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A LINEARIZED MODEL OF QUANTUM TRANSPORT IN THE ASYMPTOTIC REGIME OF QUANTUM WELLS
1 A. Mantile
1 Laboratoire de Mathématiques, Université de Reims - FR3399 CNRS, Moulin de la
Housse BP 1039, 51687 Reims, France
PACS 03.65.Xp, 02.30.Jr, 03.65.Sq DOI 10.17586/2220-8054-2015-6-1-100-112
The effects of the local accumulation of charges in resonant tunnelling heterostructures have been described using 1D Shrodinger-Poisson Hamiltonians in the asymptotic regime of quantum wells. Taking into account the features of the underling physical system, the corresponding linearized model is naturally related to the adiabatic evolution of shape resonances on a time scale which is exponentially large w.r.t. the asymptotic parameter h. A possible strategy to investigate this problem consists of using a complex dilation to identify the resonances with the eigenvalues of a deformed operator. Then, the adiabatic evolution problem for a sheet-density of charges can be reformulated using the deformed dynamical system which, under suitable initial conditions, is expected to evolve following the instantaneous resonant states.
After recalling the main technical difficulties related to this approach, we introduce a modified model where h-dependent artificial interface conditions, occurring at the boundary of the interaction region, allow one to obtain adiabatic approximations for the relevant resonant states, while producing a small perturbation of the dynamics on the scale hNo. According to these results, we finally suggest an alternative formulation of the adiabatic problem. An a posteriori justification of our method is obtained by considering an explicitly-solvable case.
Keywords: Schrodinger-Poisson equation, adiabatic evolution of resonances. Received: 25 January 2015
1. Introduction
We consider the axial transport through resonant tunneling structures like highly doped p-n semiconductor heterojunctions (Esaki diodes), multiple barriers or quantum well diodes. In such systems, the conduction band edge-profile is described by a multiple-barrier potential (see e.g. in [7]). Due to the quantum tunneling, the charge carriers, at the resonant levels, interact with metastable states and, depending on the potential geometry, a local accumulation of charges may result. The corresponding repulsive effect, which can strongly modify the transport properties, have been described in the mean-field approximation by one-particle quantum Hamiltonians of Hartree-type with Poisson nonlinearity ( [13]). In this framework, the barriers depth, fixing the time scale for the dispersion of the metastable states, can be rather large compared to the size of the wave packets. An unitarily equivalent description of the model then consists of replacing the kinetic part of the Hamiltonian with the 'semiclassical' 1D Laplacian, — h2A, while the potential is the superposition of a barrier, supported on a bounded interval [a, b], and multiple potential wells with support of size h inside (a,b). The parameter h now corresponds to a rescaled Fermi length (see for instance [6]) fixing the quantum scale of the system and, coherently with the features of the physical
model, is assumed to be small. The resulting transport model is described by a double-scale Schrodinger-Poisson Hamiltonian:
HhNL = —h2A + Vh + VhL , (1)
where the linear part of the potential is defined according to:
Vh = V + Wh , supp V = [a, b] , (2)
and
N
Wh = — ^ Wn ((x — Xn) 1 /h) , Wn e C0 (R, R+) , supp wn = [—d, d] , (3)
n= 1
for some d > 0. For a suitable choice of V, this fixes a semiclassical island with quantum wells supported around the collection of points xn e (a,b), n = 1,...,N. The nonlinear potential term, VNNl, solves a Poisson equation with a source given by the charge density of the system. The corresponding evolution problem is as follows:
' idtuh (•,t,k) = (—h2A + Vh + VNhL) uh (•,t,k) ,
(—h2A + Vh + VNhL|t=0 — k2) uh (•, 0, k) = 0 < —AVNl = dh, (4)
dh(x) = J§ng(k) |uh (x,t,k)|2 .
Here, uh (^,t, k) denotes the nonlinear evolution of a generalized eigenfunction uh (•, 0,k) related to the Hamiltonian at t = 0. The quantum state corresponding to this picture is described by a density matrix ph defined by the momenta distribution g(k), and whose kernel evolves in time according to
/dk
—g(k) uh (x,t,k) uh (y,t,k) . (5)
When g(k) selects incoming waves (i.e.: suppg C R+), the system is nourished by charges from the left, providing a non-equilibrium condition where, depending on the potential geometry, the particles may accumulate inside the wells. These systems - characterized by a very rich nonlinear characteristics, such as hysteresis phenomena and steadily oscillating currents - have attracted increasing interest, both for the challenging mathematical problems associated with them as well as the potential application perspectives.
Resonant energies, produced by the potential Vh, naturally arise in this class of models and play a central role in the description of quantum tunneling. The incoming electrons, at resonant levels below sup(a b) V, interact with resonant states which evolve in time according to a quasi-stationary dynamics. In particular, their L2-mass remains concentrated in the vicinity of the wells support for a range of time exponentially large w.r.t. h. Depending on the position of the wells, this possibly induces a local charging process; then, the nonlinear coupling in (4) generates a positive response (depending upon the charge in the wells) which modifies the potential profile and reduces the tunneling rate.
This dynamics was considered in the works of G. Jona-Lasionio, C. Presilla and J. Sjostrand ( [13], [17], [18]), within a simplified framework where the Poisson potential is replaced by an affine function multiplied by a nonlinear charge density. Using slowly varying potential assumptions, WKB expansions and a one-mode approximation for the time evolution of the quantum state, the authors discuss the behavior of the sheet density
related to the accumulation of electrons in a single well determined by a flat double-barrier potential. Rephrasing the result of their work in our scaling, they show that the relevant time scale of the problem is on the order of eT/h, corresponding to the imaginary part of the lowest resonance, and provide with an explicit equation for the evolution of the local charge density in the limit h ^ 0 (eq. 9.7 in [17]).
It is worthwhile to note that these calculations have been shown to be relevant only in some specific cases. This concerns, in particular, the adiabatic approximation for the nonlinear evolution of generalized eigenfunctions, which appears to be an essential point of the analysis: the lack of an error bound in the adiabatic formulas for resonant states prevents the control of the possible remainder terms in the asymptotic limit. Moreover, the role played by the device's geometry in the emergence of nonlinear effects remains an open problem. This was pointed out by F.Nier, Y.Patel and V.Bonaillie-Noel, in a series of works devoted to the steady state problem related to (4) under far-from-equilibrium assumptions. In [4]- [5], an accurate microlocal analysis of the tunneling effect as h ^ 0 determines the limit occupation number of resonant states. This analysis leads to a simplified equation for the Poisson problem, where the limit charge density is described by a superposition of delta-shaped distributions centered in the points xn (see the definition (3)).
The relevance of the adiabatic approximation, for the Schrodinger-Poisson equation in the regime of quantum wells, suggests that we consider as a preliminary step, a 'linearized' problem where the nonlinear term of is replaced by a potential supported on (a, b) and adiabatically dependent on time; namely we introduce the time dependent family of Schrodinger operators:
H0h (t) = -h2A + Vh (t) , (6)
where Vh (t) = V + Wh (t) fulfills the particular scaling (2)-(3) for all t. The adiabatic evolution of an observable x (corresponding to a local charge) is described by the variable Ah (t) solving the equation:
' Ah (t) = Tr [xph (t)] , Ph(t) = 12thg(k) Iuh(k,;t)) (uh(k,-,t)| ,
(7)
iedtuh(k, •.t) = H0h(t)uh(k, - ,t),
(H0h(t = 0) - k2) uh(k, •, 0) = 0 .
In this framework, the relevant resonant energies for the transport problem are determined by the resonances of H0h with the real part embedded in (0, inf [a,b] V); these are usually referred to as shape resonances. In the asymptotic regime of quantum wells (h ^ 0), it is known that the number of shape resonances is uniformly bounded w.r.t. h, while their imaginary part is on the order of eT/h for a suitable constant t > 0 depending on the potential geometry (for this point we refer to the analysis developed in [10]- [11]. Following [20], this quantity fixes the time scale for the dispersion of the corresponding resonant states. Hence, the effect of the accumulation of charges inside the wells can be investigated by choosing the adiabatic parameter e, in (7), of size e-T/h.
1.1. Detecting quantum resonances: the exterior complex dilation
The small-h analysis of the linearized problem (7), involves the following task: clarify, in an energy interval close to the shape resonances, the relation between the evolution of generalized eigenfunctions and of the resonant states in the adiabatic limit. Let us recall that
the resonances of a Schrodinger operator correspond (modulo a restriction to a dense subset of dilation-analytic functions) to the poles of the meromorphic extension of its resolvent to the second Riemann sheet (i.e.: Im^/z < 0). These are detected through the complex deformation method (see [12] for details), which can be adapted to our framework by using the exterior complex scaling: x ^ xegi*\(a>b)(x). When 9 e R, the related deformation is an unitary operator acting on L2 (R) according to
Uu (x)
e0/2u (e0 (x - b) - b)
u (x) ,
e0/2u (e0 (x — a) — a)
x > b,
(a,b) ,
x < a.
(8)
For the Hamiltonian:
Q = —A + V , with: V e (R) and supp V = [a, b] ,
the corresponding deformed operator, next denoted with: Q (9) = UgQU-1, written as (see e.g. in [8]):
Q (0) = —e
-201»
(9)
is explicitly
> Ag + V , (10)
where Ag is the non-self-adjoint point perturbation of the Laplacian defined by the interface conditions:
e-2u(b+) = u(b-), e-2gu'(b+) = u'(b-),
, 3 (11)
e-2u(a-) = u(a+), e-2gu'(a-) = u'(a+).
Namely, we have the following:
( D (Ag) = {u e H2 (R\ {a, b}) | (11) holds } , Ag : \ (12)
[ (Ag u) (x) = —u"(x), x e R\ {a, b} .
When Im 9 > 0, this deformation produces a rotation of the essential spectrum in the second Riemann sheet: aess (Q (9)) = e-2ImgR+; in the cone spanned by R+ and aess (Q (9)) the resonances of Q identify with the spectral points of Q (9). This important result was first obtained by J. Aguilar, E. Baslev and J.M. Combes in [1], [3], where the case of analytic potentials w.r.t. the uniform complex dilation x ^ xeg was considered (see also [12, Theorem 16.4]). For potentials which can be complex deformed only outside a compact region, the exterior complex scaling technique appeared in [19] in the singular version that we reconsider here.
Proposition 1.1. Let Q be defined by (9); the resonances of Q in the cone
n
with a < — , 4
(13)
Ka = {argz e (—2a, 0)} , are eigenvalues of the operators Q (9) for a ^ Im 9.
As a consequence, the resonant state associated with a resonance Eres e KaHa (Q (9)) can be defined as an eigenvector of Q (9), with a ^ Im 9, and we have
(Q (0) — Eres) =0 , in L2
(14)
It is worth remarking that the resonances in Ka do not depend on the deformation, i.e.: if Eres e Ka H a (Q (9)) for a given 9, then, Eres e Ka H a (Q (9')) for all 9' >9 and a ^ Im 9'. Nevertheless the solution of (14) possibly depends on 9 in the exterior
region, while, according to the shape of U, the cutoff 1(a,b) (x) ^Eres (usually refferred to as the quasi-resonant state) is independent of 0.
1.2. An alternative approach to the adiabatic problem
The identification of the resonances for Q with the spectral points of the corresponding deformed operator Q (0) suggest an alternative, and possibly more natural, framework to study the adiabatic problem (7). Let us define the observable x according to
X eC~ (R) , supp x CC (a, b) . (15)
With this choice, x describes the charge accumulating in a region inside (a, b). Since x commutes with the deformation map U for all 0, making use of the properties of the trace operation we have:
Ah (t) = Tr [x Ph (t)] = Tr [u*xUPh (t)] = Tr [xUPh (t) u;] . (16)
Here, UPh (t) U*e identifies with the evolution at time t of a deformed density matrix, when the dynamical system is generated by H0h (t, 0) = UH0h (t) U-1. This allows us to rephrase (7) in the equivalent form:
' Ah (t) = Tr [xPh (t,0)] , Ph(t,0) = /2thg(k) |uh(k,^,t)) <uh(k,^,t)|,
(17)
iedtuh(k, ^,t) = H0h (t, 0) uh(k, -,t),
(H0h (t = 0,0) - k2) uh(k, •, 0) = 0 .
When g selects energies close to the shape resonances, the relevant information about the solution uh(k, •, t) of the adiabatic problem in (17) are related to the evolution of the resonant states. To fix this point, assume that, for a suitable choice of Vh (t) in (6), the linearized operator ^^(t) has a shape resonance Eh (t) remaining close for all time to a limit energy A0 (in a sense which will be specified later with an explicit example). According to Proposition 1.1, we have: Eh (t) e n a (Hh (t, 0)) for some n/4 ^ a > 0 and a ^ Im 0, and the corresponding instantaneous resonant state ^Eh(t) solves the eigenvalue equation (14) with Q (0) = Hh (t, 0). As far as supp g = {k> 0 , k2 — Re Ehes (t) — A0} , it is expected that
1(a,b) (X) uh(k, X, t) - 1(a,b) (x) ^ (t) , (18)
where (t) is the solution of the problem
iedt^h (t) = Hh (t, 0) (t) ,
(19)
^ (0) = ^Eh(0) .
For e = e-T/h, the small-h asymptotics of this dynamic is related to the adiabatic evolution of an eigenvector of the initial operator H^ (t = 0,0) and standard results in adiabatic perturbation theory (see e.g. in [16]) would suggest to identify (t) with the instantaneous resonant state ^Eh(t) (multiplied by a suitable modulation factor), with an error of size e-T/h. Such an adiabatic approximation and the relation (18) could be implemented in (10) to study the asymptotic behavior of A[j (t) as h ^ 0.
The main difficulty in this approach is due to the fact that the complex scaling does not preserve the m-accretivity of the operator, i.e.: the quadratic form associated with H0h (t, 0) has an imaginary part with undetermined sign (see the eq. (1.5) in [8]); as a
consequence, the deformed dynamics may exhibit exponential growth w.r.t. t. Since uniformin-time estimates for the dynamical system are necessary to prove the adiabatic theorem, the lack of this condition in our case, prevents us from developing a rigorous approach to the study of (19) in the small-h limit.
2. A modified model
A possible strategy to overcome the lack of uniform-in-time estimates occurring in the study of the dynamical system (19) consists of modifying the physical Hamiltonian HOh according to the following:
i D (Hh) = {u e H2 (R\ {a, b}) | (11) holds } ,
Hh : { (20)
I (Hh u) (x) = —h2u''(x) + Vh(x) u(x), x e R\ {a, b} ,
where, as before, the potential is formed by the superposition of a barrier supported on [a,b], and a collection of quantum wells Wh defined by (3). It worthwhile to remark that, for 9 = 0, H h is neither self-adjoint nor symmetric and identifies with an extension of the symmetric restriction
H^ = Hh \ {u e H2 (R) | u(a) = u'(a) = 0 , a = a, b } . (21)
In this connection, H h is an explicitly solvable model w.r.t. H0h and relevant quantities, as its resolvent or generalized eigenfunctions, can be expressed in terms of corresponding non-modified quantities, related to the selfadjoint operator H0h, through non-perturbative formulas. This well-known property of point perturbations (see e.g. in [2]) provides us with a usefull tool for spectral analysis and allows us to consider the pair H h, H0h as a scattering system. In particular, if 1[a,6]Vn > 0, it has been shown that Hn has a purely continuous spectrum coinciding with R+, provided that 9 is small (depending on h; see [14]). A detailed analysis concerned with operators of this class has been developed (in a slightly more general framework) in [8]- [15]. In what follows, we resume their main features.
2.1. Shape resonances in the regime of quantum wells
Let: Vn = V + Wn, with V and Wn defined as in (2)-(3) and fufilling the following constraints:
1[«,b] V ^ c, sup{ ||V |U(R) , ||Wn||l~ (r)} ^ C, supp W n CC (a, b) , (22)
for some c > 0, uniformly w.r.t. h e (0,h0]. It is known that this particular potential's shape prevents the accumulation of the possible eigenvalues of the corresponding 'Dirichlet operator', HD,
HD = — h2dX + Vn , D (HD) = H2 ([a, b]) H Ho1 ([a, b]) , (23)
in the energy region (c, inf[ab] V) when the limit h ^ 0 is considered (e.g. in [4]). Then, we assume Vh to verify the following spectral condition.
Condition 2.1. There exists a real A0 and a cluster of eigenvalues {Ah}.?._1 C a (Hd) such that the conditions
i) c ^ A0 ^ inf M V - c ^
< 1
ii) d (a0, a (H*) \ {A*}^) ^ c
(24)
iii)
max
|Ah- A01 < h .
hold for all h G (0, h0].
Under the action of an exterior dilation Ui7 with 7 > 0 (see the definition 8), the modified Hamiltonian transforms according to (e.g. in [8])
Hh(iY) = -h2n (x)A,+i7 + Vh , n (x) = e
a,
(25)
where the operator's domain is obtained from (20) by replacing 9 by 9 + ¿7 in the interface conditions (11). In the exterior domain, the solution of the eigenvalue problem corresponding to a resonance zres G {p G C | arg p G (—27, 0)}
(Hg (iY) - Zres) ^res = 0 , ^res G L
exhibits the exponential modes
^res(x) = c+e'( res)h (x b) , x > b
1/2 i7 ,
(zres)
^res(x) = C-e' h
1/2e¿7
(26)
(27)
(a-x)
x < a.
Let Ph (0) = (-h2A0 + Vh)
(^)) = {u G H2 ((a, b)) , (hdx + zz1/2e-0) u(a) = 0 , (frdx - iz 1/2e-0) u(b) = 0} ,
(28)
where (z)1/2 is determined according to arg z G [— |, §n). Using (27), the resonance equation (26) is re-termed as a non-linear eigenvalue problem for the operator P^1 ($)
(Pzh.es (0) - Zres) ^res = 0 .
(29)
It is worthwhile to note that this equation does not depend on the deformation, but only on the interface conditions in (20). Following the approach of Helffer-Sjostrand (see [11], [21]), the resonances for the full Hamiltonian H" (¿7) can be investigated after reducing the Grushin problem modeled from the Dirichlet operator Hd for the boundary value operator (#). This analysis, developed in [8], shows that, for potentials fulfilling the conditions (24), the shape resonances are localized in small regions of the corresponding Dirichlet's eigenvalues
Ah as h ^ 0. In particular, let denotes the regions:
and dAg (x, y, V, A)
^ = {z G C, d(z, {Aj}j=1) ^ , dAg (x,y, V,A)=/ yj(V(s) - A)+ds , x ^ y
(30)
the Agmon distance between x and y related to a potential V and an energy A G R+.
c
Proposition 2.1. Let Vh = V + be defined according to (2)-(3), (22), and assume the conditions (24) to hold. Then, for all h G (0, ho] and |0| ^ ^, the operator H^ has exactly t resonances in uch , {zj1}^, possibly counted with multiciplicities. Considered as functions
„ \e
2 ^ j •< j=i
of h, after the proper labelling w.r.t. j, these fulfill the relations:
j - Ah
O ( h-3e-
2 So h
(32)
where SO = ({a, b} , U^L, } , V, A0) is the Agmon distance between the asymptotic support of and the barrier's boundary. Under the additional condition
lim h3e2ho min |Ah - Ah,
h^Q 1 j j
variations of zh (0) — zh (0) is estimated by tie f°llowmg:
max |j (0) - zh (0)| = O (|0| h-3e-^
(33)
(34)
for all e-"h0 < |0| < ^.
As a consequence of (32), the size of the imaginary part of the shape resonances is: |lmzj1 (0)| < h-3e-"S". When the quantum evolution generated by H^ is considered for initial states with energies close to zj1 (0), it is important to have a lower bound for Im zj1 (0) as h ^ 0. Providing such a lower bound is a standard result in semiclassical analysis; in [5]-[8] this problem is analyzed for Schrodinger operators depending on the scaling parameter h according to the rules prescribed for Vh. Under some additional spectral assumptions, it is shown that:
|lmzh (0)| > e
2 So " h
(35)
In what follows we will assume that this lower bound holds.
2.2. The quantum evolution problem
We next consider the time propagator generated by the operator H^. In a slightly more general framework, a detailed analysis of this problem has been developed in [15], where the modified dynamics is defined through a similarity, between H^ and the corresponding self-adjoint model H0h, holding in some spectral subspace under the assumption that the parameters 0 and h are small.
The generalized eigenfunctions of H^, next denoted with ^(^fc, Vh), solve of the boundary value problem (we refer to [14]- [15])
' (-h2ôX + Vh) u = k2u, x G R\ (a, b} , k G R. < e-iu(b+ ) = u(b-), e-"eu'(b+) = u'(b-),
k e 2u(a ) = u(a+), e 2"u/(a )= u'(a+)
(36)
and fulfill the exterior conditions
$(x,k, Vh)
^h(x,k, Vh)
x<a = e* hhx + Rh(k,0)e-i hhx
fc>Q
^h(x,k, Vh)
x>b = T h(k,0)e* hhx
fc>Q
x<a = Th(k,0)e* hhx
fc<Q
^h(x,k, Vh)
x>b = e* hx + Rh(k,0)e" fc<0
(37)
(38)
k
h
describing an incoming wave function of momentum k with reflection and transmission coefficients Rh and Th. In the case 9 = 0, the generalized Fourier transform associated with Hh(Vh) is defined by:
(Fhp) (k)= / W(x, k, Vh))* p(x), p e L2(R). (39)
Jr (2nh) 7
Recall that Fh is a bounded operator on L2(R) with a right inverse coinciding with the adjoint (Fh) *
C dk
(Fh)* f (x) = —-172 4h(x,k, Vh)f (k). (40)
(2nh) '
In particular, it results: Fh (Fh) * = I in L2(R), while the product (Fh) * Fh defines the projector on the absolutely continuous subspace of H^(Vh) (cf. [22]).
We are interested in the quantum evolution for initial states residing in a spectral subspace of energies close to A0 (the eccumulation point of the resonances); let assume the interval [Ai, A2] C R such that:
c ^ Ai < A2 ^ inf V - c, a (HD) n [Ai, A2] = {Ah}j_i , (41)
uniformly w.r.t. h e (0, h0]. The spectral projector on [Ai, A2] associated to H^ is next denoted with n[Ai,A2]; this is explicitly given by:
r dk
n[Ai,A2]^ = -—-YT2 1[Ai,a2] (k2) ^h(x,k, Vh) (Fhhp) (k). (42)
Jr (2nh) '
The similarity between H^ and H^ on the subspace n[Al,A2]pL2(R) is provided by the operators Wh defined through the integral kernel:
Wh(x, y) = jf 2nh 1[Ai,A2] (k2) k, Vh) ($(x, k, Vh)) * . (43)
The next proposition rephrases in our case the result presented in [15] .
Proposition 2.2. Let Vh = V + Wh satisfy the conditions (24) and |9| < hNo, with N0 ^ 4. If the interval [Ai, A2] verifies (41) and the lower bound (35) holds, then there exists n > 0 such that: {W^, 9 e C, B |9| < nhNo} form an analytic family of bounded operators in L2(R) fulfilling the expansion:
Wh - n[Ai,A2]P = o (hNo-2) , (44)
in the L (L2(R)) operator norm. For |9| < nhNo, the operators Wq, W^H^ and HhWh map L2 (R) into D (Hh), and it results
Hhwh = w,hH0h. (45)
As a consequence of (44)-(45), the modified quantum evolution can be defined through the unitary propagator e-iiH° by conjugation as far as the initial state is considered in n[Ai,A2]L2(R).
Theorem 2.1. Let h e (0,h0] and |9| < hNo, with N0 ^ 4. Under the assumptions of the proposition 2.2, iH^ generates a strongly continuous group of bounded operators on n[Al,A2]L2(R). For a fixed t, e-itHh is analytic w.r.t. 9 and the expansion:
(e-iiHh - e-itHh) n[Ai,A2] = RRh (t, 9) , (46)
holds with:
sup ter
(t) = O (hN0-2) . (47)
2.3. Adiabatic evolution of shape resonances
We next consider a non-autonomous Hamiltonian defined by (20) through a time dependent potential Vh (t) such that the spectral profile of H (t) corresponds to the pitcure given in the Proposition 2.1 for any time. In particular, the potential is assigned according to the following conditions:
Condition 2.2. Vh (t) = V (t)+Wh (t), with V, e CK (R,L~ (R)) for K ^ 2, supp V (t) = [a, b] and
N
(t) = — ^ (t, (x - xn) 1/h) , supp wn(t) = [—d,d] . (48)
n=1
We assume that for all t the estimates (22) and the conditions (24) hold with A0 e CK (R) and c > 0 fixed.
Under these assumptions, H^ (t) has a cluster of time dependent shape resonances {z^ (0,t)}j.=1 which, for y > 0 large enough, belong to a (Hh(iy,t)), with Hh(iy,t) denoting the exterior complex-dilation of Hh(t). Since aess (H^ (iy)) = e-2iYR+, due to the semiclassical estimates (32) and the assumption (ii)-(24), {zj1 (0,t)}^=1 is an isolated part of the spectrum of H^(iy, t). Denoting with nzh(^ the corresponding time-dependent (nonorthogonal) projetcor, the adiabatic problem for this cluster of spectral points is written as:
¿edtuh = Hh(iy, t)uh ,
(49)
u=0 e ,t)L2 (R) .
If the dynamical system generated by Hh(iy,t) allows uniform-in-time estimates, this problem can be analyzed following the standard approach of the adiabatic theorem with spectral gap condition. For a particular choice of the deformation, the modification introduced by the 0-dependent interface conditions allows to obtain such a result. Indeed, the deformation H^(iy) is characterized as follows (Lemma 3.1 in [8]):
Lemma 2.2. Let 0 = iy, y > 0. Then: iH^(iy) is maximal accretive.
It follows that, for each fixed t, (t7'i-) is a semigroup of contractions; then, the
regularity-in-time of the potential allows us to conclude that the non-autonomous Hamil-tonian HhY(iy, t) generates a quantum dynamical system of contractions (see [8, Proposition 3.7]). In the application perspectives, we need to minimize the error produced by modification of the Hamiltonian; hence, we next assume y to be polynomially small w.r.t. h. In this framework, the resonances zj1 (iy, t) are close to the essential spectrum (located in e-i2YR+) and the spectral gap condition depends on h. Then the estimates of dtk (H^iy, t) — z) in the region: |z — zj1 (0, t) | ~ y, behave as y-1. This entails a loss of some power for h (depending on y) in the adiabatic formula. Nevertheless, assuming e = e-T/h for some t > 0, it is still possible to obtain an error bound exponentially small w.r.t. h. Next, we recall, in the simplified case of a single shape resonance, the result of the adiabatic theorem provided with in [8, Theorem 7.1] (see also [9] for the explicit form of the modulation factor ^ (t) below).
Theorem 2.3. Let Vh (t) fulfill the Condition 2.2 with i =1, and let Eh (t), (t) respectively denote the shape resonance end the resonant state related to HhN (ihN, t) in the neighbourhood of size h of the limit energy A0. We set £ = e-T/h with t > 0, and assume
that
(t) - h(t)
l2(r)
^ £
1-5
holds for any 5 e (0,1). Then, the solution (t) of the problem: 0 ^ t ^ T
iedt^h (t) = H*> (ihN ,t)^h (t) ,
^ (0) = ^(0)
fulfills the estimate:
max |Uh (t) - vh (t)|
^ Ca,6,c,5,T £l 5 y^Eh(0)
te[0T ]........lL2(R)
where the auxiliary dynamics vh (t) is expressed as:
vh (t) = ^ (t) e-i So Eh(s) ds^(t) ,
with
il2(r)
^ (t) = exp
ds I + O (£1-5) ,
(50)
(51)
(52)
(53)
(54)
l2(r)
while the positive constant Ca,b,c,&,T possibly depends on the data.
3. A conjecture for the linearized transport problem
We finally reconsider the linearized verion of the transport problem introduced in (7). According to the result of Theorem 2.1, for initial states characterized by energies close to a cluster of resonaces allowing the lower bound (35), the distance between the propagators exp (—itHjhN0) and exp (—itH^) is controlled uniformely-in-time in the L2-operator norm by O (hN°-2) for N0 ^ 4. In the non-autonomous case, this suggests that, under the same initial conditions, the quantum dynamical systems generated by HjhN° (t) and H^ (t) remain close to each other. In particular, we conjecture that on the adiabatic time-scale, the result of Theorem 2.1 extends according to:
sup ||uh (t) - uh (t)| te[o,T ]
l2(r)
O (hN
2
(55)
where uh (t) and uj (t) solve the equations: ¿£Ôtuh (t) = HhNo (t)uh (t) and ¿£Ôt«h (t) =
H0h(t)uh (t) with uh (0) = uh (0) G n[Al,A2]L2(R). When suppg = {k > 0 , |k2 - A0| = O (h)}, this initial condition is almost fufilled by the modified problem
' Ah (t) = Tr [XPh (t)] ,
Ph (t) = / 2kg(k) |uh(k,-,t)> <uh(k,-,t)| ,
ï£0tuh(k, -,t) = HfhNo (t)uh(k, -,t) ,
[ (HfhNo (t = 0) - k2) uh(k, ■, 0) = 0 , and the solution (t) in (7) is expected to be related to Ah (t) by:
Ah (t) = Ah (t) + O (hNo-2) .
(56)
Let the observable x be defined by (15); proceeding as in (16), we have:
Ah (t) = Tr [x ph (t)] = Tr [U*xUph (t)] = Tr [xUph (t) U* . (58)
Noticing that U(9ph (t) U<* identifies with the time evolution generated by the deformed operators H*^ (ihNo,t) = UH^n (t)U-i, the problem (56) is rewritten as
' Ah (t) = Tr [xph (t)] ,
Ph(t) = /2thg(k) |uh(k,-,t)> <uh(k,-,t)|,
(59)
iedtUh(k, -,t) = HhhN0 (ihNo,t)Uh(k, -,t),
(HhhN0 (ihNo ,t = 0) - k2) uh(k, ■, 0) = 0 .
In this equivalent formulation, the result of Theorem 2.3 applies and the evolution of resonant states related to shape resonances is properly described by adiabatic formulas of the type (52). This clarifies the idea of our approach: the modification of the physical model (9) by h-dependent artificial interface conditions, although introducing a small error on the solution Ah (t) (controlled by a power of h), allows us to work in the complex deformed setting (59) where, under the condition (18), an adiabatic approximation holds for the deformed dynamics Uh(k,-,t).
3.1. An explicit example and final remarks
A rigorous justification of the method described in this work is still an open problem which mainly concerns the validity of conjecture (55), allowing one to compare the modified variable Ah (t) and the physical one A (t).
In [9], the small-h behavior of the solution of (56) has been investigated for a time dependent potential formed by a flat barrier of height V0 > 0 plus an attractive, time-dependent delta interaction acting in x0 e (a, b) and preserving the quantum-well scaling. Explicitly, we consider the model:
HhhNo (t) = -h2AihNo + 1M] (x) V0 + ha (t) ¿xo. (60)
Under suitable assumptions on a (t) (corresponding to the Condition 2.2), these operators have a single shape resonance Eh (t) localized close to a positive energy below V0. In this framework, an explicit formula for the deformed dynamics Uh(k, ■, t) and direct computations allow us to prove that the relations (18), (50) hold; then, the asymptotic behavior of Ah (t) as h ^ 0 can be analyzed using the result (52). In a configuration where the scattering involves only incoming from the left wave functions and the interaction point x0 is closer to the l.h.s. of the barrier (i.e.: |x0 - a| < |x0 - b|), Ah (t) exhibits the decomposition:
Ah(t) = a(t) + J(t) + O (hNo) , (61)
where the main term a(t) = O (1) solves a limit equation describing the charging process of the well, while the first error term is J(t) = O (h2) (we refer to [9, Theorem 2.1]). In particular, our limit equation is coherent with the nonlinear reduced model obtained in [13], while the interface conditions do not affect the main contributions to Ah(t) when N0 is large enough. Hence, this result appears as an indirect confirmation of the validity of the expansion conjectured in (55).
Acknowledgements
We gratefully aknowledge the financial support from the CNRS (FR3399) and from the ITMO University of Saint Petersbourg.
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