Far from equilibrium steady states of 1D-Schrödinger-Poisson systems with quantum wells II

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Far from equilibrium steady states of

1D-Schrödinger-Poisson systems with quantum wells II

Virginie Bonnaillie-Noël, Francis Nier, Mamodyasine Patel

To cite this version:

Virginie Bonnaillie-Noël, Francis Nier, Mamodyasine Patel. Far from equilibrium steady states of

1D-Schrödinger-Poisson systems with quantum wells II. 2007. �hal-00124705v2�

1D-Shrödinger-Poisson systems with quantum wells II

V.Bonnaillie-Noël

∗

,F. Nier

†

,Y. Patel

Abstrat

This artile ontinues theasymptotianalysis ofanonlinearShrödinger-Poisson system

whihmodels ina far fromequilibrium regimethe quantumtransport ineletroni devies

likeresonant tunnelingdiodes. Withinthe redutionto anh-dependentlinearproblemwith uniform regularity estimates for the potential already established in the rst part, expliit

omputationsof theasymptoti nitedimensional nonlinear system are derived. They rely

onanaurate(phase-spae)analysisofthetunneleetwhihreliesonsomekindofBreit-

WignerformulaandFermigoldenrule.

MSC (2000): 34L25;34L30;34L40;65L10;65Z05;81Q20;82D37.

Keywords: Shrödinger-Poissonsystem;Asymptotianalysis;Multisaleproblems.

Contents

1 Introdution 2

2 Assumptionsand results 4

3 Redution ofthe relevant energyinterval 6

4 Lower bound forthe imaginary parts of theresonanes 7

5 Resolvent estimatesaroundan asymptotiresonantenergy 9

6 Case ofstrong gatherness 11

7 Isolated Wells 13

7.1 Preliminaryresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7.2 Breit-Wignerformulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7.3 AFermi-Goldenrule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.4 Valuesoftheoeientst^λ_i^{0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21}

8 Expliitasymptotivalues 23

A Agmon energyidentity 27

∗

IRMAR, UMR-CNRS 6625, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, Frane,

Virginie.Noel-Bonnaillieuniv-rennes1.fr

†

IRMAR, UMR-CNRS 6625, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, Frane,

Franis.Nieruniv-rennes1.fr

1 Introdution

Weompletetheasymptotianalysisstartedin[BNP1℄ofsomeout-of-equilibrium1DShrödinger-

Poissonsystemarisingfromthemodellingofresonanttunellingdiodes. Thisproblemisanonlinear

problem whosefuntional frameworkwasonsidered in [BDM℄,[Ni3℄ withinaLandauer-Büttiker

approah[BuLa℄, [ChVi℄,[Lan℄ (seealso [NiPa℄,[Pat℄,[JLPS ℄, [PrSj ℄, [BNP℄,[BNP1℄). Wereall

that theanalysis hasbeenredued,in [BNP1℄,to anh^{-dependentlinear problemafter providing}

uniform estimates for theinitial semilinear problem. Hene weonsider for h > 0 ^{goingto zero}

andforsomexedintervalI= [a, b]^{theShrödingeroperatoronthereal line,} P^h:=− d²

dx² + ˜V^h−W^h, V˜^h:=−B+V^h, V^h∈W^1,∞(a, b), ^(1.1)

where

B(x) =−Bx−a

b−a1[a,b](x)−B·1[b,+∞)(x) ^(1.2)

and B ^{is a} non-negativeonstant. The potential B ^{simply models the applied bias. The family}

of potentials (V^h)h∈(0,h0) ^{has uniformly bounded seond} derivatives ∂_x²V^h =∂_x²V˜^{h in} M^b([a, b])

whihonvergeweaklytosomemeasureµ⁰∈ M^b([a, b])^{,withtheadditionalboundaryonditions} V^h(a) =V^h(b) = 0.

Reall that this makes abounded familyof funtions V˜^{h in} W^1,∞(a, b) ^{and whih onverges in} C^0,α(I)^, α <1^,toafuntionV˜^0,∂_x²V˜⁽⁰⁾

(a,b)=µ⁰

(a,b)^{. Weassumethat} h∈(0,hinf0),x∈I

V˜^h(x) =: Λ0>0. ^(1.3)

Finally,thepotential−W^{h desribesquantumwellsaordingto} W^h(x) =

N

X

i=1

wi

x−ci

h

, ^(1.4)

where c1 < . . . < cN ^are N ^{given points in} (a, b) ^{and the funtions} wi ^{are ontinuous1 positive}

funtions supported in [−κ, κ] ^{for some xed} κ > 0^{. We shall use the onvention} c0 = a ^and cN+1=b^{. The} HamiltonianH^{his the}self-adjointrealization ofP^{h ontherealline withdomain} H²(R)

∀u∈D(H^h) =H²(R), H^hu:=P^hu. ^(1.5)

ReallthatthenotationP ^{isusedforthedierentialoperatorwhile}H ^{isreservedforsomelosed}

nonneessaryself-adjointrealizationasanunboundedoperatoronL^2.

The potentialswi^,i = 1, . . . , N^{, is hosenso that thespetrum} σ(Hi)^{of the}HamiltoniansHi =

−∆−wi ^satises

V˜^h(ci) + infσ(Hi)≥κi>0,

with κi independent of h^{. With suh an assumption the operator} H^{h has a purely ontinuous}

spetrumequalto[−B,∞)^.

Due totheappliedbiasB≥0^{,thedispersionrelationassoiatedwiththe}HamiltonianH^{h reads} λk:=

k^{2 if}k >0,

k²−B ^ifk <0. ^(1.6)

1

In[BNP1℄,thenonlinearanalysiswasarriedoutwithonlyw_i∈L^∞(I)^.

Fork∈Rsuhthat λk ∈(−B,+∞)\ {0}^{,the inomingsatteringstate}ψ−(k, x)^{is thesolution}

of

P^hψ^h−(k,·) =λkψ−^h(k,·), ^(1.7)

with thenormalization

fork >0 ψ−(k, x) =







e^ikxh +rke^{−ikxh for} x < a , tke^i(λk+B)1

/2x

h for x > b ,

fork <0 ψ−(k, x) =







tke^−i(λk)1

/2x

h for x < a , e^ikxh +rke^{−ikxh for} x > b .

Thesquarerootz^{1/2ishosenwiththeramiationalongthehalf-line}iR₋inordertoensurethat

e^{−i(λk)1/2x deays}exponentiallyasx→ −∞^whenλk∈(−B,0)^.

This anbereduedtok^-dependenttransparentboundaryonditions

fork >0







hh∂x+iλ^1/2_k i

u(a) = 2ike^ikah, h∂x−i(λk+B)^1/2

u(b) = 0,

(1.8)

fork <0







hh∂x+iλ^1/2_k i

u(a) = 0, h∂x−i(λk+B)^1/2

u(b) = 2ike^ikbh .

(1.9)

Theoeientstk ^andrk ^arethetransmissionandreexionoeientsandsatisfyforλk >0

|rk|²+ r λk

λk+B|tk|²= 1. ^(1.10)

Denote, for i= 1, . . . , N ^byσi ^{theset of negative}eigenvaluesoftheHamiltonian Hi =−∆−wi

with D(Hi) =H²(R)

σi :={eⁱ_k}k∈Ki⊂(−∞; 0), Ki⊂N, i= 1, . . . , N . ^(1.11)

Thesetofasymptotiresonantenergiesisdened as

E0:=

N

[

i=1

Ei, Ei:=σi+ ˜V⁰(ci). ^(1.12)

Letusreallaswellthenotionofasymptotiresonantwellsassoiatedwithλ∈ E0^:

Jλ:={i∈ {1, . . . , N} ^{s. t.} λ∈ Ei}.

Themultipliitymλ ^{oftheasymptotiresonantenergy}λ^{is givenby} mλ:= #Jλ.

Likein[BNP1℄,wefousonpositiveenergies: Wexanenergydomain(Λ∗,Λ^∗)⊂(0,Λ0)^,andwe

onsider thefuntions

θ∈ Cc⁰((Λ∗,Λ^∗)), θ≥0, ^(1.13)

and g(k) =θ(λk)1R+(k). ^(1.14)

g(K₋^h)[x, y] = Z

k

g(k)ψ^h₋(k, x)ψ^h₋(k, y) dk

2πh, ^(1.15)

andweareinterestedintheasymptotiofthepartiledensityn^h(x)^{dened by} Z b

a

ϕ(x)dn^h(x) =^Tr

g(K₋^h)ϕ(x)

, ∀ϕ∈ Cc⁰((a, b)),

orequivalently

n^h(x) = Z

k

g(k)|ψ₋^h(k, x)|² dk 2πh.

Theresultof[BNP1,Theorem1.6℄statesthat,possiblyafterextratingasubsequene,themeasure

dn^{h onvergesweaklyto}dn^{0 in}M^b((a, b))^with dn⁰= X

λ∈E0

X

i∈Jλ

t^λ_i θ(λ)δx=ci, t^λ_i ∈[0,1]. ^(1.16)

Ouraimhereistheauratedetermination oftheoeientst^λ_i ^{aordingtothegeometry ofthe}

potential.

Reallthat thisresult, [BNP1,Theorem1.6℄,isessentiallyobtainedbyhekingthat thet^λ_i^'s

are equalto 1 ^{whenthe funtion} g(k)^{isreplaed by}θ(λk)^and g(K₋^h)^by θ(H^h)^{. In thisartile,}

wefousontheanisotropiasewheng(k) =θ(λk)1R+(k)^{annotbewritten asafuntionof the}

energy. Notethatdue tothedeomposition

θ(H^h) =g−(K₋^h) +g+(K₋^h), g−(k) =1k<0·θ(λk), g+(k) =1k>0·θ(λk), ^(1.17)

theresultanbetranformedintoaresultforfuntionsg− ^{supportedonnegativemomentumand}

evenarriesovertomoregeneralombination.

2 Assumptions and results

Sine(1.16)isaloalresultontheenergyaxiswhilethesetofasymptotiresonantenergiesE0^is

nite, theanalysisanbepartlysimpliedafterthenextassumption.

Assumption 1 Supposethatthe supportoffuntionθ ^andthereforeg(k) =1k>0·θ(λk),^ontains

only oneasymptotiresonantenergy

suppθ∩ E⁰={λ0}.

The nextassumptions aretehniallymoreserious. Somespei ongurationsallowto handle

aurately and quite simplythe disussion with respet to thegeometry in terms of theAgmon

distane.

Denition 2.1 With an energy λ ∈ R and a potential V ∈ L^∞(I)^{, is assoiated the Agmon}

(possibly degenerate)distaned(., .;V, λ)^denedby:

∀x, y∈I, d(x, y;V, λ) =

Z y x

p(V(t)−λ)+dt

. ^(2.1)

Notation 1 TheAgmondistaneassoiatedwith theasymptoti potentialV˜^{0 and theasymptoti}

resonantenergyλ0 ^isdenotedbyd0^{. Itisdenedby}

d0(x, y) :=

Z y x

qV˜⁰(τ)−λ0dτ ,

With this distane, let

S0:=d0(∪i∈Jλ0{ci}, ∂I), SU := max

i,j∈Jλ0

d0(ci, cj), SI :=d0(a, b) ^(2.2)

berespetively thedistanebetweenthe λ0^{-resonant wellsandthe boundary}∂I ={a, b}^{,the diam-}

eter ofthe union ofthe resonantwells, andthe diameterof the island.

Itis sometimesonvenienttointroduethe set

U ={c1, . . . , cN}.

Finally, introduefor η0>0^{the quantity} S˜U := max

τ∈[c1,cN]

qV˜⁰(τ) +η0−λ0|cN−c1|,

whih measuresthe diameter ofthe areawhih ontainsallthe wells.

Notie thatS˜U ^{iswritten intermsof some}L^{∞-norm ofthepotentialinsteadofanintegral. The}

parameterη0 ^{isintroduedinorderto ensure}S˜U > SU^{. Itan behosen}arbitrarilysmall.

Denition 2.2 We say that the λ0^{-resonant wells are gathered (resp. strongly gathered) if and}

only if

S0+SU < SI/2 (resp. S0+mλ0SU < SI/2). ^(2.3)

As S0+SU ^{is the greatest distane from the boundary of the island to the resonant wells, the}

onditionS0+SU < SI/2 ^{expressesthattheresonantwellsaregatheredin onethehalvesof the}

island. This explainstheterminology.

Denition 2.3 Wesaythat the wellsareisolatedifandonly if

S0>8 ˜SU ^and mλ0 =N. ^(2.4)

Inequality(2.4)meansthatthewellsareonnedintheentralpartoftheisland.

Theorem2.4 MakeAssumption 1. Supposethat the λ0^{-resonantwells arestrongly gathered, or}

suppose that the wells are isolated (mλ0 = N^{) and gathered with} N =mλ0^{. Then the two next}

statements hold:

i) Theoeientst^λ_i^0,i∈Jλ0^{,areallequal to}1 ^ifd0(a, ci)< d0(ci, b)^{for all}i∈Jλ0^.

ii) The oeientst^λ_i^{0 ,}i∈Jλ0^{,areallequal to}0 ^ifd0(a, ci)> d0(ci, b)^{for all} i∈Jλ0^.

In the rstase thewellsare onned in the left-handhalf of theisland, whereasin the seond

asethewellsareonned intheright-handside oftheisland,thispartition beingdoneinterms

of theAgmon distane d0^{. Thisresult anbe}interpretedin termsof tunneling eet: in ase i) thetunnelingeetiseasierfroma^{tothewellsthanfromthewellsto}b^{,thepartilesomingfrom}

−∞^(rememberg+(−|k|) = 0)^{aretrappedbythewells;inaseii),thepartileesapemoreeasily}

from thewellstob^{thantheygetintothewellsfrom}a^.

Theorem2.5 Assume that the wells are isolated aording to Denition 2.3 (mλ0 = N^{). Let} λ^h₁ < . . . < λ^hmλ0

bethe eigenvaluesoftheDirihletHamiltonianH_I^{h on}I= [a, b]^onvergingtoλ0

as h→0 ^{with the normalized}eigenvetors φ^h₁, . . . , φ^h_mλ

0

. Fix ε∈(0,1/2 min0≤i6=i^′≤N+1|ci−ci^′|)

and let ψ˜^h₋(k,·) ^{be the} generalized eigenfuntions of H˜^h = H^h+W^{h. Then the oeient} t^λ_i^0, i= 1, . . . , mλ0^{,isobtainedasthe limit ofthe quantity}

mλ0

X

j=1

Z ci+ε

ci−ε |φ^h_j(x)|² dx 1 +

q λ^h_j

D

φ^h_j , W^hψ˜₋^h(−q

λ^h_j +B,·)E

2

qλ^h_j +B

Dφ^h_j, W^hψ˜^h₋(+q λ^h_j,·)E

2

, ^(2.5)

as h→0 ^{(after possibly extrating a}subsequene).

From this result non trivial ases for whih not all the t^λ_i ^{belong to} {0,1} ^{will be exhibited in}

Setion 8,in partiularinProposition 8.5andProposition8.6.

WhenN = 1^{,wewillestablishthat,theoeient}t^λ₁^{0 belongsto}(0,1)^onlyifd0(a, c1) =d0(c1, b)^.

Intheaseof twowellsN = 2^,thevaluesoft^λ₁^{0 and}t^λ₂^{0 havetofulll thenextrules}

1. t^λ₁⁰ = 1^andt^λ₂⁰∈[0,1]^ifd0(a, c1)< d0(c2, b)^;

2. t^λ₁⁰ ∈[0,1]^andt^λ₂⁰= 0 ^ifd0(a, c1)> d0(c2, b)^;

3. 1≥t^λ₁⁰≥t^λ₂⁰≥0 ^ifd0(a, c1) =d0(c2, b)^.

Alltheseruleswhihwereprovedonlyforisolatedwellsandespeiallythegeneralonditiont^λ₁⁰ ≥ t^λ₂^{0 haveaverynatural}interpretationwithintheprobabilistipresentationofquantummehanis.

Theyareprobablyvalidin allasesalthough ourproofrequiressomespeiassumptions. They

weretakenasgrantedinthenumerialappliationstreatedin[BNP℄. Notethatourresultsprovide

essentiallyaompleteunderstandingofwhatisgoingonwhenthereisnointerationofresonanes,

orwhentheinteration ofresonantstatesinvolvesonlytwowells. Inthenalnonlinearproblem

presentedin [BNP ℄, [BNP1 ℄, theoeients t^λ_i ^{play the roleof Lagrange} multipliers whih have an arbitraryvaluein [0,1]^{when theassoiatedonstraintfor theasymptotiresonantenergy or}

theAgmondistanesissaturated.

Finallynotethattheassumptionmλ0 =N^{intheseondaseofTheorem2.4(isolatedandgathered}

wells)isnotruial. Itisassumedhereinordertoavoidsomeunessentialtehnialitieswhihhave

alreadybeenonsideredin[BNP1℄andaretreatedin thesligthlysimplerrstase.

3 Redution of the relevant energy interval

In [BNP1℄, asmall h^{-dependent energy domain around}λ0 ^{has been introdued. Let} H_I^{h denote}

the Dirihlet realization of P^{h on the interval} I = [a, b] ^{and let} {λ^h₁, . . . , λ^h_mλ

0} ^{be the ordered}

eigenvaluesonvergingtoλ^{0 as}h→0^{. Set}

Ωh:={z∈C s.t. Re(z)∈Kh, ^Im(z)∈[−4h,4h]} ^(3.1)

with Kh:= [λ0−α^h, λ0+α^h] ^(3.2)

and α^h:= 4 max

h,|λ0−λ^h_j|, j = 1, . . . , mλ0 . ^(3.3)

TheProposition 6.4of[BNP1℄yieldsthenextenergyintervalredution.

h→0lim

Tr

g(K₋^h)ϕ(x)

−g(p λ0)^Tr

1Kh(H^h)1(0,+∞)(K₋^h)ϕ(x)

= 0

holds forany ϕ∈ Cc⁰((a, b))^.

Hene wewill mainly fousonthe energieslyingin Kh ^{and onthe spetralparameterslying}

in Ωh^{in thesequel.}

4 Lower bound for the imaginary parts of the resonanes

In thissimple one-dimensionalproblem where thepotential ispieewise onstantoutside aom-

pat interval, the resonanes are easily introdued after an expliit omplexdeformation of the

transparentboundaryonditions(1.8)-(1.9). Theoperator H_ζ^{h isdened foraomplex}ζ ^lyingin

aneighborhoodofλ∈(−B,0)^by D(H_ζ^h) =

(

u∈H²(I),

h∂x+iζ^1/2

u(a) = 0, h∂x−i(ζ+B)^1/2

u(b) = 0 )

, ^(4.1)

H_ζ^hu=P^hu= [−h²∆ +V^h(x)]u , ∀u∈D(H_ζ^h). ^(4.2)

The resonanes are then exatly the omplex values z ^{for whih the operator} (H_z^h −z) ^{is not}

injetive(see[BNP1℄forthis spei aseand[BaCo℄,[HeSj1℄,[HiSi℄formoregeneralversionsof

theomplexdeformation).

It wasproved in [BNP1℄that theresonanes onvergingto λ0 ^{lie in a}O˜(e^−2S0/h)-neighborhood of theDirihleteigenvalues(see[BNP1,Proposition5.2℄). Hene wegettheusualresultthat the

imaginarypartofresonanesonvergingto λ0 ^areexponentiallysmall Im(z^h) = ˜O(e^−2Sh0).

Providing a lower bound for the imaginary part of resonanes is a standard result within the

semilassial analysis of resonanes (see [HeSj1℄). Wehek itwith a morepedestrian approah

forour1Dproblemwherethepotentialdoesnottexatlywiththesemilassialsettingandhas

alimitedregularity. Notethat thelowerboundanbemuhsmallerthantheupperboundin the

multiplewellase.

Proposition4.1 For any η >0,^{there exists apositive onstant} Cη >0 ^{suhthat for any reso-}

nanez^{h onvergingto}λ0^,onehas

Cηe^−2S0h−η ≥ −^Im(z^h)≥C_η⁻¹e^{−2(S0 +SUh )+η}. ^(4.3)

Proof: Letz^{hsuharesonaneand}u^{hanormalizedresonantstateassoiated,thatisanelement}

in thekernelofH_z^hh−z^{h with}L²(I)^{-normequalto1. Itsatises}

−h²∆u^h+V^h(x)u^h=z^hu^h, u^h

L²(I)= 1,

with the boundary onditions provided by u^h ∈ D(H_z^hh)^{. By taking the imaginarypart of the}

identity(A.1 )applied withV =V^h,u2=u1=u^h,z=z^hand ϕ≡0 ^onegets

−^Im(z^h) =h^Re(p

z^h+B)|u^h(b)|²+h^Re(√

z^h)|u^h(a)|². ^(4.4)

If theimaginarypartofz^{h istoosmall,} u^{h satises aCauhyproblem in}x=a^{withsmalldatas}

beauseoftheresonantboundaryonditionsandlimh→0z^h=λ0∈(Λ∗,Λ^∗)^{. Wenexthekthat}

suhasmallnessislimitedbythenormalizationassumption

u^h

L² = 1^{. Inordertogetthis,set}

F(x) :=





u^h(x) ihdu^h

dx (x)



. ^(4.5)

F ^{satisestheODEon}I ihdF

dx =A^h(x)F(x), A^h(x) :=

0 1 z^h− V^h 0

, V^h= ˜V^h−W^h. ^(4.6)

EndowC²withthestandardhermitiannorm. Ifρ^h(x)^{denotesthespetralradiusof}A^h(x)A^h(x)^T,

onegetstheestimate

hdF dx

2

≤ρ^h(x)|F(x)|². ^(4.7)

ByGronwall'slemmathisyields

|F(x)| ≤min

|F(a)|e^{1hRax|zh−Vh(τ)|1/2dτ};|F(b)|e^{h1Rxb|zh−Vh|1/2dτ}

, ^(4.8)

forallx∈I^{. The}transparentonditionsgivenbyu^h∈D(H_z^hh)^imply

|F(a)|²=|u^h(a)|²(1 +|z^h|), |F(b)|²=|u^h(b)|²(1 +|z^h+B|). ^(4.9)

Apply nowthe Agmon estimate tehnique like in [DiSj ℄ in order to hek that the resonant

wavefuntion onentratesin thewells: Taking thereal partoftheidentity(A.1) withV =V^h, z=z^h, u1=u2=u^handϕ(x) =d(x,^suppW^h; ˜V^h−ε0,^Rez^h)^withε0>0 ^leadsto

0 = Z b

a

h∂x(e^ϕhu^h)

2

dx+ε0

Z

I\^suppW^h

e^ϕhu^h

2

dx +

Z

suppW^h

( ˜V^h(x)−W^h(x)−^Rez^h) u^h

2 dx

+h^Im[(z^h)^1/2]e^2ϕ(a)h u^h(a)

2+h^Im[(z^h+B)^1/2]e^2ϕ(b)h u^h(b)

2 .

Sinelimh→0z^h=λ0>0^andIm(z^h) = ˜O(e^−2S0/h)^{andfrom(4.4)wededuetheestimate} Z

I\^suppW^h

h∂x(e^ϕhu^h)

2

+ε0

e^ϕhu^h

2

dx≤O˜ e^−4Sh0

maxn

e^2ϕ(a)h , e^2ϕ(b)h o

− Z

suppW^h

( ˜V^h(x)−W^h(x)−^Rez^h) u^h

2 dx .

Owing toϕ(a)≤d0(a, U)^andϕ(b)≤d0(b, U)^forh >0^{smallenoughandto} u^h

L² = 1^weget Z

I\^suppW^h

h∂x(e^ϕhu^h)

2

+ε0

e^ϕhu^h

2

dx≤C

for some onstant independent of h >0 ^{(small enough). Let} χ ^{aut-o funtion whih anels}

around the boundary of I^{. Then,} χu^{h is lose to an} eigenfuntion for the Dirihlet operator