A few aspects concerning the Schrödinger
equation
Laurent Di Menza
Laboratoire de Mathématiques de Reims, URCA

Graduate School of Mathematical Sciences, Tokyo
April 24th 2024

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Plan of the talk

Basic facts about Schrödinger equations

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Plan of the talk

Basic facts about Schrödinger equations
The nonlinear Schrödinger equation in nonlinear optics

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Plan of the talk

Basic facts about Schrödinger equations
The nonlinear Schrödinger equation in nonlinear optics
Computation of stationary solutions

Laurent Di Menza

A few aspects concerning the Schrödinger equation

A universal model used at a wide range of scales...

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Probabilistic description in quantum mechanics
Let us study the time evolution of a particle with mass m that moves
under the action of a given real potential V.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Probabilistic description in quantum mechanics
Let us study the time evolution of a particle with mass m that moves
under the action of a given real potential V.
The particule can be associated to a complex-valued function ψ(t, x);|ψ(t, x)|2 dx stands as the probability to find the particle at time t in
the elementary volume dx centered at x.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Probabilistic description in quantum mechanics
Let us study the time evolution of a particle with mass m that moves
under the action of a given real potential V.
The particule can be associated to a complex-valued function ψ(t, x);|ψ(t, x)|2 dx stands as the probability to find the particle at time t in
the elementary volume dx centered at x.

∫
Normalization:
R3|ψ(t, x)|2 dx = 1 for t ≥ 0.

(because the particule is located somewhere in space!)
Laurent Di Menza

A few aspects concerning the Schrödinger equation

Time evolution
Principle:
Time derivative of the wave function is proportional to the Hamiltonian H := T + V acting on ψ where T and V respectively denotethe kinetic energy operator and potential energy operator.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Time evolution
Principle:
Time derivative of the wave function is proportional to the Hamiltonian H := T + V acting on ψ where T and V respectively denotethe kinetic energy operator and potential energy operator.
2ℏSince T = − 2m∆· (where ℏ = h/2π, h standing for Planck’s
constant) and V = V·, one then finds the Schrödinger equation

(S)

iℏ

∂ψ
ℏ2
=−
∆ψ + Vψ∂t
2m

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Time evolution
Principle:
Time derivative of the wave function is proportional to the Hamiltonian H := T + V acting on ψ where T and V respectively denotethe kinetic energy operator and potential energy operator.
2ℏSince T = − 2m∆· (where ℏ = h/2π, h standing for Planck’s
constant) and V = V·, one then finds the Schrödinger equation

(S)

iℏ

∂ψ
ℏ2
=−
∆ψ + Vψ∂t
2m

Initialization: at t = 0, we set ψ(0, x) = ψ0 (x), with∫|ψ0 (x)|2 dx = 1.
R3

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Basic properties
The solution ψ of (S) is such that∫∫2∀ t ≥ 0,|ψ(t, x)| dx =R3R3

|ψ0 (x)|2 dx = 1.

To see this,∫ multiply (S) by ψ and integrate; the imaginary partdgives dt
( R3 |ψ(t, x)|2 dx) = 0.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Basic properties
The solution ψ of (S) is such that∫∫2∀ t ≥ 0,|ψ(t, x)| dx =R3R3

|ψ0 (x)|2 dx = 1.

To see this,∫ multiply (S) by ψ and integrate; the imaginary partdgives dt
( R3 |ψ(t, x)|2 dx) = 0.
If V is nice enough, then (S) admits a unique solution ψ
defined for all t > 0. The mathematical problem is said to bewell-posed.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Basic properties
The solution ψ of (S) is such that∫∫2∀ t ≥ 0,|ψ(t, x)| dx =R3R3

|ψ0 (x)|2 dx = 1.

To see this,∫ multiply (S) by ψ and integrate; the imaginary partdgives dt
( R3 |ψ(t, x)|2 dx) = 0.
If V is nice enough, then (S) admits a unique solution ψ
defined for all t > 0. The mathematical problem is said to bewell-posed.
Normalization of physical constants: without loss of generality, we
can assume that m = 1/2 and ℏ = 1.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Stationary solutions?
Seeking a specific solution of the Schrödinger equation of the form
ψ(t, x) = e−iEt φ(x) leads us to the time-independent equation
Hφ = Eφoften referred in the literature as the Schrödinger equation.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to calculate solutions?
The problem i ∂ψ = Hψ∂t ψ(0, .) = ψt>00

can be solved using the separation of variables method: we assume
that the solution ψ expresses as ψ(t, x) = θ(t)φ(x).

Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to calculate solutions?
The problem i ∂ψ = Hψ∂t ψ(0, .) = ψt>00

can be solved using the separation of variables method: we assume
that the solution ψ expresses as ψ(t, x) = θ(t)φ(x).
Plugging this expression into the Schrödinger equation implies thatiθ′ (t)φ(x) = θ(t)Hφ(x).

Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to calculate solutions?
The problem i ∂ψ = Hψ∂t ψ(0, .) = ψt>00

can be solved using the separation of variables method: we assume
that the solution ψ expresses as ψ(t, x) = θ(t)φ(x).
Plugging this expression into the Schrödinger equation implies thatiθ′ (t)φ(x) = θ(t)Hφ(x). After division by θφ, we find thatiθ′ (t)Hφ(x)=:= λ = constant.
θ(t)
φ(x)

Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to calculate solutions?
The problem i ∂ψ = Hψ∂t ψ(0, .) = ψt>00

can be solved using the separation of variables method: we assume
that the solution ψ expresses as ψ(t, x) = θ(t)φ(x).
Plugging this expression into the Schrödinger equation implies thatiθ′ (t)φ(x) = θ(t)Hφ(x). After division by θφ, we find thatiθ′ (t)Hφ(x)=:= λ = constant.
θ(t)
φ(x)
We thus obtain 2 decoupled problems in t and x! The spatial one issimply the spectral problem
Hφ = λφ.
Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to calculate solutions?
Asume now that it is possible to find a family (λk , φk )k≥0 satisfying
this spectral problem (that is eigenstates of H).

Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to calculate solutions?
Asume now that it is possible to find a family (λk , φk )k≥0 satisfying
this spectral problem (that is eigenstates of H).
Solving the time-dependent problem in θ, we find for each k a particular solution of the formψk (t, x) = e−iλk t φk (x)∑αk ψk (t, x).
and we seek the solution as ψ(t, x) =k≥0Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to calculate solutions?
Asume now that it is possible to find a family (λk , φk )k≥0 satisfying
this spectral problem (that is eigenstates of H).
Solving the time-dependent problem in θ, we find for each k a particular solution of the formψk (t, x) = e−iλk t φk (x)∑αk ψk (t, x).
and we seek the solution as ψ(t, x) =k≥0This solution satisfies the initial condition if we have∑∑
ψ0 (x) =αk ψk (0, x) =αk φk (x) for each x ∈ R3 .
k≥0

k≥0

Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to calculate solutions?
Asume now that it is possible to find a family (λk , φk )k≥0 satisfying
this spectral problem (that is eigenstates of H).
Solving the time-dependent problem in θ, we find for each k a particular solution of the formψk (t, x) = e−iλk t φk (x)∑αk ψk (t, x).
and we seek the solution as ψ(t, x) =k≥0This solution satisfies the initial condition if we have∑∑
ψ0 (x) =αk ψk (0, x) =αk φk (x) for each x ∈ R3 .
k≥0

k≥0

It is possible if any initial state ψ0 can express as a infinite linear
combination of eigenstates of H.
Laurent Di Menza

A few aspects concerning the Schrödinger equation

Example: potential wall in dimension 1
Consider V(x) = 0 if x ∈ [0, L] and V(x) = +∞ elsewhere: theparticle is confined into the domain ]0, L[ which is equivalent toprescribe boundary conditions
ψ(t, 0) = ψ(t, L) = 0 for each t.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Example: potential wall in dimension 1
Consider V(x) = 0 if x ∈ [0, L] and V(x) = +∞ elsewhere: theparticle is confined into the domain ]0, L[ which is equivalent toprescribe boundary conditions
ψ(t, 0) = ψ(t, L) = 0 for each t.
The spectral problem writes −φ′′ (x) = λφ(x), φ(0) = φ(L) = 0.
and admits nontrivial solutions φk (x) = sin(k πx
L ) for λ = λk =k2 π 2 /L2 .

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Example: potential wall in dimension 1
Consider V(x) = 0 if x ∈ [0, L] and V(x) = +∞ elsewhere: theparticle is confined into the domain ]0, L[ which is equivalent toprescribe boundary conditions
ψ(t, 0) = ψ(t, L) = 0 for each t.
The spectral problem writes −φ′′ (x) = λφ(x), φ(0) = φ(L) = 0.
and admits nontrivial solutions φk (x) = sin(k πx
L ) for λ = λk =k2 π 2 /L2 . Now find coeﬀicients αk such that
( πx )∑
ψ0 (x) =αk sin k.
Lk≥0

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Example: potential wall in dimension 1
Consider V(x) = 0 if x ∈ [0, L] and V(x) = +∞ elsewhere: theparticle is confined into the domain ]0, L[ which is equivalent toprescribe boundary conditions
ψ(t, 0) = ψ(t, L) = 0 for each t.
The spectral problem writes −φ′′ (x) = λφ(x), φ(0) = φ(L) = 0.
and admits nontrivial solutions φk (x) = sin(k πx
L ) for λ = λk =k2 π 2 /L2 . Now find coeﬀicients αk such that
( πx )∑
ψ0 (x) =αk sin k.
Lk≥0

It is obtained that αk are sine Fourier coeﬀicients of the 2L-periodic
odd extension of ψ0 . The desired solution is then
( πx )∑ −i k2 π2 tψ(t, x) =e L2 αk sin k.
Lk≥0

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Illustration
2

Case V(x) = 0 if |x| ≤ 5, +∞ elsewhere and ψ0 (x) = e−x :The particle is confined within the domain [−5, 5] and the boundary
generates reflections. The solution is time periodic.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Other illustrations
2

Case V(x) = x2 and ψ0 (x) = e−x :The particle is confined around the minimal value of the potentialthat exhibits a trapping effect.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Other illustrations
2

Case V(x) = 0 and ψ0 (x) = e−x :The particle freely propagates in the whole space and in this case,2
there is a continuum of stationary states ψξ (t, x) = e−iξ t eiξx .

Laurent Di Menza

A few aspects concerning the Schrödinger equation

A model in nonlinear optics
Assume that V locally depends on |ψ|2 . Example: V(ψ) = −|ψ|2 .

Laurent Di Menza

A few aspects concerning the Schrödinger equation

A model in nonlinear optics
Assume that V locally depends on |ψ|2 . Example: V(ψ) = −|ψ|2 .
Different asymptotics :Dispersion: as in the linear case, potential
cannot trap the matter (for a small enoughtotal mass).

Laurent Di Menza

A few aspects concerning the Schrödinger equation

A model in nonlinear optics
Assume that V locally depends on |ψ|2 . Example: V(ψ) = −|ψ|2 .
Different asymptotics :Dispersion: as in the linear case, potential
cannot trap the matter (for a small enoughtotal mass).
Stationary states: exact compensation between potential and dispersion. The spatial
form of the amplitude is preserved in time.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

A model in nonlinear optics
Assume that V locally depends on |ψ|2 . Example: V(ψ) = −|ψ|2 .
Different asymptotics :Dispersion: as in the linear case, potential
cannot trap the matter (for a small enoughtotal mass).
Stationary states: exact compensation between potential and dispersion. The spatial
form of the amplitude is preserved in time.
Self-focusing: potential enables to concentrate the mass at a single point (for a large
enough total mass).

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Illustration

Propagation of a laser beam inside a nonlinear medium: self-focusing
effect of the beam for a suﬀiciently large power.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to explain?
Consider a group of runners that run at constant speed (that depends on the considered individual runners) on a rigid track where
no nonlinear effect occurs.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to explain?
Consider a group of runners that run at constant speed (that depends on the considered individual runners) on a rigid track where
no nonlinear effect occurs.
For a concentrated initial density of runners, there is a spreading
effect on the whole space: this is dispersion.

START

Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to explain?
Consider a group of runners that run at constant speed (that depends on the considered individual runners) on a rigid track where
no nonlinear effect occurs.
For a concentrated initial density of runners, there is a spreading
effect on the whole space: this is dispersion.

START

Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to explain?
Assume now that the track is no more rigid: running is performed
on a mattress where surface can deform with a curvature dependingon the mass of runners (∼ nonlinear effect).

Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to explain?
Assume now that the track is no more rigid: running is performed
on a mattress where surface can deform with a curvature dependingon the mass of runners (∼ nonlinear effect).
If the mattress is stiff, the runners dynamics is also the same andthe mattress does not prevent dispersion.

START

Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to explain?
Assume now that the track is no more rigid: running is performed
on a mattress where surface can deform with a curvature dependingon the mass of runners (∼ nonlinear effect).
If the mattress is stiff, the runners dynamics is also the same andthe mattress does not prevent dispersion.

START

Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to explain?
Assume now that the track is no more rigid: running is performed
on a mattress where surface can deform with a curvature dependingon the mass of runners (∼ nonlinear effect).
If the mattress is too smooth, the runners are trapped at a single
point: the nonlinear effect becomes predominant!

START

Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to explain?
Assume now that the track is no more rigid: running is performed
on a mattress where surface can deform with a curvature dependingon the mass of runners (∼ nonlinear effect).
If the mattress is too smooth, the runners are trapped at a single
point: the nonlinear effect becomes predominant!

START

Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to explain?
Assume now that the track is no more rigid: running is performed
on a mattress where surface can deform with a curvature dependingon the mass of runners (∼ nonlinear effect).
In a equilibrum case, the runners density propagates without deformation. There is an exact compensation between dispersion andnonlinearity!

START

Laurent Di Menza

A few aspects concerning the Schrödinger equation

How to explain?
Assume now that the track is no more rigid: running is performed
on a mattress where surface can deform with a curvature dependingon the mass of runners (∼ nonlinear effect).
In a equilibrum case, the runners density propagates without deformation. There is an exact compensation between dispersion andnonlinearity!

START

Laurent Di Menza

A few aspects concerning the Schrödinger equation

The search for stationary localized states
Let us consider the general nonlinear focusing Schrödinger equationi∂ψ+ ∆ψ + |ψ|2σ ψ = 0∂twhere ψ = ψ(t, x) ∈ C, x ∈ Rd and σ > 0.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

The search for stationary localized states
Let us consider the general nonlinear focusing Schrödinger equationi∂ψ+ ∆ψ + |ψ|2σ ψ = 0∂twhere ψ = ψ(t, x) ∈ C, x ∈ Rd and σ > 0.
A natural question: how is it possible to compute solutions thatkeep their spatial profile as time evolves?

Laurent Di Menza

A few aspects concerning the Schrödinger equation

The search for stationary localized states
Let us consider the general nonlinear focusing Schrödinger equationi∂ψ+ ∆ψ + |ψ|2σ ψ = 0∂twhere ψ = ψ(t, x) ∈ C, x ∈ Rd and σ > 0.
A natural question: how is it possible to compute solutions thatkeep their spatial profile as time evolves?
A miracle in this model: The nonlinearity f(ψ) = |ψ|2σ ψ is suchthat for each θ, one hasf(ψeiθ ) = f(ψ)eiθ .

Laurent Di Menza

A few aspects concerning the Schrödinger equation

The search for stationary localized states
Let us consider the general nonlinear focusing Schrödinger equationi∂ψ+ ∆ψ + |ψ|2σ ψ = 0∂twhere ψ = ψ(t, x) ∈ C, x ∈ Rd and σ > 0.
A natural question: how is it possible to compute solutions thatkeep their spatial profile as time evolves?
A miracle in this model: The nonlinearity f(ψ) = |ψ|2σ ψ is suchthat for each θ, one hasf(ψeiθ ) = f(ψ)eiθ .
It means that even in this nonlinear case, the separation of variables
can still be used with complex exponential time dependence!!

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Computation of localized states
We thus look for stationary states ψ of the form ψ(t, x) = eiωt u(x)(ω > 0), where spatial profile u is spatially localized, that is whichtends to zero at infinity.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Computation of localized states
We thus look for stationary states ψ of the form ψ(t, x) = eiωt u(x)(ω > 0), where spatial profile u is spatially localized, that is whichtends to zero at infinity.
Plugging this form into the Schrödinger equation gives the elliptic
problem−ωu + ∆u + |u|2σ u = 0, x ∈ Rd .

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Computation of localized states
We thus look for stationary states ψ of the form ψ(t, x) = eiωt u(x)(ω > 0), where spatial profile u is spatially localized, that is whichtends to zero at infinity.
Plugging this form into the Schrödinger equation gives the elliptic
problem−ωu + ∆u + |u|2σ u = 0, x ∈ Rd .
In dimension 1: unique solution u(x) =Laurent Di Menza(ω(σ + 1))1/2σ.
√
cosh(σ ωx)1/σ

A few aspects concerning the Schrödinger equation

Computation of localized states
We thus look for stationary states ψ of the form ψ(t, x) = eiωt u(x)(ω > 0), where spatial profile u is spatially localized, that is whichtends to zero at infinity.
Plugging this form into the Schrödinger equation gives the elliptic
problem−ωu + ∆u + |u|2σ u = 0, x ∈ Rd .
In dimension 1: unique solution u(x) =(ω(σ + 1))1/2σ.
√
cosh(σ ωx)1/σ

In larger dimensions: infinite number of radially symmetric
excited states that vanish k times (for any k) and tend to zeroat infinity.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Computation of localized states
If we then look for smooth radially symmetric solutions (that isu(x) = u(r), r = ∥x∥), then it turns out that u solves the secondorder nonlinear differential equation
 u′′ (r) + d − 1 u′ (r) − ωu(r) + |u(r)|2σ u(r) = 0, r > 0r u(0) = β, u′ (0) = 0.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Computation of localized states
If we then look for smooth radially symmetric solutions (that isu(x) = u(r), r = ∥x∥), then it turns out that u solves the secondorder nonlinear differential equation
 u′′ (r) + d − 1 u′ (r) − ωu(r) + |u(r)|2σ u(r) = 0, r > 0r u(0) = β, u′ (0) = 0.
The main problem is then to choose initial data β at the origin, suchthat the corresponding solution vanishes k times and tends to zeroat infinity : use of the shooting method.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Computation of localized states
If we then look for smooth radially symmetric solutions (that isu(x) = u(r), r = ∥x∥), then it turns out that u solves the secondorder nonlinear differential equation
 u′′ (r) + d − 1 u′ (r) − ωu(r) + |u(r)|2σ u(r) = 0, r > 0r u(0) = β, u′ (0) = 0.
The main problem is then to choose initial data β at the origin, suchthat the corresponding solution vanishes k times and tends to zeroat infinity : use of the shooting method.
Enables to compute the sequence (uk )k≥0 with use of a dichotomy
algorithm for d ≥ 2. It only requires to solve numerically a differential system.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

What makes it work ?
We use a very convenient property of the solutions of the nonlinear
ODE: the number of zeros is an increasing function of the initial
data β.
7654No zero
3
2
1
0
−1
−2
−3
0

1

2

3

4

Laurent Di Menza

5
r

6

7

8

9

10

A few aspects concerning the Schrödinger equation

What makes it work ?
We use a very convenient property of the solutions of the nonlinear
ODE: the number of zeros is an increasing function of the initial
data β.
7654One zero
3
2
1
0
−1
−2
−3
0

1

2

3

4

Laurent Di Menza

5
r

6

7

8

9

10

A few aspects concerning the Schrödinger equation

What makes it work ?
We use a very convenient property of the solutions of the nonlinear
ODE: the number of zeros is an increasing function of the initial
data β.
7
6
5
Two zeros

4
3
2
1
0
−1
−2
−3
0

1

2

3

4

Laurent Di Menza

5
r

6

7

8

9

10

A few aspects concerning the Schrödinger equation

What makes it work ?
We use a very convenient property of the solutions of the nonlinear
ODE: the number of zeros is an increasing function of the initial
data β.
765Three zeros
4
3
2
1
0
−1
−2
−3
0

1

2

3

4

Laurent Di Menza

5
r

6

7

8

9

10

A few aspects concerning the Schrödinger equation

What makes it work ?
We use a very convenient property of the solutions of the nonlinear
ODE: the number of zeros is an increasing function of the initial
data β.
76Four zeros

5
4
3
2
1
0
−1
−2
−3
0

1

2

3

4

Laurent Di Menza

5
r

6

7

8

9

10

A few aspects concerning the Schrödinger equation

Example for k = 0, d = 2, σ = 1
4

3

2

1

0

−1

−2
0

2

4

6

8

10
r

12

14

16

18

20

β = 4 : β is too big !

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Example for k = 0, d = 2, σ = 1
4

3

2

1

0

−1

−2
0

2

4

6

8

10
r

12

14

16

18

20

β = 2 : β is too small !

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Example for k = 0, d = 2, σ = 1
4

3

2

1

0

−1

−2
0

2

4

6

8

10
r

12

14

16

18

20

β = 2.21 : β is too big !

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Example for k = 0, d = 2, σ = 1
4

3

2

1

0

−1

−2
0

2

4

6

8

10
r

12

14

16

18

20

β = 2.20617 : β is too small !

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Example for k = 0, d = 2, σ = 1
4

3

2

1

0

−1

−2
0

2

4

6

8

10
r

12

14

16

18

20

β = 2.20620273691129 : Correct on the interval [0, 20].

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Illustration in dimension 2
Determination of the profile of uk with respect to r for the bidimensional cubic case σ = 1, ω = 1 (solutions exponentially decreasing
to zero at infinity).
5

5

0 node1 node2 nodes

0 node1 node2 nodes
4

0
3

2
−5
1

0
−10

−1

−15

−2
0

5

10

15

0

5

10

r

r

Plot of uk , k = 0, 1, 2.

Plot of log |uk |, k = 0, 1, 2.

Laurent Di Menza

15

A few aspects concerning the Schrödinger equation

Asymptotics with large number of nodes
Our goal : compute excited states when k is large and σ is large inorder to guess accurate asymptotic laws for the maximal amplitude∥uk ∥L∞ of these solutions (this is a physically relevant quantity).
Example in the case d = 2
14

sigma = 0.5
12sigma = 1.0
sigma = 1.5
sigma = 2.0

10

8

6

4

2

0

5

10

15

20

25

k

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Asymptotics for large k
These results suggest that we may have a power law of the form
Amplitude ∼ kγ(d,σ) ,where γ has to be determined in terms of d and σ.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Asymptotics for large k
These results suggest that we may have a power law of the form
Amplitude ∼ kγ(d,σ) ,where γ has to be determined in terms of d and σ.
How to do ?

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Asymptotics for large k
These results suggest that we may have a power law of the form
Amplitude ∼ kγ(d,σ) ,where γ has to be determined in terms of d and σ.
How to do ?
Compute an approximation, say λ, of the derivative of the logof amplitude with respect to log k as k ≫ 1.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Asymptotics for large k
These results suggest that we may have a power law of the form
Amplitude ∼ kγ(d,σ) ,where γ has to be determined in terms of d and σ.
How to do ?
Compute an approximation, say λ, of the derivative of the logof amplitude with respect to log k as k ≫ 1.
Main problem: it requires a lot of computations for large k,
the values d and σ being independently considered in order tofind out a “correct” power law for γ(d, σ).

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Dimension d = 2

0.5

0.45
sigma = 0.5
sigma = 1.0
sigma = 1.5

0.4

sigma = 2.0

0.35

0.3

0.25

0.2

0

5

10

15

20

25

k

λ = f(k), d = 2.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Dimensions d = 3 and d = 4

2

5.5
5

1.8

4.5

1.6

sigma = 0.25

4sigma = 0.5

1.4

sigma = 0.75

3.5
1.2
3
1
2.5
0.8

sigma = 0.25

2sigma = 0.5
0.6

sigma = 0.75

1.5

sigma = 1.0
0.4

1

0.2

0

0.5

0

5

10

15

20

25

0

0

5

10

15

k

k

λ = f(k), d = 3.

λ = f(k), d = 4.

Laurent Di Menza

20

A few aspects concerning the Schrödinger equation

And finally, after many and many computations...
18

16

Theoretical, d = 3
Numerical, d = 3
Theoretical, d = 4
Numerical, d = 4
Theoretical, d = 5
Numerical, d = 5

14

12

10

8

6

4

2

0

0

0.2

0.4

0.6

0.8

1
σ

1.2

1.4

1.6

1.8

2

Comparison between the numerical exponent and the conjectured powerd−1
law γ(d, σ) = 2+σ(2−d), d = 3, 4, 5.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

What about the computations?

Laurent Di Menza

A few aspects concerning the Schrödinger equation

What about the computations?
To compute a given excited state, one has to solve the same
nonlinear ODE, adjusting at each iteration the value of thesolution at the origin.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

What about the computations?
To compute a given excited state, one has to solve the same
nonlinear ODE, adjusting at each iteration the value of thesolution at the origin.
To compute γ(d, σ) (d and σ being prescribed), one has tocalculate a sequence of stationary states for k larger and larger.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

What about the computations?
To compute a given excited state, one has to solve the same
nonlinear ODE, adjusting at each iteration the value of thesolution at the origin.
To compute γ(d, σ) (d and σ being prescribed), one has tocalculate a sequence of stationary states for k larger and larger.
To guess a accurate expression of the power law depending on
both d and σ, one has to make vary independently parameters
d and σ.

Laurent Di Menza

A few aspects concerning the Schrödinger equation

What about the computations?
To compute a given excited state, one has to solve the same
nonlinear ODE, adjusting at each iteration the value of thesolution at the origin.
To compute γ(d, σ) (d and σ being prescribed), one has tocalculate a sequence of stationary states for k larger and larger.
To guess a accurate expression of the power law depending on
both d and σ, one has to make vary independently parameters
d and σ.
=⇒ Very costly computations (several billions of independent resolutions of nonlinear ODE’s !!!). This can be achieved using parallel
computing (no exchange between processors since the resolutions
are independent !).

Laurent Di Menza

A few aspects concerning the Schrödinger equation

Thank you for your attention!

Laurent Di Menza

A few aspects concerning the Schrödinger equation