Derivation of the nonlinear Schr odinger equation with ...Introduction i @ @t = @2 @x2 0j j 2 >0 I 0...

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$Page 1: Derivation of the nonlinear Schr odinger equation with ...Introduction i @ @t = @2 @x2 0j j 2 >0 I 0 Dirac-delta distribution, 2R I The nonlinearity is \concentrated" in x= 0 I For$

Derivation of the nonlinear Schrodinger

equation with point-like nonlinearityA joint work in collaboration with D. Finco, D. Noja, and A. Teta

Claudio Cacciapuoti

Dipartimento di Scienzae Alta TecnologiaUniversita dell’Insubria

&

Stochastic and Analytic Methods in Mathematical PhysicsYerevan, Armenia, September 4-11, 2016

Summary

1D ModelThe NLS with delta-like nonlinearityApproximation through local and nonlocal nonlinearity

3D ModelThe NLS with delta-like nonlinearityApproximation through nonlocal nonlinearity

Conclusions and Perspectives

Introduction

i∂

∂tψ = − ∂2

∂x2ψ + γδ0|ψ|2µψ µ > 0

I δ0 Dirac-delta distribution, γ ∈ RI The nonlinearity is “concentrated” in x = 0

I For x 6= 0 the dynamics is “free”

Applications:

I Solid state physics [Malomed, Azbel PRB ’93 (starting from a model byPresilla, Jona-Lasinio, and Capasso ’91); Sun, Tsironis ’95; Bulashenko,Kochelap, and Bonilla PRB ’96]

I Bose-Einstein condensates trapped into optical lattices [Dror-MalomedPRA ’11]

I Nonlinear Kronig-Penney model [Hennig, Tsironis, Molina, Gabriel ’94;Gaididei, Christiansen, Rasmussen, Johansson ’97]

Introduction

i∂

∂tψ = − ∂2

∂x2ψ + γδ0|ψ|2µψ µ > 0

I δ0 Dirac-delta distribution, γ ∈ RI The nonlinearity is “concentrated” in x = 0

Applications:

I Solid state physics [Malomed, Azbel PRB ’93 (starting from a model byPresilla, Jona-Lasinio, and Capasso ’91); Sun, Tsironis ’95; Bulashenko,Kochelap, and Bonilla PRB ’96]

I Bose-Einstein condensates trapped into optical lattices [Dror-MalomedPRA ’11]

I Nonlinear Kronig-Penney model [Hennig, Tsironis, Molina, Gabriel ’94;Gaididei, Christiansen, Rasmussen, Johansson ’97]

Introduction

Goal: Derivation of equations of the form

“ i∂

∂tψ = −∆ψ + γδ0|ψ|2µψ ”

From regular, scaled, local or nonlocal NLS of the form

i∂

∂tψε = −∆ψε + Gε(µ, x, ψε)

Introduction

Two-dimensional effects in nonlinear Kronig-Penney models

Yu. B. Gaididei,* P. L. Christiansen, K. O” . Rasmussen, and M. JohanssonDepartment of Mathematical Modelling, The Technical University of Denmark, DK-2800 Lyngby, Denmark

~Received 27 November 1996!

An analysis of two-dimensional ~2D! effects in the nonlinear Kronig-Penney model is presented. We estab-lish an effective one-dimensional description of the 2D effects, resulting in a set of pseudodifferential equa-tions. The stationary states of the 2D system and their stability is studied in the framework of these equations.In particular it is shown that localized stationary states exist only in a finite interval of the excitation power.@S0163-1829~97!52220-X#

There is a growing interest in the subject of wave propa-gation in nonlinear photonic band-gap materials and in peri-odic nonlinear dielectric superlattices.1 The basic dynamicsin these systems is described by the fundamental nonlinearSchrodinger ~NLS! equation

i] tc1π2c1 f ~rW ,ucu2!c50, ~1!

where c(rW ,t) is the complex amplitude of quasimonochro-matic wave trains, the variable t is time, and rW is the spatialcoordinate. The function f (rW ,ucu2) characterizes the nonlin-earity of the medium, e.g., the nonlinear corrections of therefractive index of the photonic band-gap materials or theself-interaction of the quasiparticles in the superlattices. Inthe case of periodic nonlinear superlattices consisting of al-ternating layers of two dielectrics, it is usually assumed thatthe nonlinearity of one of the dielectrics is much larger thanthe nonlinearity of the other, so that the latter can be consid-ered linear. If the thickness of the nonlinear layer is smallcompared to the de Broglie wavelength within the layer, theproblem can be described by the nonlinear Kronig-Penneymodel2 with the nonlinearity f (rW ,ucu2) in the form

f ~rW ,ucu2!5(n

d~x2xn!uc~rW ,t !u2 ~2!

corresponding to a focusing medium with cubic nonlinearity.Here xn5nl is the coordinate of the nth nonlinear layer andl is the distance between adjacent nonlinear layers. Wavepropagation in the framework of the one-dimensional ~1D!nonlinear Kronig-Penney model was studied in detail inRefs. 3–5, but in these works the coupling between the lon-gitudinal and transversal degrees of freedom was ignored. Itwas shown that the transmission properties depend criticallyon the injected wave power. Additionally, it was shown thatthese systems exhibit bistability and multistability.In the present paper we consider 2D effects in the nonlin-

ear Kronig-Penney model given by Eqs. ~1! and ~2!, wherethe complex amplitude depends on the coordinate x transver-sal to the nonlinear layers and the longitudinal coordinatez . Denoting with an overbar c , the Fourier transform withrespect to t and z , one can represent Eqs. ~1! and ~2! in theform

2~v1k2!c1]x2c1(

nd~x2xn!ucu2c50. ~3!

Similarly to the approach used in Refs. 3 and 5, we can solvethese equations in the linear medium and thereby express thefield c(x ,z;t) for nl<x<(n11)l in terms of the complexamplitudes cn(z ,t)[c(xn ,z;t) at the nonlinear layers,

c~x ,z;t !5sinh$k@~n11 !l2x#%

sinh~ kl !cn~z ,t !

1sinh@ k~x2nl !#

sinh~ kl !cn11~z ,t !, ~4!

where the complex amplitude cn(z ,t) satisfies the set ofpseudodifferential equations

k

sinhlk~cn111cn21!2

2k

tanhlkcn1ucnu2cn50, ~5!

with periodic boundary conditions cn1N5cn , where N isthe number of layers. In Eqs. ~4! and ~5! the operator k isdefined as kc5A2i] t2]z

2c or expressed in the Fourier do-main kc5Av1k2c .In passing it is worth noting the following two limits

where the system ~5! reduces to systems previously dis-cussed in the literature. First, considering the ordering

] t;]z2;e , cn;Ae , cn111cn2122cn;ecn

for e�0, Eq. ~5! reduces to

l2~ i] t1]z2!cn1cn111cn2122cn1lucnu2cn50, ~6!

which is the so-called discrete-continuum NLS equation in-troduced by Aceves et al.6 to describe soliton dynamics innonlinear optical fiber arrays. Second, increasing the distancel between the nonlinear layers, the interlayer coupling @thefirst term on the left-hand side of Eq. ~5!# vanishes and theequation takes the form

2A2i] t2]z2cn2ucnu2cn50, ~7!

which for static distributions ] tc50 reduces to the nonlinearHilbert NLS equation introduced recently by Gaididei et al.7

RAPID COMMUNICATIONS

PHYSICAL REVIEW B 15 MAY 1997-IIVOLUME 55, NUMBER 20

Two-dimensional effects in nonlinear Kronig-Penney models

Yu. B. Gaididei,* P. L. Christiansen, K. O” . Rasmussen, and M. JohanssonDepartment of Mathematical Modelling, The Technical University of Denmark, DK-2800 Lyngby, Denmark

~Received 27 November 1996!

An analysis of two-dimensional ~2D! effects in the nonlinear Kronig-Penney model is presented. We estab-lish an effective one-dimensional description of the 2D effects, resulting in a set of pseudodifferential equa-tions. The stationary states of the 2D system and their stability is studied in the framework of these equations.In particular it is shown that localized stationary states exist only in a finite interval of the excitation power.@S0163-1829~97!52220-X#

There is a growing interest in the subject of wave propa-gation in nonlinear photonic band-gap materials and in peri-odic nonlinear dielectric superlattices.1 The basic dynamicsin these systems is described by the fundamental nonlinearSchrodinger ~NLS! equation

i] tc1π2c1 f ~rW ,ucu2!c50, ~1!

where c(rW ,t) is the complex amplitude of quasimonochro-matic wave trains, the variable t is time, and rW is the spatialcoordinate. The function f (rW ,ucu2) characterizes the nonlin-earity of the medium, e.g., the nonlinear corrections of therefractive index of the photonic band-gap materials or theself-interaction of the quasiparticles in the superlattices. Inthe case of periodic nonlinear superlattices consisting of al-ternating layers of two dielectrics, it is usually assumed thatthe nonlinearity of one of the dielectrics is much larger thanthe nonlinearity of the other, so that the latter can be consid-ered linear. If the thickness of the nonlinear layer is smallcompared to the de Broglie wavelength within the layer, theproblem can be described by the nonlinear Kronig-Penneymodel2 with the nonlinearity f (rW ,ucu2) in the form

f ~rW ,ucu2!5(n

d~x2xn!uc~rW ,t !u2 ~2!

corresponding to a focusing medium with cubic nonlinearity.Here xn5nl is the coordinate of the nth nonlinear layer andl is the distance between adjacent nonlinear layers. Wavepropagation in the framework of the one-dimensional ~1D!nonlinear Kronig-Penney model was studied in detail inRefs. 3–5, but in these works the coupling between the lon-gitudinal and transversal degrees of freedom was ignored. Itwas shown that the transmission properties depend criticallyon the injected wave power. Additionally, it was shown thatthese systems exhibit bistability and multistability.In the present paper we consider 2D effects in the nonlin-

ear Kronig-Penney model given by Eqs. ~1! and ~2!, wherethe complex amplitude depends on the coordinate x transver-sal to the nonlinear layers and the longitudinal coordinatez . Denoting with an overbar c , the Fourier transform withrespect to t and z , one can represent Eqs. ~1! and ~2! in theform

2~v1k2!c1]x2c1(

nd~x2xn!ucu2c50. ~3!

Similarly to the approach used in Refs. 3 and 5, we can solvethese equations in the linear medium and thereby express thefield c(x ,z;t) for nl<x<(n11)l in terms of the complexamplitudes cn(z ,t)[c(xn ,z;t) at the nonlinear layers,

c~x ,z;t !5sinh$k@~n11 !l2x#%

sinh~ kl !cn~z ,t !

1sinh@ k~x2nl !#

sinh~ kl !cn11~z ,t !, ~4!

where the complex amplitude cn(z ,t) satisfies the set ofpseudodifferential equations

k

sinhlk~cn111cn21!2

2k

tanhlkcn1ucnu2cn50, ~5!

with periodic boundary conditions cn1N5cn , where N isthe number of layers. In Eqs. ~4! and ~5! the operator k isdefined as kc5A2i] t2]z

2c or expressed in the Fourier do-main kc5Av1k2c .In passing it is worth noting the following two limits

where the system ~5! reduces to systems previously dis-cussed in the literature. First, considering the ordering

] t;]z2;e , cn;Ae , cn111cn2122cn;ecn

for e�0, Eq. ~5! reduces to

l2~ i] t1]z2!cn1cn111cn2122cn1lucnu2cn50, ~6!

which is the so-called discrete-continuum NLS equation in-troduced by Aceves et al.6 to describe soliton dynamics innonlinear optical fiber arrays. Second, increasing the distancel between the nonlinear layers, the interlayer coupling @thefirst term on the left-hand side of Eq. ~5!# vanishes and theequation takes the form

2A2i] t2]z2cn2ucnu2cn50, ~7!

which for static distributions ] tc50 reduces to the nonlinearHilbert NLS equation introduced recently by Gaididei et al.7

RAPID COMMUNICATIONS

PHYSICAL REVIEW B 15 MAY 1997-IIVOLUME 55, NUMBER 20

1

1D Model

Rigorous analysis [Adami-Teta ’01], also [Komech-Komech ’07, Holmer-Liu ’15]i∂

∂tψ = − ∂2

∂x2ψ + γδ0|ψ|2µψ

ψ∣∣∣t=0

= ψ0

I Conservation lawsI Mass conservation: ‖ψ(t)‖2 = ‖ψ0‖2I Energy conservation: E[ψ(t)] = E[ψ0] with

E[ψ] = ‖ψ′‖2 +γ|ψ(0)|2+2µ

1 + µ

I Global Well-Posedness in H1(R): If γ ≥ 0 for all µ > 0

1D Model

∂tψ = − ∂2

ψ∣∣∣t=0

= ψ0

E[ψ] = ‖ψ′‖2 +γ|ψ(0)|2+2µ

1 + µ

1D Model

∂tψ = − ∂2

ψ∣∣∣t=0

= ψ0

E[ψ] = ‖ψ′‖2 +γ|ψ(0)|2+2µ

1 + µ

1D Model

∂tψ = − ∂2

ψ∣∣∣t=0

= ψ0

E[ψ] = ‖ψ′‖2 +γ|ψ(0)|2+2µ

1 + µ

I Global Well-Posedness in H1(R): If γ < 0 for all 0 < µ < 1

I Gagliardo–Nirenberg inequality

‖ψ‖∞ ≤ C‖ψ′‖12 ‖ψ‖

12

together with Mass and Energy Conservation

E0 = E[ψ] ≥ ‖ψ′‖2 − C|γ|‖ψ′‖1+µ‖ψ‖1+µ

I Indeed, if γ < 0 and µ ≥ 1 there exist blow-up solutions

1D Model

∂tψ = − ∂2

ψ∣∣∣t=0

= ψ0

E[ψ] = ‖ψ′‖2 +γ|ψ(0)|2+2µ

1 + µ

I Global Well-Posedness in H1(R): If γ < 0 for all 0 < µ < 1I Gagliardo–Nirenberg inequality

‖ψ‖∞ ≤ C‖ψ′‖12 ‖ψ‖

12

E0 = E[ψ] ≥ ‖ψ′‖2 − C|γ|‖ψ′‖1+µ‖ψ‖1+µ

1D Model

∂tψ = − ∂2

ψ∣∣∣t=0

= ψ0

E[ψ] = ‖ψ′‖2 +γ|ψ(0)|2+2µ

1 + µ

I Global Well-Posedness in H1(R): If γ < 0 for all 0 < µ < 1I Gagliardo–Nirenberg inequality

‖ψ‖∞ ≤ C‖ψ′‖12 ‖ψ‖

12

E0 = E[ψ] ≥ ‖ψ′‖2 − C|γ|‖ψ′‖1+µ‖ψ‖1+µ

1D Model

Connection with the dirac-delta potential

The solution ψ(x, t) satisfies the boundary conditions

ψ(0+, t) = ψ(0−, t)

ψ′(0+, t)− ψ′(0−, t) = γ|ψ(0, t)|2µψ(0, t)

A non-linear version of the dirac-delta potential in dimension one

Hα = − d2

dx2+ αδ0 α ∈ R

Defined by

D(Hα) ={ψ ∈ H1(R) ∩H2(R\{0}) : ψ′(0+)− ψ′(0−) = αψ(0), α ∈ R

}Hαψ = −ψ′′ ∀x 6= 0

The NLS with concentrated nonlinearity is obtained by setting

α→ γ|ψ(0)|2µ

1D Model

ψ(0+, t) = ψ(0−, t)

ψ′(0+, t)− ψ′(0−, t) = γ|ψ(0, t)|2µψ(0, t)

Hα = − d2

dx2+ αδ0 α ∈ R

Defined by

D(Hα) ={ψ ∈ H1(R) ∩H2(R\{0}) : ψ′(0+)− ψ′(0−) = αψ(0), α ∈ R

}Hαψ = −ψ′′ ∀x 6= 0

α→ γ|ψ(0)|2µ

1D Model

ψ(0+, t) = ψ(0−, t)

ψ′(0+, t)− ψ′(0−, t) = γ|ψ(0, t)|2µψ(0, t)

Hα = − d2

dx2+ αδ0 α ∈ R

Defined by

D(Hα) ={ψ ∈ H1(R) ∩H2(R\{0}) : ψ′(0+)− ψ′(0−) = αψ(0), α ∈ R

}Hαψ = −ψ′′ ∀x 6= 0

α→ γ|ψ(0)|2µ

1D Model

Local approximation [CC-Finco-Noja-Teta 2014]

Consider i∂

∂tψε = − ∂2

∂x2ψε +

1

εV( ·ε

)|ψε|2µψε

ψε∣∣∣t=0

= ψ0

V ∈ C∞0 (R)

and i∂

∂tψ = − ∂2

ψ∣∣∣t=0

= ψ0

with γ =

∫RV dx

Then, for any ψ0 ∈ H1(R) and T > 0,

supt∈[0,T ]

‖ψε(t)− ψ(t)‖H1 → 0 as ε→ 0

I for all µ > 0 if V ≥ 0

I for all 0 < µ < 1 if V has a negative part

1D Model

Consider i∂

∂tψε = − ∂2

∂x2ψε +

1

εV( ·ε

)|ψε|2µψε

ψε∣∣∣t=0

= ψ0

V ∈ C∞0 (R)

and i∂

∂tψ = − ∂2

ψ∣∣∣t=0

= ψ0

with γ =

∫RV dx

supt∈[0,T ]

‖ψε(t)− ψ(t)‖H1 → 0 as ε→ 0

1D Model

Consider i∂

∂tψε = − ∂2

∂x2ψε +

1

εV( ·ε

)|ψε|2µψε

ψε∣∣∣t=0

= ψ0

V ∈ C∞0 (R)

and i∂

∂tψ = − ∂2

ψ∣∣∣t=0

= ψ0

with γ =

∫RV dx

supt∈[0,T ]

‖ψε(t)− ψ(t)‖H1 → 0 as ε→ 0

1D Model

Consider i∂

∂tψε = − ∂2

∂x2ψε +

1

εV( ·ε

)|ψε|2µψε

ψε∣∣∣t=0

= ψ0

V ∈ C∞0 (R)

and i∂

∂tψ = − ∂2

ψ∣∣∣t=0

= ψ0

with γ =

∫RV dx

supt∈[0,T ]

‖ψε(t)− ψ(t)‖H1 → 0 as ε→ 0

1D Model

Consider i∂

∂tψε = − ∂2

∂x2ψε +

1

εV( ·ε

)|ψε|2µψε

ψε∣∣∣t=0

= ψ0

V ∈ C∞0 (R)

and i∂

∂tψ = − ∂2

ψ∣∣∣t=0

= ψ0

with γ =

∫RV dx

Heuristically

1

εV( ·ε

)ε→0−−−→ γδ0

1D Model

Non-local (mean-field) approximation [Komech-Komech 2007]

Take ρε(x) = 1ερ(xε

), with ρ ∈ C∞0 (R), ρ > 0, and

∫R ρ dx = 1

i∂

∂tψε = − ∂2

∂x2ψε − ρεF (〈ρε, ψε〉)

ψ∣∣∣t=0

= ψ0

and i∂

∂tψ = − ∂2

∂x2ψ − δ0F (〈δ0, ψ〉)

ψ∣∣∣t=0

= ψ0

with F (z) = −∇zU(|z|2), U ∈ C2(R), and U(|z|2) ≥ a− b|z|2, a ∈ R, b > 0.

Then ψε(t)→ ψ(t) as ε→ 0.

Heuristically

ρε =1

ερ( ·ε

)→ δ0

1D Model

), with ρ ∈ C∞0 (R), ρ > 0, and

∫R ρ dx = 1

i∂

∂tψε = − ∂2

ψ∣∣∣t=0

= ψ0

and i∂

∂tψ = − ∂2

∂x2ψ − δ0F (〈δ0, ψ〉)

ψ∣∣∣t=0

= ψ0

Heuristically

ρε =1

ερ( ·ε

)→ δ0

1D Model

), with ρ ∈ C∞0 (R), ρ > 0, and

∫R ρ dx = 1

i∂

∂tψε = − ∂2

ψ∣∣∣t=0

= ψ0

and i∂

∂tψ = − ∂2

∂x2ψ − δ0F (〈δ0, ψ〉)

ψ∣∣∣t=0

= ψ0

Heuristically

ρε =1

ερ( ·ε

)→ δ0

3D Model

In 3D the Dirac-delta in the equation

“ i∂

∂tψ = −∆ψ + γδ0|ψ|2µψ ”

cannot be understood in distributional sense

This is true (and well known) also for the linear equation (µ = 0)[Albeverio et al. 1988]

“ i∂

∂tψ = −∆ψ + γδ0ψ

”

I The Green’s function, −∆G = δ0, is

G(x) =1

4π|x|

I Too strong (local) singularity in x = 0 to make sense of the term

δ0ψ(as well as δ0|ψ|2µψ

)I In 1D, G ∈ H1

loc(R)

3D Model

“ i∂

∂tψ = −∆ψ + γδ0|ψ|2µψ ”

“ i∂

∂tψ = −∆ψ + γδ0ψ

”

G(x) =1

4π|x|

)I In 1D, G ∈ H1

loc(R)

3D Model

“ i∂

∂tψ = −∆ψ + γδ0|ψ|2µψ ”

“ i∂

∂tψ = −∆ψ + γδ0ψ

”

G(x) =1

4π|x|

)I In 1D, G ∈ H1

loc(R)

3D Model

“ i∂

∂tψ = −∆ψ + γδ0|ψ|2µψ ”

“ i∂

∂tψ = −∆ψ + γδ0ψ

”

G(x) =1

4π|x|

)I In 1D, G ∈ H1

loc(R)

3D Model

How to define a NLS with concentrated nonlinearity in 3D?[Adami-Dell’Antonio-Figari-Teta ’03] based on previous results by Sayapova-Yafaev ’84

Start with the linear equation

id

dtψ(t) = Hαψ(t)

with“Hα = −∆ + αδ0”

Rigorous definition of Hα

D(Hα) =

{ψ ∈ L2(R3)|ψ = φ+ qG; φ ∈ H2(R3), q ∈ C; φ(0) = αq, α ∈ R

}Hαψ = −∆φ

Or equivalentlyHαψ = −∆ψ ∀x 6= 0

Then set α→ γ|q|2µ

3D Model

id

dtψ(t) = Hαψ(t)

D(Hα) =

}Hαψ = −∆φ

3D Model

id

dtψ(t) = Hαψ(t)

D(Hα) =

}Hαψ = −∆φ

3D Model

id

dtψ(t) = Hαψ(t)

D(Hα) =

}Hαψ = −∆φ

3D Model

A distributional approach

i∂

∂tψ(t) = −∆ψ(t)− q(t)δ0

I For given q(t) : R+ → C this describes a non-autonomous problem

3D Model

i∂

∂tψ(t) = −∆ψ(t)− q(t)δ0

I Solutions of the formψ(t) = φ(t) + q(t)G

I G is the Green’s function: −∆G = δ0I φ(t) regular function

i∂

∂tφ(t) = −∆φ(t)− iq(t)G,

I Initial data of the formψ0 = φ0 + q0G

I By Duhamel formula

φ(t) = ei∆tφ0 −∫ t

0

ds(ei∆(t−s)G)q(s)

3D Model

i∂

∂tψ(t) = −∆ψ(t)− q(t)δ0

i∂

∂tφ(t) = −∆φ(t)− iq(t)G,

φ(t) = ei∆tφ0 −∫ t

0

3D Model

i∂

∂tψ(t) = −∆ψ(t)− q(t)δ0

i∂

∂tφ(t) = −∆φ(t)− iq(t)G,

φ(t) = ei∆tφ0 −∫ t

0

3D Model

i∂

∂tψ(t) = −∆ψ(t)− q(t)δ0

i∂

∂tφ(t) = −∆φ(t)− iq(t)G,

φ(t) = ei∆tφ0 −∫ t

0

3D Model

Settingφ(0, t) = γ|q(t)|2µq(t) γ ∈ R

in

φ(t) = ei∆tφ0 −∫ t

0

We obtain the following equation fo q(t)

q(t) + 4√πi γ

∫ t

0

ds|q(s)|2µq(s)√

t− s= 4√πi

∫ t

0

ds(ei∆sψ0)(0)√

t− s

3D Model

i∂

∂tψ(t) = −∆ψ(t)− q(t)δ0

ψ∣∣t=0

= ψ0

q(t) + 4√πi γ

∫ t

0

ds|q(s)|2µq(s)√

t− s= 4√πi

∫ t

0

ds(ei∆sψ0)(0)√

t− s

Well-posedness in the energy domain:

E :={ψ ∈ L2(R3)|ψ = φ+ qG; φ ∈ H1(R3), q ∈ C

}I Mass and Energy

M [ψ] = ‖ψ‖2 and E[ψ] = ‖∇φ‖2 +γ

µ+ 1|q|2µ+2

are conserved for ψ0 ∈ EI Global Well-Posedness in E [AdAFT ’03]

I If γ ≥ 0 for all µ > 0I If γ < 0 for all 0 < µ < 1

(from the bound |q| ≤ C‖∇φ‖1/2‖ψ‖1/2 for all ψ ∈ E)I Indeed, if γ < 0 and µ ≥ 1 there exist blow-up solutions [AdAFT ’04]

3D Model

i∂

∂tψ(t) = −∆ψ(t)− q(t)δ0

ψ∣∣t=0

= ψ0

q(t) + 4√πi γ

∫ t

0

ds|q(s)|2µq(s)√

t− s= 4√πi

∫ t

0

ds(ei∆sψ0)(0)√

t− sWell-posedness in the energy domain:

E :={ψ ∈ L2(R3)|ψ = φ+ qG; φ ∈ H1(R3), q ∈ C

}

I Mass and Energy

M [ψ] = ‖ψ‖2 and E[ψ] = ‖∇φ‖2 +γ

µ+ 1|q|2µ+2

3D Model

i∂

∂tψ(t) = −∆ψ(t)− q(t)δ0

ψ∣∣t=0

= ψ0

q(t) + 4√πi γ

∫ t

0

ds|q(s)|2µq(s)√

t− s= 4√πi

∫ t

0

ds(ei∆sψ0)(0)√

E :={ψ ∈ L2(R3)|ψ = φ+ qG; φ ∈ H1(R3), q ∈ C

}I Mass and Energy

M [ψ] = ‖ψ‖2 and E[ψ] = ‖∇φ‖2 +γ

µ+ 1|q|2µ+2

are conserved for ψ0 ∈ E

I Global Well-Posedness in E [AdAFT ’03]I If γ ≥ 0 for all µ > 0I If γ < 0 for all 0 < µ < 1

3D Model

i∂

∂tψ(t) = −∆ψ(t)− q(t)δ0

ψ∣∣t=0

= ψ0

q(t) + 4√πi γ

∫ t

0

ds|q(s)|2µq(s)√

t− s= 4√πi

∫ t

0

ds(ei∆sψ0)(0)√

E :={ψ ∈ L2(R3)|ψ = φ+ qG; φ ∈ H1(R3), q ∈ C

}I Mass and Energy

M [ψ] = ‖ψ‖2 and E[ψ] = ‖∇φ‖2 +γ

µ+ 1|q|2µ+2

(from the bound |q| ≤ C‖∇φ‖1/2‖ψ‖1/2 for all ψ ∈ E)

I Indeed, if γ < 0 and µ ≥ 1 there exist blow-up solutions [AdAFT ’04]

3D Model

i∂

∂tψ(t) = −∆ψ(t)− q(t)δ0

ψ∣∣t=0

= ψ0

q(t) + 4√πi γ

∫ t

0

ds|q(s)|2µq(s)√

t− s= 4√πi

∫ t

0

ds(ei∆sψ0)(0)√

E :={ψ ∈ L2(R3)|ψ = φ+ qG; φ ∈ H1(R3), q ∈ C

}I Mass and Energy

M [ψ] = ‖ψ‖2 and E[ψ] = ‖∇φ‖2 +γ

µ+ 1|q|2µ+2

3D Model

[CC-Finco-Noja-Teta]i∂

∂tψ(t) = −∆ψ(t)− q(t)δ0

ψ∣∣t=0

= ψ0

q(t) + 4√πi γ

∫ t

0

ds|q(s)|2µq(s)√

t− s= 4√πi

∫ t

0

ds(ei∆sψ0)(0)√

t− s

Well-posedness in the Operator Domain

D :=

{ψ ∈ L2(R3)|ψ = φ+ qG; φ ∈ H2(R3), q ∈ C; φ(0) = γ|q|2µq

}

I Under the assumptionsI If γ ≥ 0 for all µ ≥ 0I If γ < 0 for all 0 ≤ µ < 1

the limit problem is globally well-posed in D:

for all T > 0 and ψ0 ∈ D, the map ψ0t−→ ψ(t) belongs to C([0, T ],D)

3D Model

[CC-Finco-Noja-Teta]i∂

∂tψ(t) = −∆ψ(t)− q(t)δ0

ψ∣∣t=0

= ψ0

q(t) + 4√πi γ

∫ t

0

ds|q(s)|2µq(s)√

t− s= 4√πi

∫ t

0

ds(ei∆sψ0)(0)√

t− s

Well-posedness in the Operator Domain

D :=

}

I Under the assumptionsI If γ ≥ 0 for all µ ≥ 0I If γ < 0 for all 0 ≤ µ < 1

the limit problem is globally well-posed in D:

for all T > 0 and ψ0 ∈ D, the map ψ0t−→ ψ(t) belongs to C([0, T ],D)

3D Model

What scaled problem approximates the concentrated nonlinearity in 3D?

i∂

∂tψε = −∆ψε +

1

ε3V( ·ε

)|ψε|2µψε

and

i∂

∂tψε = −∆ψε + γρε|〈ρε, ψε〉|2µ〈ρε, ψε〉

with ρε(x) = 1ε3ρ(x/ε)

Certainly not!

3D Model

ρ ∈ C∞0 (R3), ρ > 0, ρ(x) = ρ(|x|), and∫ρ(x)dx = 1

` =

∫ρ(x)ρ(y)

4π|x− y|dx dy > 0 and ρε(x) =1

ε3ρ(x/ε)

Regular problemi∂

∂tψε = −∆ψε − ρε ε

`〈ρε, ψε〉+ γρε

ε2µ+2

`2µ+2|〈ρε, ψε〉|2µ 〈ρε, ψε〉

ψε∣∣∣t=0

= ψε0

Singular problemi∂

∂tψ(t) = −∆ψ(t)− q(t)δ0

ψ∣∣t=0

= ψ0

q(t) + 4√πi γ

∫ t

0

ds|q(s)|2µq(s)√

t− s= 4√πi

∫ t

0

ds(ei∆sψ0)(0)√

t− s

Then ψε(t)→ ψ(t) as ε→ 0

3D Model

` =

∫ρ(x)ρ(y)

ε3ρ(x/ε)

Regular problemi∂

ε2µ+2

`2µ+2|〈ρε, ψε〉|2µ 〈ρε, ψε〉

ψε∣∣∣t=0

= ψε0

∂tψ(t) = −∆ψ(t)− q(t)δ0

ψ∣∣t=0

= ψ0

q(t) + 4√πi γ

∫ t

0

ds|q(s)|2µq(s)√

t− s= 4√πi

∫ t

0

ds(ei∆sψ0)(0)√

t− s

3D Model

` =

∫ρ(x)ρ(y)

ε3ρ(x/ε)

Regular problemi∂

ε2µ+2

`2µ+2|〈ρε, ψε〉|2µ 〈ρε, ψε〉

ψε∣∣∣t=0

= ψε0

∂tψ(t) = −∆ψ(t)− q(t)δ0

ψ∣∣t=0

= ψ0

q(t) + 4√πi γ

∫ t

0

ds|q(s)|2µq(s)√

t− s= 4√πi

∫ t

0

ds(ei∆sψ0)(0)√

t− s

3D Model

How to chose the initial data.

“Operator Domain”

Scaled problemψε0 ∈ H2(R3)

Limit problem ψ0 ∈ D

D :=

}

We fix ψ0 ∈ D and choose

ψε0 = φ0 + q0ρε ∗G

for all ε > 0 one has ψε0 ∈ H2(R3) and

‖ψε0 − ψ0‖ ≤ Cε1/2

3D Model

D :=

}

ψε0 = φ0 + q0ρε ∗G

‖ψε0 − ψ0‖ ≤ Cε1/2

3D Model

D :=

}

ψε0 = φ0 + q0ρε ∗G

‖ψε0 − ψ0‖ ≤ Cε1/2

3D Model

D :=

}

ψε0 = φ0 + q0ρε ∗G

‖ψε0 − ψ0‖ ≤ Cε1/2

3D Model

TheoremLet ψ0 ∈ D and ψε0 as before. Moreover fix T > 0, γ ∈ R, and µ > 0 if γ ≥ 0,or 0 < µ < 1 if γ < 0. Then there exist positive constants ε0, C andδ ∈ (0, 1/4) such that, for all 0 < ε < ε0,

supt∈[0,T ]

‖ψε(t)− ψ(t)‖ ≤ Cεδ

3D Model: Idea of the proof

〈ρε, ψε(t)〉 ε→0−−−→∞

butqε(t) :=

ε

`〈ρε, ψε(t)〉 ε→0−−−→ q(t)

Comparison of the eqs. for ψε and ψ

i∂

∂tψε = −∆ψε − ε

`〈ρε, ψε〉ρε + γ

ε2µ+2

`2µ+2|〈ρε, ψε〉|2µ 〈ρε, ψε〉ρε

i∂

∂tψ = −∆ψ − qδ0

The approximating equation reads

i∂

∂tψε = −∆ψε − qερε + γ

ε

`|qε|2µ qερε

Recall ρε → δ0, hence, if qε → q then

ψε(t)→ ψ(t)

Why qε → q?

〈ρε, ψε(t)〉 ε→0−−−→∞

butqε(t) :=

ε

`〈ρε, ψε(t)〉

ε→0−−−→ q(t)

i∂

ε2µ+2

i∂

∂tψ = −∆ψ − qδ0

i∂

ε

`|qε|2µ qερε

ψε(t)→ ψ(t)

Why qε → q?

〈ρε, ψε(t)〉 ε→0−−−→∞

butqε(t) :=

ε

`〈ρε, ψε(t)〉 ε→0−−−→ q(t)

i∂

ε2µ+2

i∂

∂tψ = −∆ψ − qδ0

i∂

ε

`|qε|2µ qερε

ψε(t)→ ψ(t)

Why qε → q?

〈ρε, ψε(t)〉 ε→0−−−→∞

butqε(t) :=

ε

`〈ρε, ψε(t)〉 ε→0−−−→ q(t)

i∂

ε2µ+2

i∂

∂tψ = −∆ψ − qδ0

i∂

ε

`|qε|2µ qερε

ψε(t)→ ψ(t)

Why qε → q?

〈ρε, ψε(t)〉 ε→0−−−→∞

butqε(t) :=

ε

`〈ρε, ψε(t)〉 ε→0−−−→ q(t)

i∂

ε2µ+2

i∂

∂tψ = −∆ψ − qδ0

i∂

ε

`|qε|2µ qερε

ψε(t)→ ψ(t)

Why qε → q?

〈ρε, ψε(t)〉 ε→0−−−→∞

butqε(t) :=

ε

`〈ρε, ψε(t)〉 ε→0−−−→ q(t)

i∂

ε2µ+2

i∂

∂tψ = −∆ψ − qδ0

i∂

ε

`|qε|2µ qερε

ψε(t)→ ψ(t)

Why qε → q?

〈ρε, ψε(t)〉 ε→0−−−→∞

butqε(t) :=

ε

`〈ρε, ψε(t)〉 ε→0−−−→ q(t)

i∂

ε2µ+2

i∂

∂tψ = −∆ψ − qδ0

i∂

ε

`|qε|2µ qερε

ψε(t)→ ψ(t)

Why qε → q?

By Duhamel formula

ψε(t) = ei∆tψε0 + i

∫ t

0

ds(ei∆(t−s)ρε)qε(s)

− iγ ε`

∫ t

0

ds(ei∆(t−s)ρε) |qε(s)|2µ qε(s)

⇓

`

εqε(t) = 〈ρε, ei∆tψε0〉+ i

∫ t

0

ds〈ρε, ei∆(t−s)ρε〉qε(s)

− iγ ε`

∫ t

0

ds〈ρε, ei∆(t−s)ρε〉 |qε(s)|2µ qε(s)

By using

i

∫ t

0

〈ρε, ei∆(t−s)ρε〉f(s)ds =`

εf(t)− 1

4π√πi

f(0)√t− 1

4π√πi

∫ t

0

f(s)√t− s

ds+o(1)

By Duhamel formula

∫ t

0

− iγ ε`

∫ t

0

⇓

`

∫ t

0

− iγ ε`

∫ t

0

By using

i

∫ t

0

εf(t)− 1

4π√πi

f(0)√t− 1

4π√πi

∫ t

0

f(s)√t− s

ds+o(1)

By Duhamel formula

∫ t

0

− iγ ε`

∫ t

0

⇓

`

∫ t

0

− iγ ε`

∫ t

0

By using

i

∫ t

0

εf(t)− 1

4π√πi

f(0)√t− 1

4π√πi

∫ t

0

f(s)√t− s

ds+o(1)

How to prove

i

∫ t

0

〈ρε, ei∆(t−s)ρε〉f(s)ds ' `

εf(t)− 1

4π√πi

d

dt

∫ t

0

f(s)√t− s

ds

Since

ei∆tρε = ei∆tρε ∗ δ0 = ei∆tρε ∗ (−∆G) = −∆ei∆tρε ∗G = id

dtei∆tρε ∗G

we have

i〈ρε, ei∆tρε〉 = − d

dt〈ρε, ei∆tρε ∗G〉

Hence

i

∫ t

0

〈ρε, ei∆(t−s)ρε〉f(s)ds

=〈ρε, ρε ∗G〉f(t)− 〈ρε, ei∆tρε ∗G〉f(0)−∫ t

0

〈ρε, ei∆(t−s)ρε ∗G〉f(s)ds

To conclude

〈ρε, ρε ∗G〉 =`

εand

〈ρε, ei∆(t−s)ρε ∗G〉 =1

4π√πi

1√t− s

+O

(εη

(t− s)(1+η)/2

), 0 < η < 1

Eε[ψε] = ‖∇ψε‖2 − ε

`|〈ρε, ψε〉|2 +

γ

µ+ 1

ε2+2µ

`2+2µ|〈ρε, ψε〉|2+2µ

Setqε :=

ε

`〈ρε, ψε〉

Then

Eε[ψε] = ‖∇ψε‖2 − `

ε|qε|2 +

γ

µ+ 1|qε|2+2µ

Setting

φε := ψε − qερε ∗G one has ψε = φε + qερε ∗G

and

‖∇ψε‖2 = ‖∇φε‖2 +`

ε|qε|2

ThenEε[ψε] = ‖∇φε‖2 +

γ

1 + µ|qε|2+2µ

From mass and energy conservation, under the usual assumptions on γ and µ,one obtains the a priori bounds

‖∇φε‖ ≤ C and |qε| ≤ C

By the mass conservation

‖ψε(t)− ψ(t)‖2 =‖ψε0‖2 + ‖ψ0‖2 − 2 Re 〈ψ(t), ψε(t)〉

=‖ψε0‖2 − ‖ψ0‖2 − 2 Re 〈ψ(t), ψε(t)− ψ(t)〉

Hence we need to prove

limε→0|〈ψ?, ψε(t)− ψ(t)〉| = 0

for all ψ? ∈ D and t ∈ [0, T ].

limε→0|〈ψ?, ψε(t)− ψ(t)〉| = 0

exploiting

ψε(t)−ψ(t) = ei∆t(ψε0−ψ0)+i

∫ t

0

ds(

(ei∆(t−s)ρε)qε(s)− (ei∆(t−s)δ0)q(s))

− iγ ε`

∫ t

0

one is left with terms of the form

limε→0

∣∣∣∣∫ t

0

ds(qε(s)− q(s))(ei∆(t−s)ψ?)(0)

∣∣∣∣ = 0

It turns out that it is enough to prove

limε→0

∣∣∣∣∫ t

0

dsqε(s)− q(s)√

t− s

∣∣∣∣ = 0

We are not able to prove limε→0 |qε(t)− q(t)| = 0!!

Both in 1D and 3D the approximating problems were modeled on the linearcounterparts:

I In 1D

−∆ +λ

εV( ·ε

)→ Hα with α = λ

∫RV dx

and

−∆ +λ

ερ( ·ε

)(ρε, ·)→ Hα with α = λ

I In 3D

−∆− 1

`

1

ε2ρ( ·ε

)(ρε, ·) + λ

1

`21

ερ( ·ε

)(ρε, ·)→ Hα with α = λ

I The approximating problem in the nonlinear case is obtained by setting λequal to a (properly chosen) function of ψ

I In the linear case it is more convenient to prove the convergence of theresolvent

Zero energy resonance. We say that −∆ + V has a zero energy resonance ifthere exists a distributional solution of (−∆ + V )ψ = 0, i.e., ψ ∈ H2, ψ /∈ L2

Local approximations of Hα.

Both for d = 1 [Albeverio-C.-Finco, C.-Exner, Hryniv-Golovaty-Manko] andd = 3 [Albeverio et al. 2005], if −∆ + V has a zero energy resonance then

−∆ +1

ε2V( ·ε

)+λ

εV( ·ε

)→ Hα

with α ∝ λI For d = 1 the limit operator is not necessarily a point interaction of delta

type, but the “strength” parameter is still proportional to λ

Conjecture: If λ = λ(ψε) then the limit is a non-linear point interaction