 1 but βs may be ∼ k!.
s=0
s=0
Definition
A norm in H(r,k) is kFk := inf
k
P
zs s=0
Remark
kFk :=
k
P
s=0
|βs |, where F =
k
P
s=0
βs zs .
|αs |
A. Anikin
Normal forms and spectral problems.
Application 1: Growth of coefficients
Definition
An element F =
kF(r,k) k ≤
∞
P
F(r,k), where F(r,k) ∈ H(r,k)
r,k=0
βγ r+k for some constants β, γ > 0.
is analytic, if
Remark
If F is analytic, then πF is a convergent Taylor series.
Theorem (A., 2009)
In one degree of freedom a normal form of an analytic element is
analytic.
Corollary
k
Coefficients of the normal form in primitive basis: |α2k
s | ≤ βγ
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Normal forms and spectral problems.
Application 1: Growth of coefficients
Recall that
Eν (h) ∼
∞
X
hk Ekν ,
Ekν
k
= k!(−i)
k X
ν+k−s
s=0
k=1
k
α2k
s
Corollary
|Ekν | ≤ (ν + 1) . . . (ν + k)βγ k ≤ k!βγ k .
Example: Ek0 = k!(−i)k α2k
k .
Question
N
P
hk Ekν ?
What can be said about the remainder Eν (h) −
k=1
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Normal forms and spectral problems.
Application 2: Inverse spectral problem
Recall that V (x) is an even degree polynomial s.t.
V (0) = V ′ (0) = 0 and V ′′ (0) = 1.
Proposition
ν =
Denote γk,s
Ekν
ν+k−s
k
k
s
and τsk = 2k2 + 7k + 16s(k − s), then
k
k
X
X
(2k)!
k u3 u2k−1
ν
ν
γk,s τs
γk,s −
u2k
+ Eekν
= 2k
2 k!
3k
s=0
s=0
where Eekν depends only on u3 , . . . , u2k−2 .
Proposition
k
P
ν
γk,s
s=0
P
k
µ
γk,s
s=0
k
P
s=0
k
P
s=0
ν τk
γk,s
s
µ k
γk,s
τs
6= 0 for all µ 6= ν and k ≥ 2.
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Normal forms and spectral problems.
Application 2: Inverse spectral problem
Theorem
Fix ν 6= µ. In the class of potentials s.t. V (3) (0) 6= 0, sequences
Ekµ and Ekν determine V (x) uniquely up to the x → −x symmetry.
Remark
Known results: V (x) is determined by Ekµ for all µ ≥ 0 and k ≥ 2
(Y. Colin de Verdiere, H. Hezari, 2008). (The assumption
V (3) (0) 6= 0 is also imposed).
Remark
In the degenerate case V (3) (0) = 0 one should consider first
V (5) (0) 6= 0. This leads also to a 2 × 2 determinant but more
complicated.
A. Anikin
Normal forms and spectral problems.