**1.3 Hamiltonian perturbation theory**

**1.3.1 Application to quantum mechanics**

The formalism of Hamiltonian perturbation theory just developed can be applied to quantum
mechanics once one identifies the relevant quantum objects such as Hamiltonian flows and
canonical transformations. To such a purpose, we start by observing that unitary
transforma-tions of the wave functionΨ, the unknown of the Schrödinger equation, are canonical
trans-formations of the latter equation. Indeed, given any unitary operator ˆUindependent of time,
and definingΨ^{0}:= ˆU^{†}Ψ^{, ˆ}H^{0}= ˆU^{†}H^{ˆ}U^{ˆ}, one has

ıħΨt= ˆHΨ ⇐⇒ ıħΨ^{0}t= ˆH^{0}Ψ^{0}^{,} ^{(1.75)}
the equation on the right being identical in form to that on the left. Thus, in the perturbative
context where ˆH_{= ˆ}h* _{+ λ ˆ}*V+ · · · , one can try to remove the leading order perturbation ˆV (and
then the higher order contributions as well) by looking for a particular unitary operator ˆU

*that conjugates ˆH to its normal form, to leading order. In particular, by analogy with the classical case one looks for a unitary operator ˆU*

_{λ}*that is the Schrödinger flow at time*

_{λ}*λ of some*unknown Hamiltonian (Hermitian) operator ˆG, namely

Uˆ_{λ}_{= e}^{−ıλ ˆ}^{G}^{/ħ} (1.76)

(we hare recall that the solution of the Schrödinger equation ıħΨt= ˆHΨis formally given by
Ψ(t) = e^{−ıt ˆ}^{H}^{/ħ}Ψ(0)). To any Hermitian operator ˆGone can associate the operator

LˆGˆ := −ı

ħ£ , ˆG¤ , (1.77)

i.e. the quantum Lie derivative associated to ˆG. One easily proves the following
**Lemma 1.5. For any pair of Hermitian operators ˆ**Fand ˆGindependent of*λ, one has*

e^{+ıλ ˆ}^{G}^{/ħ}Fˆe^{−ıλ ˆ}^{G}^{/ħ}= e^{λ ˆ}^{L}^{G}^{ˆ}Fˆ . (1.78)
CPROOF. Define ˆF(*λ) the left hand side of (1.78) and take its derivative with respect to λ,*
getting

d

d*λ*Fˆ(*λ) =* ı
ħ

¡GˆF^{ˆ}(*λ) − ˆ*F(*λ) ˆ*G_{¢ = ˆ}L_{G}_{ˆ}F^{ˆ}(*λ) .*

The latter differential equation can be formally integrated with the initial condition ˆF_{(0) = ˆ}F, to
yield (1.78). BIn particular, from the latter Lemma it follows that ˆFis invariant with respect
to the flow of ˆGiff [ ˆF, ˆG] = 0 (prove it). Now, supposing that ˆH_{λ}_{= ˆ}h* _{+ λ ˆ}*Vand transforming it by
Uˆ

*= e*

_{λ}

^{−ıλ ˆ}^{G}

^{/ħ}, one gets

Hˆ^{0}_{λ}_{:= ˆ}U^{†}* _{λ}*H

^{ˆ}U

^{ˆ}

_{λ}_{= e}

^{λ ˆ}^{L}

^{G}

^{ˆ}

^{¡}h

^{ˆ}

*V*

_{+ λ ˆ}_{¢ = ˆ}h

_{+ λ}^{¡}V

^{ˆ}

_{+ ˆ}L

_{G}

_{ˆ}h

^{ˆ}

_{¢ + O(λ}^{2}).

One now requires that the latter expression be in normal form with respect to ˆhto first order,
i.e. that ˆH^{0}_{λ}_{= ˆ}h* _{+λ ˆ}*S

_{+O(λ}^{2}), with ˆSinvariant with respect to the flow of ˆh: ˆL

_{h}

_{ˆ}S

^{ˆ}= 0. Thus, taking into account that ˆL

_{G}

_{ˆ}h

^{ˆ}

_{= − ˆ}L

_{h}

_{ˆ}G

^{ˆ}, one writes down the (first order) quantum homological equation

Sˆ_{= ˆ}V_{− ˆ}LhˆGˆ (1.79)

and solves it for ˆSand ˆGexactly as done in the classical case. The result is
Sˆ_{=}^{}V^{ˆ}^{®}_{h}_{ˆ} _{= lim}

t→+∞

1 t

Z _{t}

0

e^{ħ}^{ı}^{s ˆ}^{h}Vˆe^{−}^{ħ}^{ı}^{s ˆ}^{h}ds ; (1.80)

Gˆ _{= ˆ}G+ lim

t→+∞

1 t

Z t

0 (s − t)e^{ħ}^{ı}^{s ˆ}^{h}¡Vˆ_{−}^{}V^{ˆ}^{®}hˆ¢ e^{−}^{ħ}^{ı}^{s ˆ}^{h}ds. (1.81)
**Exercise 1.13. Prove formulas (1.80) and (1.81).**

It follows that the averaging principle, Theorem 1.1, holds in quantum mechanics to all orders, with a formulation that, up to the replacement of the Poisson bracket with the commu-tator divided by ıħ, is completely analogous to the classical one. In particular, to first order, the perturbation in normal form is the time average of the perturbation along the flow of the unperturbed system.

**Exercise 1.14. Consider the classical anharmonic oscillator defined by the Hamiltonian**

H(x, p) = p^{2}
2m+kx^{2}

2 *+ µ*x^{4}

4 . (1.82)

Quantize the problem, introduce the quantum analogue of the Birkhoff variables and compute the normal form to first order with respect to the quadratic Hamiltonian. Compare the result with the corresponding classical one.

**Chapter 2**

**Applications to Hamiltonian PDEs**

In the sequel we will consider PDEs defined by Hamiltonians of the form
H[u,*π] :=*

Z

ΛH ^{(}*π, u, u*x, u_{xx}, . . . ) dx , (2.1)
where u(t, x) and *π(t, x) are defined on* R ×Λ^{,} Λ being a real domain to be discussed below
together with its related boundary conditions. The Hamiltonian “density”H in the integral
(2.1) is an ordinary function of its arguments, the dots denoting derivatives of u with respect
to x up to some finite order. A dependence on the derivatives of*π with respect to x might be*
included as well. Concerning the space domainΛone typically considers one out of the three
possible cases.

• Λ*= [0, L] with fixed ends at the boundary: u, π|*0,L= 0.

• Λ= [0, L] with periodic boundary conditions: u|0= u|L and *π|*0*= π|*L. Otherwise stated,
Λ= R/(LZ).

• Λ= R, with boundary conditions ensuring the existence of the integral (2.1). For example,
a dependence ofH ^{on}*π proportional to π*^{2}, the typical physical case, requires*π → 0 as*
x → ±∞.

The Hamilton equations of motion associated to (2.1) read µ u

where the functional derivatives*δH/δu and δH/δπ, are defined through the weak or Gateaux*
differential of H by the formula

*δH :=* d

In the latter formula the increments *δu and δπ satisfy the same boundary conditions of the*
fields u and *π. In terms of the Hamiltonian density* H defining the Hamiltonian (2.1), one
easily checks that

where the dots stand for the successive terms of the obvious form. In many applications one
hasH *= π*^{2}/2+V^{(u, u}x, u_{xx}, . . . ), so that the Hamiltonian system (2.2) is equivalent to the wave
equation of second order (with respect to time)

u_{tt}= −*δH*

*δu* = −*∂*V

*∂u* + d
dx

*∂*V

*∂u*x− d^{2}
dx^{2}

*∂*V

*∂u*xx+ · · · ,

**Example 2.1. A simple classical example is that of the so-called Klein-Gordon (KG) equation**

u_{tt}= −m^{2}u + c^{2}u_{xx} , (2.4)

wherem and c are two parameters. One easily proves that the Hamiltonian of such an equation is given by

H = Z

Λ

*π*^{2}+ m^{2}u^{2}+ c^{2}u^{2}_{x}

2 dx. (2.5)

In the case m = 0 the KG equation (2.4) becomes the vibrating string, or wave- equation: utt=
c^{2}u_{xx}.

**Example 2.2. A generalized KG equation (gKG) is given by**

u_{tt}= c^{2}u_{xx}−U^{0}^{(u)}^{,} ^{(2.6)}

whereU is a given function of its argument. The Hamiltonian of the gKG equation is given by

H = Z

Λ

·*π*^{2}+ c^{2}u^{2}_{x}

2 +U^{(u)}

¸

dx. (2.7)

**Exercise 2.1. Consider the wave equation u**_{tt}= c^{2}u_{xx} and its Hamiltonian formulation.
Per-form the first change of variables (u,*π) 7→ (v,π), where v := u*x. Write down the equations of
motion in the new variables; then, determine the new Poisson tensor and the new Hamiltonian,
checking the correctness of the equations of motion. Do the same performing the second change
of variables (v,*π) 7→ (U*^{+},U^{−}), where U^{±}*:= (cv ± π)/*p

2c. Finally, solve the Hamilton equations of motion in the latter case.

**Exercise 2.2. Consider the Boussinesq equation**
u_{tt}= uxx*+ λ*£

*γu*xxxx*+ α(u*^{2}_{x})_{x}¤ , (2.8)

where*γ and α are real parameters, whereas λ is a small parameter. Write down the *
Hamilto-nian H* _{λ}*[u,

*π] = h + λP of the system. Perform the change of variables (u,π) 7→ (v*

^{+}, v

^{−}), where v

^{±}:= (ux

*± π)/*p

2, and write the transformed Poisson tensor and the transformed Hamiltonian.

Compute the normal form Hamiltonian to first order with respect toh. Prove that the Hamilton equations corresponding to the normal form Hamiltonian are the Korteweg - de Vries equations

w^{±}_{t} = ±w^{±}_{x} *± λ*
µ*γ*

2w^{±}_{xxx}+ *α*

p2w^{±}w^{±}_{x}

¶

. (2.9)

31

**Exercise 2.3. Consider the nonlinear, non homogeneous KG equation**

u_{tt}= −m^{2}*u + λ*£u_{xx}− gu^{3}− V (x)u¤ , (2.10)
wherem and g are real parameters, V (x) is a given function and*λ is a small parameter. Write*
the Hamiltonian H* _{λ}*[u,

*π] = h + λP of the system. Perform the change of variables (u,π) 7→*

(*ψ,ψ*^{∗}), where *ψ := (mu + ıπ)/*p

2m and ı denotes the imaginary unit; write the transformed Poisson tensor and the transformed Hamiltonian. Compute the normal form Hamiltonian to first order with respect to h. Prove that the Hamilton equations corresponding to the normal form Hamiltonian are the Gross-Pitaevskii equation

ı*ϕ*t*= mϕ +* *λ*
m

·

*−ϕ*xx*+ V (x)ϕ +* 3g
2m*|ϕ|*^{2}*ϕ*

¸

(2.11) and its complex conjugate.

**Exercise 2.4. Consider the Euler-Bernoulli (EB) equation**

u_{tt}= −b^{2}u_{xxxx} , (2.12)

where b is a real parameter (the equation describes the flexural oscillations of a beam). Write
the Hamiltonian of the system. Perform the change of variables (u,*π) 7→ (v,π), where v := u*xx,
and write the equations of motion in the new variables. Compute the transformed Poisson tensor
and the transformed Hamiltonian and check the Hamilton equations. Do the same performing
the further change of variables (v,*π) 7→ (ψ,ψ*^{∗}), where*ψ := (π+ıbv)/*p

2b. Show that in the latter variables the EB equation transforms into the Schrödinger equation for the free particle.

**Exercise 2.5. Consider the EB equation of the previous exercise with imaginary constant b 7→**

ıb or, which is the same, with imaginary time t 7→ ıt, namely

u_{tt}= b^{2}u_{xxxx} . (2.13)

Write the Hamiltonian of the system, and perform the same change of variables considered
in the previous exercise. Then, perform the change of variables (v,*π) 7→ (ψ*^{+},*ψ*^{−}), where *ψ*^{±}:=

(*π ± bv)/*p

2b. Show that, in the new variables, equation (2.13) transforms into *ψ*^{±}_{t} *= ±bψ*^{±}_{xx}.
Solve the heat (or diffusion) equation satisfied by*ψ*^{+}on the real line, with a Dirac delta function
as initial condition: *ψ*^{+}(0*, x) = Aδ(x). Show that, as a consequence, the time reversed heat*
equation satisfied by *ψ*^{−}, with a smooth initial condition, develops a singularity at a finite,
positive time.

**Exercise 2.6. Consider the gKG equation (2.6) on the real line (**Λ= R). Discuss the existence of
traveling (or solitary) wave solutions of the form*u(t, x) = σ(x − vt), where v is a real parameter*
and*σ a real function of a single real variable. Assume boundary conditions of the form σ(ξ) →*
*σ*± as *ξ → ±∞, the approach to the limit value being smooth: σ*^{0}(*ξ) → 0, σ*^{00}(*ξ) → 0 and so on*
as*ξ → ±∞. Show that such solutions correspond either to the homoclinic connections or to the*
heteroclinic connections of the local potentialU of the equation (2.6), corresponding to the case
*σ*+*= σ*−or*σ*+*6= σ*−, respectively.

**Appendix A**

**Hierarchy of structures in topology and** **their utility**

**Definition A.1. A metric space is a pair (X**, d) where X is a set and d : X ×X → R+is a distance,
i.e. a nonnegative function which satisfies

1. d(x, y) = 0 iff x = y (nondegeneracy);

2. d(x, y) = d(y, x) ∀x, y ∈ X (symmetry);

3. d(x, z) ≤ d(x, y) + d(y, z) ∀x, y, z ∈ X (triangle inequality).

In metric spaces one can define limits of functions and thus continuity. Indeed, given two
metric spaces (X, d) and (X^{0}, d^{0}), one says that a function f : X → X^{0}has (finite) limit*` ∈ X*^{0} as
x tends to x_{0}, and writes lim_{x→x}_{0}*f (x) = `, if for any ² > 0 there exists a δ**²*> 0 such that for all
x ∈ X \ {x0} satisfying d(x, x_{0}*) < δ** _{²}*one has d

^{0}( f (x),

*`) < ². The definition of continuous function*is obtained by setting

*` = f (x*0).

Notice that a metric space is not necessarily a linear (or vector) space. On the other hand,
if one wants to build up differential calculus, one needs the linear structure. Indeed, the
following definition of (weak, or Gateaux) differential of a function f : X → X^{0} in x_{0}∈ X with
increment h ∈ X is quite natural:

d f (x_{0}; h) := lim

t→0

1

t[ f (x0+ th) − f (x0)] .

Observe that one needs the linear combination x_{0}+ th ∈ X and the differential d f (x0; h) ∈ X^{0},
the convergence of the above limit being meant in the metric d^{0} of X^{0}. Actually, one needs
to measure the size of the objects in X and X^{0}, in order to have some control on the above
formula (for example, one would like to have that d f is “small” when h is “small”). This is
achieved when the distances d on the linear spaces X is induced by a norm k · k, such that
d(x, y) = kx − yk.

**Definition A.2. A normed space is a pair (X**, k · k) where X is a linear space and k · k : X → R_{+}
is a norm, i.e. a nonnegative function satisfying

1. kxk = 0 iff x = 0 (nondegeneracy);

33

*2. kλxk = |λ|kxk ∀λ ∈ R and ∀x ∈ X (homogeneity of degree one);*

3. kx − zk ≤ kx − yk + ky − zk (triangle inequality).

One easily checks that a normed linear space is a metric space with the distance

d(x, y) := kx − yk . (A.1)

Differential calculus is performed in normed spaces, but in order to set up all the machinery
of the differential equations one needs a stronger property. More precisely, in order to have
the uniqueness of the solution to a given differential equation one needs the completeness of
the space, namely that every Cauchy sequence converges (recall that a sequence {x_{n}}_{n∈N}⊂ X
*is a Cauchy sequence if ∀² > 0 there exist N** _{²}*∈ N such that for any pair n, m ∈ N satisfying
n ≥ m > N

*²*one has d(xn, xm

*) < ²).*

**Definition A.3. A complete normed space is called Banach space.**

Almost all the classical results in the theory of differential equations is (existence, unique-ness and regularity) hold in Banach spaces. Completeunique-ness enters the game in the proof of the Picard existence and uniqueness theorem, as follows. We recall that a differential equation

˙x(t) = u(x(t)), with initial condition x(0) = x0, is naturally solved by iterating the map P : X → X defined by

x^{0}(t) = P(x) := x0+
Z _{t}

0

u(x(s)) ds ,

with initial point x(s) ≡ x0. Now, P acts on the space B(I, X ) of bounded functions t 7→ x(t),
defined on a suitable real interval I, with value in the Banach space X . Such a space, endowed
with the metric d(x, y) := sup_{t∈I}kx(t)− y(t)k, k·k being the norm on X , turns out to be complete.

A local Lipschitz condition on u guarantees (and is optimal) that P : B → B is a contraction,
i.e. that there exists*ρ ∈]0,1[ such that d(P(x), P(y)) ≤ ρd(x, y) for any x, y close enough to x*0.
Since a contraction in a complete metric space admits a unique fixed point, then there exists a
unique ˆx(t) ∈ B(I, X ) such that ˆx(t) = P( ˆx(t)), i.e. ˆx(t) is the local solution of the given differential
equation.

It may happen that the linear space where the problem at hand is set up possesses a Eu-clidean structure, i.e. a scalar (or internal) product between its elements. When this happens it is very useful, since then concepts like angle, direction and projection become meaningful.

**Definition A.4. A Euclidean space is a pair (X**, 〈,〉) where X is a linear space and 〈,〉 : X ×X → R
is a scalar product, i.e. a real function satisfying

1. 〈x, x〉 > 0 ∀x ∈ X \ {0}, 〈0,0〉 = 0 (nondegeneracy);

2. 〈x, y〉 = 〈y, x〉 ∀x, y ∈ X (symmetry);

3.

*λx + µy, z® = λ〈x, z〉 + µ〈y, z〉 ∀λ,µ ∈ R and ∀x, y, z ∈ X (linearity in the first entry).*

35 One easily checks that a Euclidean space is a normed (and thus a metric, see (A.1)) space with the norm

kxk :=p

〈x, x〉 . (A.2)

Two vectors x, y of (X , 〈,〉) are said to be mutually orthogonal if 〈x, y〉 = 0. In a Euclidean space,
for example, the Pythagoras theorem holds, namely kx + yk^{2}= kxk^{2}+ kyk^{2} for all pairs x, y of
mutually orthogonal vectors.

Of course, the Euclidean structure is very useful in dealing with differential equations, whose appropriate environment, as explained above, is the Banach space. This justifies the introduction of a stronger topological structure, namely that of a Euclidean space which is complete with respect to the norm (A.2) naturally induced by the scalar product, which one can shortly refer as to a Euclidean-Banach space.

**Definition A.5. A Euclidean-Banach space is called a Hilbert space.**

**Remark A.1. Wherever needed, the linear structure of the space X can be considered on the**
complex fieldC, which only requires the symmetry property 2. of the scalar product to change
as follows: 〈x, y〉 = 〈y, x〉^{∗}, a star denoting complex conjugation.

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[4] V. I. Arnol’d, Lectures on Partial Differential Equations, Springer, 2004.

[5] A. Fasano and S. Marmi, Analytical Mechanics, Oxford University Press, 2006.

[6] J. N. Franklin, Matrix Theory, Dover, 1968.

[7] F. R. Gantmacher, Lezioni di Meccanica Analitica, Editori Riuniti, 1980.

[8] A. N. Kolmogorov and S. V. Fomin, Elementi di teoria delle funzioni e di analisi funzionale, Mir, 1980.

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37