Effect of magnetic field on resonant tunneling in 3D waveguides of variable cross-section
11 Effect of magnetic field on resonant tunneling in 3Dwaveguides of variable cross-section
L.M. Baskin, B.A. Plamenevskii, O.V. Sarafanov
Abstract
We consider an infinite three-dimensional waveguide that far from the coordinateorigin coincides with a cylinder. The waveguide has two narrows of diameter ε . Thenarrows play the role of effective potential barriers for the longitudinal electron motion.The part of waveguide between the narrows becomes a ”resonator” and there can ariseconditions for electron resonant tunneling. A magnetic field in the resonator can changethe basic characteristics of this phenomenon. In the presence of a magnetic field, thetunneling phenomenon is feasible for producing spin-polarized electron flows consistingof electrons with spins of the same direction.We assume that the whole domain occupied by a magnetic field is in the resonator.An electron wave function satisfies the Pauli equation in the waveguide and vanishes atits boundary. Taking ε as a small parameter, we derive asymptotics for the probability T ( E ) of an electron with energy E to pass through the resonator, for the ”resonant en-ergy” E res , where T ( E ) takes its maximal value, and for some other resonant tunnelingcharacteristics. In this paper, we consider a three-dimensional waveguide that, far from the coordinate origin,coincides with a cylinder G containing the axis x . The cross-section of G is a two-dimensionaldomain (of an arbitrary form) with smooth boundary. The waveguide has two narrows ofsmall diameter ε . The waveguide part between the narrows plays the role of a resonatorand there can arise conditions for electron resonant tunneling. This phenomenon consistsof the fact that, for an electron with energy E , the probability T ( E ) to pass from one partof the waveguide to the other through the resonator has a sharp peak at E = E res , where E res denotes the ”resonant” energy. To analyse the operation of devices based on resonanttunneling, it is important to know E res , the behavior of T ( E ) for E close to E res , the heightof the resonant peak, etc.The presence of a magnetic field can essentially affect the basic characteristics of theresonant tunneling and bring new possibilities for applications in electronics. In particular,in the presence of a magnetic field, the tunneling phenomenon is feasible for producingspin-polarized electron flows consisting of electrons with spins of the same direction. Wesuppose that a part of the resonator has been occupied by the magnetic field generatedby an infinite solenoid with axis orthogonal to the axis x . Electron wave function satisfiesthe Pauli equation in the waveguide and vanishes at its boundary (the work function ofthe waveguide is supposed to be sufficiently large, so such a boundary condition has beenjustified). Moreover, we assume that only one incoming wave and one outgoing wave canpropagate in each cylindrical outlet of the waveguide. In other words, we do not discuss the a r X i v : . [ m a t h - ph ] N ov multichannel electron scattering and consider only electrons with energy between the firstand the second thresholds. We take ε as small parameter and obtain asymptotic formulas forthe aforementioned characteristics of the resonant tunneling as ε →
0. It turns out that suchformulas depend on the limiting form of the narrows. We suppose that, in a neighborhoodof each narrow, the limiting waveguide coincides with a double cone symmetric about thevertex.The asymptotic description of electron resonant tunneling in the absence of externalfields was presented in [1] for 3D quantum waveguides of similar geometry. Previously therewere only episodic studies of the phenomenon by numerical methods, see [2], [3]. Theextensive literature on the resonant tunneling in 1D waveguides was mainly based on theWKB-method; for our problem the method does not work. In [1], the study was based onthe compound asymptotic method; the general theory of the method was elaborated in [4].In the present paper, we modify the approach in [1] not only analysing the effect of magneticfields but also developing a more general and simple scheme of study.Section 2 contains statement of the problem. In Section 3, we introduce so-called”limit” boundary value problems, which are independent of the parameter ε . Some modelsolutions to the problems are studied in Section 4. The solutions will be used in Section5 to construct asymptotic formulas for appropriate wave functions. In the same section,we investigate the asymptotics of the wave functions and derive asymptotic formulas formain characteristics of the resonant tunneling. Remainders in the asymptotic formulas areestimated in Section 6. To describe the domain G ( ε ) in R occupied by the waveguide we first introduce domains G and Ω in R independent of ε . The domain G is the cylinder G = R × D = { ( x, y, z ) ∈ R : x ∈ R = ( −∞ , + ∞ ); ( y, z ) ∈ D ⊂ R } whose cross-section D is a bounded two-dimensional domain with smooth boundary. Let usdefine Ω. Denote by K a double cone with vertex at the coordinate origin O that containsFigure 1: The domain Ω.the axis x and is symmetric about the origin. The set K ∩ S with S standing for the unitFigure 2: The waveguide G ( ε ) . sphere consists of two non-overlapping one-connected domains symmetric about the centerof sphere. Assume that the domain Ω contains the cone K together with a neighborhoodof its vertex. Moreover, Ω coincides with K outside a sufficiently large ball centered at theorigin. The boundary ∂ Ω of Ω is supposed to be smooth.Let us turn to the waveguide G ( ε ). We denote by Ω( ε ) the domain obtained from Ωby the contraction with center at O and coefficient ε . In other words, ( x, y, z ) ∈ Ω( ε ) if andonly if ( x/ε, y/ε, z/ε ) ∈ Ω. Let K j and Ω j ( ε ) stand for K and Ω( ε ) shifted by the vector r j = ( x j , , j = 1 ,
2. The value | x − x | is assumed to be sufficiently large so that thedistance between ∂K ∩ ∂K and G is positive. We set G ( ε ) = G ∩ Ω ( ε ) ∩ Ω ( ε ) . The wave function Ψ = (Ψ + , Ψ − ) T of an electron with energy E = k (cid:126) / m in amagnetic field H satisfies the Pauli equation( − i ∇ + A ) Ψ + ( (cid:98) σ, H ) Ψ = k Ψ in G ( ε ) , (2.1)where (cid:98) σ = ( σ , σ , σ ) with the Pauli matrices σ = (cid:18) (cid:19) , σ = (cid:18) − ii (cid:19) , σ = (cid:18) − (cid:19) , and H = − ( e/c (cid:126) H ) = rot A . If the magnetic field is directed along the axis z that is H = H k , H being a scalar function, then (2.1) decomposes into the two scalar equations( − i ∇ + A ) Ψ ± ± H Ψ ± = k Ψ ± . (2.2)Let the function H depend only on ρ = (( x − x ) + ( y − y ) ) / with H ( ρ ) = 0 for ρ > R , R being a fixed positive number. Such a field is generated by an infinite solenoid with radius R and axis parallel to the axis z . Then A = A e ψ , where e ψ = ρ − ( − y + y , x − x ,
0) and A ( ρ ) = 1 ρ (cid:26) (cid:82) ρ tH ( t ) dt, ρ < R ; (cid:82) R tH ( t ) dt, ρ > R. The equality rot A = H determines A up to a term of the form ∇ f . We neglect the waveguideboundary permeability to the electrons and consider the equations (2.2) supplemented bythe homogeneous boundary conditionΨ ± = 0 on ∂G ( ε ) . (2.3)The obtained boundary value problems are self-adjoint with respect to the Green formulas(( − i ∇ + A ) u ± Hu − k u, v ) G ( ε ) − ( u, ( − i ∇ + A ) v ± Hv − k v ) G ( ε ) +( u, ( − ∂ n − A n ) v ) ∂G ( ε ) − (( − ∂ n − A n ) u, v ) ∂G ( ε ) = 0 , where A n is the projection of A onto the outward normal to ∂G ( ε ) and u, v ∈ C ∞ c ( G ( ε ))(which means that u and v are smooth functions vanishing outside a bounded set). Besides,Ψ ± must satisfy some radiation conditions at infinity. To formulate such conditions, we haveto introduce incoming and outgoing waves. From the requirements on H and the choice of A , it can be seen that the coefficients of equations (2.2) stabilize at infinity with a powerrate. Such a slow stabilization offers difficulties in defining these waves. Therefore we willmodify A by a gauge transformation so that the coefficients in (2.2) become constant forlarge | x | .Let ( ρ, ψ ) be polar coordinate on the plane xy centered at ( x , y ) and ψ = 0 onthe ray of the same direction as the axis x . We introduce f ( x, y, z ) = cψ , where c = (cid:82) R tH ( t ) dt . For definiteness, assume that − π/ < ψ < π/
2. The function f is uniquelydetermined in the waveguide for | x − x | >
0, moreover, ∇ f = A for | x − x | > R . Let τ be a cut-off function on R + equal to 1 for t > R + 2 and 0 for t < R + 1. We set A (cid:48) ( x, y, z ) = A ( x, y, z ) − ∇ ( τ ( | x − x | ) f ( x, y, z )). Then rot A (cid:48) = rot A = H while A (cid:48) = 0for | x − x | > R + 2. The wave functions Ψ (cid:48)± = Ψ ± exp { iτ f } satisfy (2.2) with A replacedby A (cid:48) . For | x − x | > R + 2, the coefficients of the equations (2.2) with new vector potential A (cid:48) coincide with the coefficients of the Helmholtz equation −(cid:52) Ψ (cid:48)± = k Ψ (cid:48)± . In order to formulate the radiation conditions, we consider the problem∆ v ( y, z ) + λ v ( y, z ) = 0 , ( y, z ) ∈ D, (2.4) v ( y, z ) = 0 , ( y, z ) ∈ ∂D. The values of parameter λ that correspond to the nontrivial solutions of this problem formthe sequence λ < λ < . . . with λ >
0. These numbers are called the thresholds. Assumethat k in (2.2) coincides with none of the thresholds and take up the equation in (2.2) withΨ + . For a fixed k > λ there exist finitely many bounded solutions (wave functions) linearlyindependent modulo L ( G ( ε )); in other words, a linear combination of such solutions belongsto L ( G ( ε )) if and only if all coefficients are equal to zero. The number of wave functions withsuch properties remains constant for k ∈ ( λ q , λ q +1 ), q = 1 , , . . . and step-wise increases atthe thresholds.In the present paper, we discuss only the situation, where k ∈ ( λ , λ ). In such a case,there exist two independent wave functions. A basis in the space spanned by such functionscan be composed of the wave functions u +1 and u +2 satisfying the radiation conditions u +1 ( x, y, z ) = (cid:40) e iν x Ψ ( y, z ) + s +11 ( k ) e − iν x Ψ ( y, z ) + O ( e δx ) , x → −∞ ,s +12 ( k ) e iν x Ψ ( y, z ) + O ( e − δx ) , x → + ∞ ; (2.5) u +2 ( x, y ) = (cid:40) s +21 ( k ) e − iν x Ψ ( y, z ) + O ( e δx ) , x → −∞ ,e − iν x Ψ ( y, z ) + s +22 ( k ) e iν x Ψ ( y, z ) + O ( e − δx ) , x → + ∞ ;here ν = (cid:112) k − λ and Ψ stands for an eigenfunction of problem (2.4) corresponding to λ and being normalized by the equality ν (cid:90) D | Ψ ( y, z ) | dy dz = 1 . (2.6)The function U ( x, y, z ) = e iν x Ψ ( y, z ) in the cylinder G is a wave incoming from −∞ andoutgoing to + ∞ , while U ( x, y, z ) = e − iν x Ψ ( y, z ) is a wave going from + ∞ to −∞ . Thematrix S + = (cid:107) s + mj (cid:107) m,j =1 , with entries determined by (2.5) is called the scattering matrix; it is unitary. The quantities R +1 := | s +11 | , T +1 := | s +1 2 | are called the reflection coefficient and the transition coefficient for the wave U coming in G ( ε ) from −∞ . (Similar definitions can be given for the wave U , incoming from + ∞ .) Inthe same manner we introduce the scattering matrix S − and the reflection and transitioncoefficients R − and T − for the equation in (2.2) with Ψ − .We consider only the scattering of the wave going from −∞ and denote the reflectionand transition coefficients by R ± = R ± ( k, ε ) = | s ± ( k, ε ) | , T ± = T ± ( k, ε ) = | s ± ( k, ε ) | . (2.7)We intend to find a ”resonant” value k ± r = k ± r ( ε ) of the parameter k which correspondsto the maximum of the transition coefficient and to describe the behavior of T ± ( k, ε ) near k ± r ( ε ) as ε → To derive the asymptotics of a wave function (i.e. a solution to problem (2.2)) as ε → ε . Let the vector potential A (cid:48) and, in particular, the magnetic field H differ from zero only in the resonator, which is the part of waveguide between the narrows.Then, outside the resonator and in a neighborhood of the narrows, the wave function underconsideration satisfies the Helmholtz equation. We set G (0) = G ∩ K ∩ K (Fig. 3), so G (0) consists of three parts G , G , and G . Theboundary value problems∆ v ( x, y, z ) + k v ( x, y, z ) = f ( x, y, z ) , ( x, y, z ) ∈ G j , (3.1) v ( x, y, z ) = 0 , ( x, y, z ) ∈ ∂G j , where j = 1 ,
3, and( − i ∇ + A (cid:48) ) v ( x, y, z ) ± H ( ρ ) v ( x, y, z ) − k v ( x, y, z ) = f ( x, y, z ) , ( x, y, z ) ∈ G , (3.2) v ( x, y, z ) = 0 , ( x, y, z ) ∈ ∂G , Figure 3: The domain G (0) . are called the first kind limit problems.We introduce function spaces for the problem (3.2) in G . Denote by O and O theconical points of the boundary ∂G and by φ and φ smooth real functions on the closure G of G such that φ j = 1 in a neighborhood of O j while φ + φ = 1. For l = 0 , , γ ∈ R , we denote by V lγ ( G ) the completion in the norm (cid:107) v ; V lγ ( G ) (cid:107) = (cid:90) G l (cid:88) | α | =0 2 (cid:88) j =1 φ j ( x, y, z ) r γ − l + | α | ) j | ∂ α v ( x, y, z ) | dx dy dz / (3.3)of the set of smooth functions on G vanishing near O and O ; here r j is the distance betweenthe points ( x, y, z ) and O j , α = ( α , α , α ) is the multiindex, and ∂ α = ∂ | α | /∂x α ∂y α ∂z α .Let K j be the tangent cone to ∂G at O j and S ( K j ) the domain that K j cuts outon the unit sphere centered at O j . We denote by µ ( µ + 1) and µ ( µ + 1) the first andsecond eigenvalues of the Dirichlet problem for the Laplace-Beltrami operator in S ( K ),0 < µ ( µ + 1) < µ ( µ + 1). Moreover, let Φ stand for an eigenfunction corresponding to µ ( µ + 1) and normalized by (2 µ + 1) (cid:90) S ( K ) | Φ ( ϕ ) | dϕ = 1 . The next proposition follows from the general results, e.g. see [5, Chapters 2 and 4, §§ Proposition 3.1.
Assume that | γ − | < µ + 1 / . Then for f ∈ V γ ( G ) and any k exceptthe positive increasing sequence { k p } ∞ p =1 of eigenvalues k p → ∞ , there exists a uniquesolution v ∈ V γ ( G ) to the problem (3.2) in G . The estimate (cid:107) v ; V γ ( G ) (cid:107) ≤ c (cid:107) f ; V γ ( G ) (cid:107) (3.4) holds with a constant c independent of f . If f vanishes in a neighborhood of O and O ,then v admits the asymptotics v ( x, y, z ) = (cid:40) b r − / (cid:101) J µ +1 / ( kr )Φ ( ϕ ) + O (cid:0) r µ +1 / (cid:1) , r → ; b r − / (cid:101) J µ +1 / ( kr )Φ ( − ϕ ) + O (cid:0) r µ +1 / (cid:1) , r → near O and O , where ( r j , ϕ j ) are ”polar coordinates” centered at O j , r j > and and ϕ j ∈ S ( K j ) ; b j are certain constants; (cid:101) J µ denotes the Bessel function multiplied by a constantsuch that (cid:101) J µ ( kr ) = r µ + o ( r µ ) .Let k = k be an eigenvalue of problem (3.2) , then the problem (3.2) is solvableif and only if ( f, v ) G = 0 for any eigenfunction v corresponding to k . Under suchconditions there exists a unique solution v to problem (3.2) that is orthogonal to all theseeigenfunctions and satisfies (3.4) . We turn to problems (3.1) for j = 1 ,
3. Let χ ,j and χ ∞ ,j be smooth real functions onthe closure G j of G j such that χ ,j = 1 in a neighborhood of O j , χ ,j = 0 outside a compactset, and χ ,j + χ ∞ ,j = 1. We also assume that the support supp χ ∞ ,j is in the cylindrical partof G j . For γ ∈ R , δ >
0, and l = 0 , ,
2, the space V lγ, δ ( G j ) is the completion in the norm (cid:107) v ; V lγ, δ ( G j ) (cid:107) = (cid:90) G j l (cid:88) | α | =0 (cid:0) χ ,j r γ − l + | α | ) j + χ ∞ ,j exp(2 δx ) (cid:1) | ∂ α v | dx dy dz / (3.5)of the set of functions with compact support smooth on G j and equal to zero in a neighbor-hood of O j .By assumption, k is between the first and second thresholds, so in every domain G j there is only one outgoing wave; let U − = U be the outgoing wave in G and U − = U thatin G (the definition of the waves U j in G see in Section 2). The next proposition followsfrom Theorem 5.3.5 in [5]. Proposition 3.2.
Let | γ − | < µ + 1 / and let the homogeneous problem (3.1) (with f = 0 ) have no nontrivial solutions in V γ, ( G j ) . Then for any right-hand side f ∈ V γ, δ ( G j ) there exists a unique solution v to the problem (3.1) that admits the representation v = u + A j χ ∞ ,j U − j , where A j = const , u ∈ V γ, δ ( G j ) and δ is sufficiently small. Moreover there holds the estimate (cid:107) u ; V γ, δ ( G j ) (cid:107) + | A j | ≤ c (cid:107) f ; V γ, δ ( G j ) (cid:107) , (3.6) with a constant c independent of f . If the function f vanishes in a neighborhood of O j , thenthe solution v in G admits the decomposition v ( x, y, z ) = a r − / (cid:101) J µ +1 / ( kr )Φ ( − ϕ ) + O (cid:0) r µ +1 / (cid:1) , r → , and for the solution in G there holds v ( x, y ) = a r − / (cid:101) J µ +1 / ( kr )Φ ( ϕ ) + O (cid:0) r µ +1 / (cid:1) , r → , where a j are certain constants and µ l are the same as in the preceding proposition. In the domains Ω j , j = 1 ,
2, introduced in Section 2, we consider the boundary value problems (cid:52) w ( ξ j , η j , ζ j ) = F ( ξ j , η j , ζ j ) , ( ξ j , η j , ζ j ) ∈ Ω j ; w ( ξ j , η j , ζ j ) = 0 , ( ξ j , η j , ζ j ) ∈ ∂ Ω j , (3.7)which are called the second kind limit problems; here ( ξ j , η j , ζ j ) denote Cartesian coordinateswith origin at O j .Let ρ j = dist(( ξ j , η j , ζ j ) , O j ) and let ψ ,j , ψ ∞ ,j be smooth real functions on Ω j suchthat ψ ,j = 1 for ρ j < N/ ψ ,j = 0 for ρ j > N , and ψ ,j + ψ ∞ ,j = 1 with sufficiently largepositive N . For γ ∈ R and l = 0 , ,
2, the space V lγ (Ω j ) is the completion in the norm (cid:107) v ; V lγ (Ω j ) (cid:107) = (cid:90) Ω j l (cid:88) | α | =0 (cid:0) ψ ,j + ψ ∞ ,j ρ γ − l + | α | ) j (cid:1) | ∂ α v | dξ j dη j dζ j / (3.8)of the set C ∞ c (Ω j ) of smooth functions with compact support in Ω j . The next propositionis a corollary of Theorem 4.3.6 in [5]. Proposition 3.3.
Assume that | γ − | < µ + 1 / . Then for F ∈ V γ (Ω j ) there exists aunique solution w ∈ V γ (Ω j ) of the problem (3.7) such that (cid:107) w ; V γ (Ω j ) (cid:107) ≤ c (cid:107) F ; V γ (Ω j ) (cid:107) , (3.9) with a constant c independent of F . If F ∈ C ∞ c (Ω j ) , then the function w is smooth on Ω j and admits the representation w ( ξ j , η j , ζ j ) = (cid:40) α j ρ − µ − j Φ ( − ϕ j ) + O (cid:0) ρ − µ − j (cid:1) , ξ j < ,β j ρ − µ − j Φ ( ϕ j ) + O (cid:0) ρ − µ − j (cid:1) , ξ j > , (3.10) with ρ j → ∞ ; here ( ρ j , ϕ j ) are polar coordinates on Ω j centered at O j while µ l and Φ arethe same as in Proposition . The constants α j and β j are given by α j = − ( F, w lj ) Ω , β j = − ( F, w rj ) Ω , where w lj and w rj are unique solutions to the homogeneous problem (3.7) that satisfy, for ρ j → ∞ , the conditions w lj = (cid:40)(cid:0) ρ µ j + αρ − µ − j (cid:1) Φ ( − ϕ j ) + O (cid:0) ρ − µ − j (cid:1) , ξ j < βρ − µ − j Φ ( ϕ j ) + O (cid:0) ρ − µ − j (cid:1) , ξ j >
0; (3.11) w rj = (cid:40) βρ − µ − j Φ ( − ϕ j ) + O (cid:0) ρ − µ − j (cid:1) , ξ j < (cid:0) ρ µ j + αρ − µ − j (cid:1) Φ ( ϕ j ) + O (cid:0) ρ − µ − j (cid:1) , ξ j > . (3.12) The coefficients α and β depend only on the domain Ω . In each domain G j , j = 1 , ,
3, we introduce special solutions to the homogeneous problems(3.1). Such solutions will be needed in the next section for constructing the asymptotics ofa wave function. From Propositions 3.1 and 3.2 it follows that the bounded solutions of thehomogeneous problems (3.1) are trivial (except the eigenfunctions of the problem in G ), sowe will consider solutions unbounded in a neighborhoods of the points O j .Let us consider the problem in the cone K , which is, as in Proposition 3.1, thetangent cone to ∂G at O :∆ u + k u = 0 in K, u = 0 on ∂K. (4.1)The function v ( r, ϕ ) = r − / (cid:101) N µ +1 / ( kr )Φ ( ϕ ) (4.2)satisfies problem (4.1); here (cid:101) N µ is the Neumann function multiplied by such a constant that (cid:101) N µ ( kr ) = r − µ + o ( r − µ ) , and Φ is the same function as in Proposition 3.1. Let t (cid:55)→ Θ( t ) be a cut-off function on R equal to 1 for t < δ/ t > δ with a small positive δ . We introduce the solution v ( x, y, z ) = Θ( r ) v ( r , ϕ ) + (cid:101) v ( x, y, z ) (4.3)to the homogeneous problem (3.1) in G , whereas (cid:101) v satisfies (3.1) with f = − [ (cid:52) , Θ] v ; theexistence of (cid:101) v is provided by Proposition 3.2. Thus v ( x, y, z ) = (cid:40) r − / (cid:0) (cid:101) N µ +1 / ( kr ) + a (cid:101) J µ +1 / ( kr ) (cid:1) Φ ( − ϕ ) + O ( r µ ) , r → ,AU − ( x, y, z ) + O ( e δx ) , x → −∞ , (4.4)where (cid:101) J µ is the same function as in Proposition 3.1 and 3.2 and the constant A (cid:54) = 0 dependsonly on the domain G .In the domain G , we introduce the solution v to the homogeneous problem (3.1), v ( x, y, z ) := v ( d − x, − y, − z ), where d = dist( O , O ). Then v ( x, y, z ) = (cid:40) r − / (cid:0) (cid:101) N µ +1 / ( kr ) + a (cid:101) J µ +1 / ( kr ) (cid:1) Φ ( ϕ ) + O ( r µ ) , r → ,Ae − iν d U − ( x, y, z ) + O ( e − δx ) , x → + ∞ . (4.5) Lemma 4.1.
There holds the equality | A | = Im a .Proof. Let ( u, v ) Q stand for the integral (cid:82) Q uv dx dy dz , and let G N, δ be the truncated domain G ∩ { x > − N } ∩ { r > δ } . By the Green formula0 = ( (cid:52) v + k v , v ) G N, δ − ( v , (cid:52) v + k v ) G N, δ = ( ∂ v /∂n, v ) ∂G N, δ − ( v , ∂ v /∂n ) ∂G N, δ = 2 i Im ( ∂ v /∂n, v ) E , E = ( ∂G N, δ ∩{ x = − N } ) ∪ ( ∂G N, δ ∩{ r = δ } ). Taking into account (4.4) for x → + ∞ and (2.6), we haveIm ( ∂ v /∂n, v ) ∂G N, δ ∩{ x = − N } = − Im (cid:90) D A ∂U − ∂x ( x, y, z ) AU − ( x, y, z ) (cid:12)(cid:12)(cid:12) x = − N dy dz + o (1)= | A | ν (cid:90) l/ − l/ | Ψ ( y, z ) | dy dz + o (1) = | A | + o (1) . With (4.4) as r → (see Proposition 3.1), we obtainIm ( ∂ v /∂n, v ) ∂G N, δ ∩{ r = δ } = Im (cid:90) S ( K ) (cid:20) − ∂∂r r − / (cid:0) (cid:101) N µ +1 / ( kr ) + a (cid:101) J µ +1 / ( kr ) (cid:1)(cid:21) × r − / (cid:0) (cid:101) N µ +1 / ( kr ) + a (cid:101) J µ +1 / ( kr ) (cid:1) | Φ ( − ϕ ) | r (cid:12)(cid:12)(cid:12) r = δ dϕ + o (1)= − (Im a )(2 µ + 1) (cid:90) S ( K ) | Φ( − ϕ ) | dϕ + o (1) = − Im a + o (1) . Thus | A | − Im a + o (1) = 0 as N → ∞ and δ → k , ± be a simple eigenvalue of the problem (3.2) in the resonator G and v ± is aneigenfunction corresponding to k , ± and normalized by the condition (cid:82) G | v ± | dx dy dz = 1.By virtue of Proposition 3.1 v ± ( x, y, z ) ∼ (cid:40) b ± r − / (cid:101) J µ +1 / ( k , ± r )Φ( ϕ ) , r → ,b ± r − / (cid:101) J µ +1 / ( k , ± r )Φ( − ϕ ) , r → . (4.6)We consider that b ± j (cid:54) = 0. If H = 0, then it is true, for instance, for the eigenfunctionscorresponding to the minimal eigenvalue of the resonator. For nonzero H this condition canbe violated owing to the Aharonov-Bohm effect; here we do not discuss this phenomenon. For k in a punctured neighborhood of k , ± separated from the other eigenvalues, we introducethe solutions v ± j to the homogeneous problem (3.2) by the relations v ± j ( x, y, z ) = Θ( r j ) v ( r j , ϕ j ) + (cid:101) v ± j ( x, y, z ) , j = 1 , , (4.7)where v is defined by (4.2) and (cid:101) v ± j is a bounded solution to the problem (3.2) with f j ( x, y, z ) = − [ (cid:52) , Θ( r j )] v ( r j , ϕ j ). Lemma 4.2.
In a neighborhood V ⊂ C of k , ± containing no eigenvalues of the problem (3.2) in G distinct from k , ± , there hold the equalities (cid:101) v ± j = − b ± j ( k − k , ± ) − v ± + (cid:98) v ± j , where b ± j are the same as in (4.6) and the functions (cid:98) v ± j are analytic in k ∈ V .Proof. We first verify that ( v ± j , v ± ) G = − b ± j / ( k − k , ± ), where v ± j are defined by (4.7). Wehave ( (cid:52) v ± j + k v ± j , v ± ) G δ − ( v ± j , (cid:52) v ± + k v ± ) G δ = − ( k − k , ± )( v ± j , v ± ) G δ ;the domain G δ is obtained from G by cutting out the balls of radius δ with centers at O and O . Applying the Green formula in the same way as in the proof of Lemma 4.1, wearrive at − ( k − k , ± )( v ± j , v ± ) G δ = b ± j + o (1). It remains to let δ → k , ± is a simple eigenvalue, we have (cid:101) v ± j = B ± j ( k ) k − k , ± v ± + (cid:98) v ± j , (4.8)where B ± j ( k ) is independent of ( x, y, z ) and (cid:98) v ± j are certain functions analytic in k near k = k , ± . Multiplying (4.7) by v ± and taking into account (4.8), the obtained function for( v ± j , v ± ) G , and the normalized condition ( v ± , v ± ) G = 1, we arrive at B ± j ( k ) = − b ± j + ( k − k , ± ) (cid:101) B ± j ( k ), (cid:101) B ± j are being certain analytic functions. Together with (4.8), this completesthe proof.In view of Lemma 4.2, the expressions v ± = ( k − k , ± ) v ± and v ± = b ± v − b ± v ± can be extended by continuity to k , ± . According to Proposition 3.1, v ± ( x, y ) ∼ (cid:40) r − / (cid:0) ( k − k , ± ) (cid:101) N µ +1 / ( kr ) + c ± ( k ) (cid:101) J µ +1 / ( kr ) (cid:1) Φ ( ϕ ) , r → ,c ± ( k ) r − / (cid:101) J µ +1 / ( kr )Φ ( − ϕ ) , r → , (4.9) v ± ( x, y ) ∼ (cid:40) r − / (cid:0) b ± (cid:101) N µ +1 / ( kr ) + d ± ( k ) (cid:101) J µ +1 / ( kr ) (cid:1) Φ ( ϕ ) , r → ,r − / (cid:0) − b ± (cid:101) N µ +1 / ( kr ) + d ± ( k ) (cid:101) J µ +1 / ( kr ) (cid:1) Φ ( − ϕ ) , r → . (4.10)From the proof of Lemma 4.2 it follows that c ± j ( k , ± ) = − b ± b ± j . In Section 5.1, we present an asymptotic formula for a wave function (see (5.1)), explain itsstructure, and describe the solutions of the first kind limit problems involved in the formula.We complete deriving the formula (5.1) in 5.2, where we describe the involved solutions ofthe second kind limit problems and calculate some coefficients in the expressions for thesolutions of the first kind problems. In Section 5.3, when analysing the expression for (cid:101) s obtained in 5.2, we derive formal asymptotics of the resonant tunneling characteristics. Notethat the remainders in (5.20) – (5.22) have arisen at the intermediate stage of considerationduring simplification of the principal part of the asymptotics; they are not the remaindersin the final asymptotic formulas. The ”final” remainders are estimated in the next Section6, see Theorem 6.3. First, we derive the integral estimate (6.13) of the remainder in (5.1),which proves to be sufficient to obtain more simplified estimates of the remainders in theformulas for the characteristics of resonant tunneling. The formula (5.1) and the estimate(6.13) are auxiliary and are analysed only to that extent, which is needed for deriving theasymptotics of tunneling. For ease of notations, we shall in this section drop the symbol” ± ”, meaning that we deal with one of the equations (2.2).2 In the waveguide G ( ε ), we consider the scattering of the wave U ( x, y, z ) = e iν x Ψ ( y, z )incoming from −∞ (see (2.6)). The corresponding wave function admits the representation u ( x, y, z ; ε ) = χ , ε ( x, y, z ) v ( x, y, z ; ε ) ++Θ( r ) w ( ε − x , ε − y , ε − z ; ε ) + χ , ε ( x, y, z ) v ( x, y, z ; ε ) + (5.1)+Θ( r ) w ( ε − x , ε − y , ε − z ; ε ) + χ , ε ( x, y, z ) v ( x, y, z ; ε ) + R ( x, y, z ; ε ) . Let us explain the notation and structure of this formula. When constructing the asymp-totics, we first describe the behavior of the wave function u outside the narrows approximat-ing u by the solutions v j of the homogeneous problems (3.1) and (3.2) in G j . As v j we takecertain linear combinations of the special solutions introduced in the preceding section; indoing so we subject v and v to the same radiation conditions at infinity as u : v ( x, y, z ; ε ) = 1 A v ( x, y, z ) + (cid:101) s ( ε ) A v ( x, y, z ) ∼ U +1 ( x, y, z ) + (cid:101) s ( ε ) U − ( x, y, z ) , x → −∞ ; (5.2) v ( x, y, z ; ε ) = C ( ε ) v ( x, y, z ) + C ( ε ) v ( x, y, z ); (5.3) v ( x, y, z ; ε ) = (cid:101) s ( ε ) Ae − iν d v ( x, y, z ) ∼ (cid:101) s ( ε ) U − ( x, y, z ) , x → + ∞ ; (5.4)for the time being the approximations (cid:101) s ( ε ), (cid:101) s ( ε ) for the entries s ( ε ), s ( ε ) of thescattering matrix and the coefficients C ( ε ), C ( ε ) are unknown. Here χ j,ε stand for thecut-off functions defined by the equalities χ , ε ( x, y, z ) = (1 − Θ( r /ε )) G ( x, y, z ) , χ , ε ( x, y, z ) = (1 − Θ( r /ε )) G ( x, y, z ) ,χ , ε ( x, y, z ) = (1 − Θ( r /ε ) − Θ( r /ε )) G ( x, y, z ) , where r j = (cid:113) x j + y j + z j and ( x j , y j , z j ) are the coordinates of a point ( x, y, z ) in the systemwith the origin shifted to O j ; G j is the indicator of the set G j (equal to 1 in G j and 0 outside G j ); Θ( ρ ) is the same cut-off function as in (4.3) (equal to 1 for 0 (cid:54) ρ (cid:54) δ/ ρ (cid:62) δ with a fixed sufficiently small positive δ ). Thus χ j, ε are defined on the whole waveguide G ( ε )as well as the functions χ j, ε v in (5.1).When substituting (cid:80) j =1 χ j, ε v j in (2.2), we obtain the discrepancy in the right-handside of the Helmholtz equation supported near the narrows. We compensate the principalpart of the discrepancy making use of the second kind limit problems. In more detail, werewrite the discrepancy supported near O j in the coordinates ( ξ j , η j , ζ j ) = ( ε − x j , ε − y j , ε − z j )in the domain Ω j and take it as right-hand side for the Laplace equation. Then we rewritethe solution w j of the corresponding problem (3.7) in the coordinates ( x , y , z ) and multiplyit by the cut-off function. As a result, there arises the term Θ( r j ) w j ( ε − x j , ε − y j , ε − z j ; ε )in (5.1).The existence of solutions w j vanishing as O ( ρ − µ − j ) at infinity follows from Propo-sition 3.3 (see (3.10)). However choosing such solutions and then substituting (5.1) in (2.2),we obtain the discrepancy of high order that has to be compensated again. Therefore we3require w j = O ( ρ − µ − j ) as ρ j → ∞ . According to 3.3, such a solution exists if the right-handside of the problem (3.7) satisfies the additional conditions( F, w lj ) Ω j = 0 , ( F, w rj ) Ω j = 0 . Such conditions (two at each narrow) uniquely define the coefficients (cid:101) s ( ε ), (cid:101) s ( ε ), C ( ε ),and C ( ε ). The remainder R ( x, y, z ; ε ) is small in comparison with the principal part of (5.1)as ε → (cid:101) s , (cid:101) s , C , and C We are now going to define the right-hand side F j of problem (3.7) and to find (cid:101) s ( ε ), (cid:101) s ( ε ), C ( ε ), and C ( ε ). We substitute χ , ε v in (2.2) and obtain the discrepancy(∆ + k ) χ , ε v = [∆ , χ ε, ] v + χ ε, (∆ + k ) v = [∆ , − Θ( ε − r )] v , distinct from zero only near the point O , where v can be replaced by the asymptotics; theboundary condition (2.3) is fulfilled. According to (5.2) and (4.4), v ( x, y, z ; ε ) = r − / (cid:0) a − ( ε ) (cid:101) N µ +1 / ( kr ) + a +1 ( ε ) (cid:101) J µ +1 / ( kr ) (cid:1) Φ ( − ϕ ) + O ( r µ ) , r → , with a − ( ε ) = 1 A + (cid:101) s ( ε ) A , a +1 = aA + (cid:101) s ( ε ) aA . (5.5)We single out the principal part of each term and put ρ = r /ε , then(∆ + k ) χ ε, v ∼ [∆ , − Θ( ε − r )] (cid:0) a − r − µ − + a +1 r µ (cid:1) Φ ( − ϕ )= ε − [∆ ( ρ ,ϕ ) , − Θ( ρ )] (cid:0) a − ε − µ − ρ − µ − + a +1 ε µ ρ µ (cid:1) Φ ( − ϕ ) . (5.6)In the same way using (5.3) and (4.9)–(4.10), we obtain the principal part of the discrepancygiven by χ ε, v supported near O :(∆ + k ) χ ε, v ∼ ε − [∆ ( ρ ,ϕ ) , − Θ( ρ )] (cid:0) b − ε − µ − ρ − µ − + b +1 ε µ ρ µ (cid:1) Φ ( ϕ ) , (5.7)where b − = C ( ε )( k − k ) + C ( ε ) b , b +1 = C ( ε ) c + C ( ε ) d . (5.8)As right-hand side F of the problem (3.7) in Ω we take the function F ( ξ , η , ζ ) = − [∆ , θ − ] (cid:0) a − ε − µ − ρ − µ − + a +1 ε µ ρ µ (cid:1) Φ ( − ϕ ) − [∆ , θ + ] (cid:0) b − ε − µ − ρ − µ − + b +1 ε µ ρ µ (cid:1) Φ ( ϕ ) , (5.9)where θ + (respectively θ − ) stands for the function 1 − Θ first restricted to the domain ξ > ξ <
0) and then extended by zero to the whole domain Ω . Let w be the corresponding solution then the term Θ( r ) w ( ε − x , ε − y , ε − z ; ε ) in (5.1) beingsubstituted in (2.2) compensate the discrepancies (5.6) – (5.7).4In a similar manner, making use of (5.3) – (5.4), (4.9) – (4.10), and (4.5), we find theright-hand side of the problem (3.7) for j = 2: F ( ξ , η , ζ ) = − [∆ , θ − ] (cid:0) a − ε − µ − ρ − µ − + a +2 ε µ ρ µ (cid:1) Φ ( − ϕ ) − [∆ , θ + ] (cid:0) b − ε − µ − ρ − µ − + b +2 ε µ ρ µ (cid:1) Φ ( ϕ ); a − ( ε ) = − C ( ε ) b , a +2 ( ε ) = C ( ε ) c + C ( ε ) d , b − ( ε ) = (cid:101) s ( ε ) Ae − iν d , b +2 ( ε ) = a (cid:101) s ( ε ) Ae − iν d . (5.10) Lemma 5.1.
If the solution w j of the problem (3.7) with right-hand side F j ( ξ j , η j , ζ j ) = − [∆ , θ − ] (cid:0) a − j ε − µ − ρ − µ − j + a + j ε µ ρ µ j (cid:1) Φ ( − ϕ j ) − [∆ , θ + ] (cid:0) b − j ε − µ − ρ − µ − j + b + j ε µ ρ µ j (cid:1) Φ ( ϕ j ) ,j = 1 , , admits the estimate O ( ρ − µ − j ) as ρ j → ∞ , then a − j ε − µ − − αa + j ε µ − βb + j ε µ = 0 , b − j ε − µ − − αb + j ε µ − βa + j ε µ = 0 , (5.11) where α and β are the coefficients in (3.11) – (3.12).Proof. By Proposition 3.3, w j = O ( ρ − µ − j ) as ρ j → ∞ , if and only if the right-hand side ofthe problem (3.7) satisfies the conditions( F j , w lj ) Ω j = 0 , ( F j , w rj ) Ω j = 0 , (5.12)where w lj and w rj are the solutions to the homogeneous problem (3.7) with expansions (3.11) –(3.12). We introduce functions f ± on Ω j by the equalities f ± ( ρ j , ϕ j ) = ρ ± ( µ +1 / − / j Φ ( ϕ j ).In order to derive (5.11) from (5.12), it suffices to verify that([∆ , θ − ] f − , w lj ) Ω j = ([∆ , θ + ] f − , w rj ) Ω j = − , ([∆ , θ − ] f + , w lj ) Ω j = ([∆ , θ + ] f + , w rj ) Ω j = α, ([∆ , θ + ] f − , w lj ) Ω j = ([∆ , θ − ] f − , w rj ) Ω j = 0 , ([∆ , θ + ] f + , w lj ) Ω j = ([∆ , θ − ] f + , w rj ) Ω j = β. Let us check the first equalities, the other ones can be considered in a similar way. Thesupport of [∆ , θ + ] f − is compact, so when calculating ([∆ , θ − ] f − , w lj ) Ω j , one can replace Ω j byΩ Rj = Ω j ∩{ ρ j < R } with sufficiently large R . Let E denote the set ∂ Ω Rj ∩{ ρ j = R }∩{ ξ j > } .By the Green formula,([∆ , θ − ] f − , w lj ) Ω j = (∆ θ − f − , w lj ) Ω Rj − ( θ − f − , ∆ w lj ) Ω Rj = ( ∂f − /∂n, w lj ) E − ( f − , ∂w lj /∂n ) E . Taking into account (3.11) for ξ j < in Proposition 3.1, we obtain([∆ , θ − ] f − , w lj ) Ω j = (cid:34) ∂ρ − µ − j ∂ρ j ( ρ µ j + αρ − µ − j ) − ρ − µ − j ∂∂ρ j ( ρ µ j + αρ − µ − j ) (cid:35) ρ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ j = R × (cid:90) S ( K ) Φ( − ϕ j ) dϕ j + o (1) = − (2 µ + 1) (cid:90) S ( K ) Φ( − ϕ j ) dϕ j + o (1) = − o (1) . It remains to let R → ∞ .5 Remark 5.2.
The solutions w j mentioned in Lemma 5.1 can be written as linear combina-tions of certain model functions independent of ε . We present the corresponding expressions,which will be needed in the next section for estimating the remainders of asymptotic formu-las. Let w lj and w rj be the solutions to problem (3.7) defined by (3.11) – (3.12) and θ + , θ − the same cut-off functions as in (5.9). We set w lj = w lj − θ − (cid:0) ρ µ j + αρ − µ − j (cid:1) Φ ( − ϕ j ) − θ + βρ − µ − j Φ ( ϕ j ) , w rj = w rj − θ − βρ − µ − j Φ ( − ϕ j ) − ζ + (cid:0) ρ µ j + αρ − µ − j (cid:1) Φ ( ϕ j ) . A straightforward verification shows that w j = a + j ε µ w lj + 1 β (cid:0) a − j ε − µ − − αa + j ε µ (cid:1) w rj = 1 β (cid:0) b − j ε − µ − − αb + j ε µ (cid:1) w lj + b + j ε µ w rj . (5.13)We use (5.5) and (5.8) to rewrite (5.11) for j = 1 in the form γ ( ε ) (cid:101) s ( ε ) + γ ( ε ) = C ( ε ) c + C ( ε ) d , δ ( ε ) (cid:101) s ( ε ) + δ ( ε ) = C ( ε )( k − k ) + C ( ε ) b , (5.14)where γ ( ε ) = 1 Aβ (cid:0) ε − µ − − aα (cid:1) , δ ( ε ) = 1 Aβ (cid:0) α + a ( β − α ) ε µ +1 (cid:1) . (5.15)Moreover, taking account of (5.10), we rewrite (5.11) with j = 2 in the form γ ( ε ) (cid:101) s ( ε ) = ( C ( ε ) c + C ( ε ) d ) e − iν d , δ ( ε ) (cid:101) s ( ε ) = − C ( ε ) b e − iν d . (5.16)From (5.14) and (5.16), by means of Lemma 4.1, we obtain C ( ε ), C ( ε ), (cid:101) s ( ε ), and (cid:101) s ( ε ): C ( ε ) =( b c ) − (cid:0) γ ( ε ) b + δ ( ε ) d (cid:1) (cid:101) s ( ε ) e iν d , C ( ε ) = − b − δ ( ε ) (cid:101) s ( ε ) e iν d , (5.17) (cid:101) s ( ε ) =(2 ib c ) − (cid:0) ( k − k ) b | γ ( ε ) | + (( k − k ) d − b c ) γ ( ε ) δ ( ε ) − b c γ ( ε ) δ ( ε ) − ( c d − c d ) | δ ( ε ) | (cid:1)(cid:101) s ( ε ) e iν d , (5.18) (cid:101) s ( ε ) =2 ib c e − iν d (cid:0) − ( k − k ) b γ ( ε ) − (( k − k ) d − b c − b c ) γ ( ε ) δ ( ε )+ ( c d − c d ) δ ( ε ) (cid:1) − . (5.19) The solutions of the first limit problems involved in (5.1) are defined for the complex k aswell. The expression (5.19) obtained for (cid:101) s has a pole at k p in the lower half-plane. To find k p , we equate 2 ib c e − iν d / (cid:101) s to zero and solve this equation with respect to k − k : k − k = (cid:0) ( b c + b c ) γ ( ε ) δ ( ε ) + ( c d − c d ) δ ( ε ) (cid:1) (cid:0) b γ ( ε ) + d γ ( ε ) δ ( ε ) (cid:1) − . Since the right-hand side of this equation behaves as O ( ε µ +1 ) for ε →
0, its solution can befound by the successive approximation method. Taking into account (5.15), c j ( k ) = − b b j ,and Lemma 4.1 and neglecting the low order terms, we obtain k p = k r − ik i , k r = k − α ( | b | + | b | ) ε µ +1 + O ( ε µ +2 ) , k i = β ( | b | + | b | ) | A ( k ) | ε µ +2 + O ( ε µ +3 ) . (5.20)6For small k − k p , (5.19) takes the form (cid:101) s ( k, ε ) = − ε µ +2 iβ A ( k ) c ( k ) e − iν d k − k p (cid:0) O ( | k − k p | + ε µ +1 ) (cid:1) . Let k − k = O ( ε µ +1 ), then | k − k p | = O ( ε µ +1 ), A ( k ) = A ( k ) + O ( ε µ +1 ), c ( k ) = − b b + O ( ε µ +1 ), ν ( k ) = ν ( k ) + O ( ε µ +1 ), and (cid:101) s ( k, ε ) = ε µ +2 iβ b b A ( k ) e − iν ( k ) d k − k p (cid:0) O ( ε µ +1 ) (cid:1) = b | b | b | b | (cid:18) A ( k ) | A ( k ) | (cid:19) e − iν ( k ) d (cid:18) | b || b | + | b || b | (cid:19) − iP k − k r ε µ +2 (cid:0) O ( ε µ +1 ) (cid:1) , where P = (2 | b || b | β | A ( k ) | ) − . Thus (cid:101) T ( k, ε ) = | (cid:101) s | = 114 (cid:18) | b || b | + | b || b | (cid:19) + P (cid:18) k − k r ε µ +2 (cid:19) (1 + O ( ε µ +1 )) . (5.21)The obtained approximation (cid:101) T for the transition coefficient has a peak at k = k r whosewidth at its half-height is equal to (cid:101) Υ( ε ) = (cid:18) | b || b | + | b || b | (cid:19) P − ε µ +2 . (5.22) As in the preceding section, here we drop the symbol ” ± ” in notations and do not mentionwhich of the two equations in (2.2) is under consideration. We will return to the detailednotation in the formulation of Theorem 6.3.We introduce the function spaces for the problem( − i ∇ + A ) u ± Hu = k u in G ( ε ) , u = 0 on ∂G ( ε ) . (6.1)Recall that the functions A and H are compactly supported and differ from zero only inthe resonator at a distance from the narrows. Let Θ be the same function as in (4.3). Weassume that the cut-off functions η j j = 1 , ,
3, are distinct from zero only in G j and satisfy η ( x, y, z ) + Θ( r ) + η ( x, y, z ) + Θ( r ) + η ( x, y, z ) = 1 in G ( ε ). With γ ∈ R , δ >
0, and l = 0 , , V lγ,δ ( G ( ε )) is the completion in the norm (cid:107) u ; V lγ,δ ( G ( ε )) (cid:107) = (cid:32)(cid:90) G ( ε ) l (cid:88) | α | =0 (cid:32) (cid:88) j =1 Θ ( r j ) ( r j + ε j ) γ − l + | α | + η e δ | x | + η + η e δ | x | (cid:33) | ∂ α v | dx dy dz (cid:33) / (6.2)7of the set of smooth functions on G ( ε ) with compact supports. Denote by V , ⊥ γ,δ the spaceof functions f that are analytic in k , take values in V γ,δ ( G ( ε )), and, at k = k , satisfy( χ ,ε σ f, v ) G = 0 with a small σ > Proposition 6.1.
Assume that k r is a resonant energy, k r → k as ε → , and | k − k r | = O ( ε µ +1 ) . We also suppose that γ satisfies µ − / < γ − < µ + 1 / , f ∈ V , ⊥ γ,δ ( G ( ε )) ,and u a solution to problem (6.1) that admits the representation u = (cid:101) u + η A − U − + η A − U − ; here A − j = const and (cid:101) u ∈ V γ,δ ( G ( ε )) with small δ > . Then (cid:107) (cid:101) u ; V γ,δ ( G ( ε )) (cid:107) + | A − | + | A − | ≤ c (cid:107) f ; V γ,δ ( G ( ε )) (cid:107) , (6.3) where c is a constant independent of f and ε .Proof. Step A. We first construct an auxiliary function u p . As was mentioned, (cid:101) s has thepole k p = k r − ik i (see (5.20)). Let us multiply the solutions of limit problems involvedin (5.1), by A ( k ) b βε µ +1 /s ( ε, k ) e iν d , set k = k p , and re-denote the obtained functionsendowing them with the index p .Then v p ( x, y, z ; ε ) = ε µ +1 ( b β + O ( ε µ +1 )) v ( x, y, z ; k p ) , (6.4) v p ( x, y, z ; ε ) = ε µ +1 b β v ( x, y, z ; k p ); v p ( x, y, z ; ε ) = (cid:18) − b + O (cid:0) ε µ +1 (cid:1)(cid:19) v ( x, y, z ; k p )+ ε µ +1 (cid:18) − α b b + O (cid:0) ε µ +1 (cid:1)(cid:19) v ( x, y, z ; k p ) ,w p ( ξ , η , ζ ; ε ) = b ε µ +1 (cid:0) ε µ +1 (cid:0) a ( k p ) β + O ( ε µ +1 ) (cid:1) w l ( ξ , η , ζ )+ (cid:0) O ( ε µ +1 ) (cid:1) w r ( ξ , η , ζ ) (cid:1) , (6.5) w p ( ξ , η , ζ ; ε ) = b ε µ +1 (cid:0)(cid:0) O ( ε µ +1 ) (cid:1) w l ( ξ , η , ζ )+ a ( k p ) βε µ +1 w r ( ξ , η , ζ ) (cid:1) ; (6.6)the dependence of k p on ε has not been indicated. We set u p ( x, y, z ; ε ) = Ξ( x, y, z ) (cid:2) χ ,ε ( x, y, z ) v p ( x, y, z ; ε ) + Θ( ε − σ r ) w p ( ε − x , ε − y , ε − z ; ε )+ χ ,ε ( x, y, z ) v p ( x, y, z ; ε ) + Θ( ε − σ r ) w p ( ε − x , ε − y , ε − z ; k, ε )+ χ ,ε ( x, y, z ) v p ( x, y, z ; k, ε )] , (6.7)where Ξ is a cut-off function on G ( ε ) equal to 1 on G ( ε ) ∩ {| x | < R } and 0 on G ( ε ) ∩ {| x | >R + 1 } with sufficiently large R >
0, ( x j , y j , z j ) are the coordinates of a point ( x, y, z ) in thesystem with origin shifted to O j . The term χ ,ε v p gives the main contribution in the normof u p . In view of the definitions of v p and v (see Section 4) and Lemma 4.2, we obtain (cid:107) χ ,ε v p (cid:107) = (cid:107) v (cid:107) + o (1). Step
B. We show that (cid:107) (( − i ∇ + A ) ± H − k p ) u p ; V γ, δ ( G ( ε )) (cid:107) ≤ cε µ + κ , (6.8)8where κ = min { µ + 1 , µ + 1 − σ , γ + 3 / } , σ = 2 σ ( µ − γ + 3 / µ − / < γ − σ is sufficiently small so that µ − µ > σ , then κ = µ + 1.By virtue of (6.7)(( − i ∇ + A ) ± H − k p ) u p ( x, y, z ; ε )= [ (cid:52) , χ ,ε ] (cid:0) v ( x, y, z ; ε ) − b βε µ +1 ( r − µ − + a ( k p ) r µ )Φ ( − ϕ ) (cid:1) + [ (cid:52) , Θ] w p ( ε − x , ε − y , ε − z ; ε ) − k Θ( ε − σ r ) w p ( ε − x , ε − y , ε − z ; ε )+ [ (cid:52) , χ ,ε ] (cid:0) v ( x, y, z ; ε ) − Θ( r ) (cid:0) b − p ( ε ) r − µ − + b +1 p ( ε ) r µ (cid:1) Φ ( − ϕ ) − Θ( r ) (cid:0) a − p ( ε ) r − µ − + a +2 p ( ε ) r µ (cid:1) Φ ( ϕ ) (cid:1) + [ (cid:52) , Θ] w p ( ε − x , ε − y , ε − z ; ε ) − k Θ( ε − σ r ) w p ( ε − x , ε − y , ε − z ; ε )+ [ (cid:52) , χ ,ε ] (cid:0) v ( x, y, z ; ε ) − b βε µ +1 ( r − µ − + a ( k p ) r µ )Φ ( ϕ ) (cid:1) + [ (cid:52) , Ξ] v ( x, y, z ; ε ) + [ (cid:52) , Ξ] v ( x, y, z ; ε ) , where b − p = O ( ε µ +1 ), b +1 p = b + O ( ε µ +1 ), a − p = O ( ε µ +1 ), a +2 p = b + O ( ε µ +1 ).Taking account of the asymptotics v as r → ξ , η , ζ ) =( ε − x , ε − y , ε − z ), we arrive at (cid:13)(cid:13) ( x, y, z ) (cid:55)→ [ (cid:52) , χ ,ε ] (cid:0) v ( x, y, z ) − ( r − µ − + a ( k p ) r µ )Φ ( − ϕ ) (cid:1) ; V γ,δ ( G ( ε )) (cid:13)(cid:13) ≤ c (cid:90) G ( ε ) ( r + ε ) γ (cid:12)(cid:12) [ (cid:52) , χ ,ε ] r − µ +11 Φ( − ϕ ) (cid:12)(cid:12) dx dy dz ≤ cε γ − µ +1 / . This and (6.4) imply that (cid:13)(cid:13) ( x, y, z ) (cid:55)→ [ (cid:52) , χ ,ε ] (cid:0) v ( x, y, z ) − ( r − µ − + a ( k p ) r µ )Φ( − ϕ ) (cid:1) ; V γ,δ ( G ( ε )) (cid:13)(cid:13) ≤ cε γ + µ +3 / . Similarly, (cid:13)(cid:13) ( x, y, z ) (cid:55)→ [ (cid:52) , χ ,ε ] (cid:0) v ( x, y, z ) − Θ( r ) (cid:0) b − p ( ε ) r − µ − + b +1 p ( ε ) r µ (cid:1) Φ ( − ϕ ) − Θ( r ) (cid:0) a − p ( ε ) r − µ − + a +2 p ( ε ) r µ (cid:1) Φ ( ϕ ) (cid:1)(cid:13)(cid:13) ≤ cε γ + µ +3 / , (cid:13)(cid:13) ( x, y, z ) (cid:55)→ [ (cid:52) , χ ,ε ] (cid:0) v ( x, y, z ) − ( r − µ − + a ( k p ) r µ )Φ ( ϕ ) (cid:1) ; V γ,δ ( G ( ε )) (cid:13)(cid:13) ≤ cε γ + µ +3 / . It is clear that (cid:13)(cid:13) [ (cid:52) , Ξ] v l ; V γ,δ ( G ( ε )) (cid:13)(cid:13) ≤ cε µ +1 , l = 1 , . Further, since w lj behaves as O ( ρ − µ − j ) at infinity, we have (cid:90) G ( ε ) ( r j + ε ) γ (cid:12)(cid:12) [ (cid:52) , Θ] w lj ( ε − x j , ε − y j , ε − z j ) (cid:12)(cid:12) dx j dy j dz j ≤ c (cid:90) K j ( r j + ε ) γ (cid:12)(cid:12) [ (cid:52) , Θ]( ε − r j ) − µ − Φ ( ϕ j ) (cid:12)(cid:12) dx j dy j dz j ≤ cε µ +1 − σ ) , where σ = 2 σ ( µ − γ + 3 / w lj changed for w rj . Inview of (6.5) and (6.6), we obtain (cid:13)(cid:13) [ (cid:52) , Θ] w jp ; V γ,δ ( G ( ε )) (cid:13)(cid:13) ≤ cε µ + µ +1 − σ . (cid:90) G ( ε ) ( r j + ε ) γ (cid:12)(cid:12) Θ( ε − σ r j ) w lj ( ε − x j , ε − y j , ε − z j ) (cid:12)(cid:12) dx j dy j dz j = ε γ +3 (cid:90) Ω ( ρ j + 1) γ (cid:12)(cid:12) Θ( ε − σ ρ j ) w lj ( ξ j , η j , ζ j ) (cid:12)(cid:12) dξ j dη j dζ j ≤ cε γ +3 , and a similar estimate for w rj , we derive (cid:13)(cid:13) ( x, y ) (cid:55)→ Θ( ε − σ r j ) w jp ( ε − x j , ε − y j , ε − z j ); V γ,δ ( G ( ε )) (cid:13)(cid:13) ≤ cε µ + γ +3 / . Combining the obtained inequalities, we arrive at (6.8).
Step
C. This part contains a somewhat modified argument in the proof of Theorem5.5.1 [4]. Let us rewrite the right-hand side of problem (6.1) in the form f ( x, y, z ) = f ( x, y, z ; ε ) + f ( x, y, z ; ε ) + f ( x, y, z ; ε )+ ε − γ − / F ( ε − x , ε − y , ε − z ; ε ) + ε − γ − / F ( ε − x , ε − y , ε − z ; ε ) , (6.9)where f l ( x, y, z ; ε ) = χ l,ε σ ( x, y, z ) f ( x, y, z ) ,F j ( ξ j , η j , ζ j ; ε ) = ε γ +3 / Θ( ε − σ ρ j ) f ( x O j + εξ j , y O j + εη j , z O j + εζ j );( x, y, z ) are arbitrary Cartesian coordinates; ( x O j , y O j , z O j ) denote the coordinates of thepoint O j in the system ( x, y, z ); x j , y j , z j were introduced in Section 2. From the definitionsof the norms it follows that (cid:107) f ; V γ, δ ( G ) (cid:107) + (cid:107) f ; V γ ( G ) (cid:107) + (cid:107) f ; V γ, δ ( G ) (cid:107) + (cid:107) F j ; V γ (Ω j ) (cid:107) ≤ (cid:107) f ; V γ, δ ( G ( ε )) (cid:107) . (6.10)We consider solutions v l and w j of the limit problems − ( − i ∇ + A ) v ± Hv + k v = f in G , v = 0 on ∂G , (cid:52) v + k v = f l in G l , v = 0 on ∂G l , l = 1 , , (cid:52) w = F j in Ω j , w = 0 on ∂ Ω j , respectively; besides, v l with l = 1 , v is subject to the condition ( v , v ) G = 0. According to Proposition 3.1, 3.2, and3.3, the problems in G l and Ω j are uniquely solvable and (cid:107) v ; V γ ( G ) (cid:107) ≤ c (cid:107) f ; V γ ( G ) (cid:107) , (cid:107) v l ; V γ,δ, − ( G l ) (cid:107) ≤ c l (cid:107) f l ; V γ,δ ( G l ) (cid:107) , l = 1 , (cid:107) w j ; V γ (Ω j ) (cid:107) ≤ C j (cid:107) F j ; V γ (Ω j ) (cid:107) , j = 1 , , (6.11)where c l and C j are independent of ε . We set U ( x, y, z ; ε ) = χ ,ε ( x, y, z ) v ( x, y, z ; ε ) + ε − γ +3 / Θ( r ) w ( ε − x , ε − y , ε − z ; ε )+ χ ,ε ( x, y, z ) v ( x, y, z ; ε ) + ε − γ +3 / Θ( r ) w ( ε − x , ε − y , ε − z ; ε )+ χ ,ε ( x, y, z ) v ( x, y, z ; ε ) . (cid:107) U ; V γ, δ, − ( G ( ε )) (cid:107) ≤ c (cid:107) f ; V γ,δ ( G ( ε )) (cid:107) (6.12)with constant c independent of ε . Denote the operator f (cid:55)→ U by R ε . Arguing as in theproof of [4, Theorem 5.5.1], we obtain ( − ( − i ∇ + A ) ± H + k ) R ε = I + S ε , where S ε is anoperator with small norm in V γ,δ ( G ( ε )). Step
D. Recall that the operator S ε is defined on the subspace V , ⊥ γ, δ ( G ( ε )). We needthat the range of S ε would also be in V , ⊥ γ, δ ( G ( ε )). To this end we change R ε for (cid:101) R ε : f (cid:55)→ U ( f ) + a ( f ) u p , where u p was constructed at step A , a ( f ) being a constant. Then ( − ( − i ∇ + A ) ± H + k ) (cid:101) R ε = I + (cid:101) S ε with (cid:101) S ε = S ε + a ( · )( − ( − i ∇ + A ) ± H + k ) u p . The condition( χ ,ε σ (cid:101) S ε f, v ) G = 0 with k = k implies that a ( f ) = − ( χ ,ε σ S ε f, v ) G / ( χ ,ε σ ( − ( − i ∇ + A ) ± H + k ) u p , v ) G . We show that (cid:107) (cid:101) S ε (cid:107) ≤ c (cid:107) S ε (cid:107) , where c is independent of ε and k . We have (cid:107) (cid:101) S ε f (cid:107) ≤ (cid:107) S ε f (cid:107) + | a ( f ) | (cid:107) ( − ( − i ∇ + A ) ± H + k ) u p (cid:107) . The estimate (6.8) (with γ > µ − / µ − µ > σ ), the formula for k p , and thecondition k − k = O (cid:0) ε µ +1 (cid:1) lead to the inequality (cid:107) ( − ( − i ∇ + A ) ± H + k ) u p ; V γ,δ (cid:107)≤ | k − k p | (cid:107) u p ; V γ,δ (cid:107) + (cid:107) ( − ( − i ∇ + A ) ± H + k p ) u p ; V γ,δ (cid:107) ≤ cε µ +1 . The supports of the functions ( − ( − i ∇ + A ) ± H + k p ) u p and χ ,ε σ are disjoint, so | ( χ ,ε σ ( − ( − i ∇ + A ) ± H + k ) u p , v ) G | = | ( k − k p )( u p , v ) G | ≥ cε µ +1 . Further, γ − < µ + 1 /
2, therefore | ( χ ,ε σ S ε f, v ) G | ≤ (cid:107) S ε f ; V γ,δ ( G ( ε )) (cid:107) (cid:107) v ; V − γ ( G ) (cid:107) ≤ c (cid:107) S ε f ; V γ,δ ( G ( ε )) (cid:107) . Hence | a ( f ) | ≤ cε − µ − (cid:107) S ε f ; V γ,δ ( G ( ε )) (cid:107) and (cid:107) (cid:101) S ε f (cid:107) ≤ c (cid:107) S ε f (cid:107) . It follows that the operator I + (cid:101) S ε in V , ⊥ γ,δ ( G ( ε )) invertible as well asthe operator of problem (6.1): A ε : u (cid:55)→ − ( − i ∇ + A ) u ± Hu + k u : ˚V , ⊥ γ,δ, − ( G ( ε )) (cid:55)→ V , ⊥ γ,δ ( G ( ε ));here ˚V , ⊥ γ,δ, − ( G ( ε )) stands for the space of functions in V γ,δ, − ( G ( ε )) that vanish at ∂G ( ε ) andare sent by the operator − ( − i ∇ + A ) ± H + k to V , ⊥ γ,δ . The inverse operator A − ε = (cid:101) R ε ( I + (cid:101) S ε ) − has been bounded uniformly with respect to ε and k . Therefore, (6.3) holdswith a constant c independent of ε and k .We consider the solution u to the homogeneous problem (2.2) satisfying u ( x, y, z ) = (cid:40) U +1 ( x, y, z ) + s U − ( x, y, z ) + O (exp ( δx )) , x → −∞ ,s U − ( x, y, z ) + O (exp ( − δx )) , x → + ∞ . Let s and s be the entries of the scattering matrix determined by this solution. Denoteby (cid:101) u ,σ the function given by (5.1) changing Θ( r j ) for Θ( ε − σj r j ) and dropping the remainder R , while (cid:101) s , and (cid:101) s stand for the quantities defined in (5.18) and (5.19).1 Theorem 6.2.
Let the assumptions of Proposition be fulfilled. Then the inequality | s − (cid:101) s | + | s − (cid:101) s | ≤ c | (cid:101) s | ε τ holds with constant c independent of ε, k ; τ = min { − δ, µ − µ } and with arbitrarily smallpositive δ .Proof. The difference R = u − (cid:101) u ,σ belongs to V γ, δ, − ( G ( ε )), whereas f := ( − ( − i ∇ + A ) ± H + k )( u − (cid:101) u ,σ ) is in V , ⊥ γ, δ ( G ( ε )). By Proposition 6.1, (cid:107) R ; V γ, δ, − ( G ( ε )) (cid:107) ≤ c (cid:107) f ; V γ,δ ( G ( ε )) (cid:107) . (6.13)We show that (cid:107) f ; V γ, δ ( G ( ε )) (cid:107) ≤ c | (cid:101) s | ( ε γ − µ +1 / + ε µ − µ − σ ) , (6.14)where σ = 2 σ ( µ − γ + 3 / γ = µ + 3 / − δ and σ = δ .Arguing as in the step B of the proof of Proposition 6.1 we obtain (cid:107) f ; V γ, δ ( G ( ε )) (cid:107) ≤ c ( ε γ +3 / + ε µ +1 − σ ) × max j =1 , ( | a − j ( ε ) | ε − µ − + | a + j ( ε ) | ε µ + | b − j ( ε ) | ε − µ − + | b + j ( ε ) | ε µ ) . From (5.11) it follows that( | a − j ( ε ) | ε − µ − + | a + j ( ε ) | ε µ ) ≤ c ( | b − j ( ε ) | ε − µ − + | b + j ( ε ) | ε µ ) . Taking into account (5.8) and (5.10) for b ± j and also (5.17) and (5.15), we derive | b − j ( ε ) | ε − µ − + | b + j ( ε ) | ε µ ≤ cε − µ − | (cid:101) s ( ε ) | . Combining the obtained estimates, we arrive at (6.14).Theorem 6.2 together with (5.21) and (5.22) lead to the following assertion. We returnhere to the detailed notations introduced in Sections 2 - 4.
Theorem 6.3.
For | k − k r, ± | = O ( ε µ +1 ) there hold the asymptotic representations T ± ( k, ε ) = 114 (cid:18) | b ± || b ± | + | b ± || b ± | (cid:19) + P ± (cid:18) k − k r, ± ε µ +2 (cid:19) (1 + O ( ε τ )) ,k r, ± = k , ± − α ( | b ± | + | b ± | ) ε µ +1 + O (cid:0) ε µ +1+ τ (cid:1) , Υ ± ( ε ) = (cid:18) | b ± || b ± | + | b ± || b ± | (cid:19) P − ± ε µ +2 (cid:0) O ( ε τ ) (cid:1) , where Υ ± ( ε ) is the width of the resonant peak at its half-height (the so-called resonant qualityfactor), P ± = (2 | b ± || b ± | β | A ( k ) | ) − , and τ = min { − δ, µ − µ } , δ being an arbitrary smallpositive number. References [1] L. Baskin, P. Neittaanm¨aki, B. Plamenevskii, and O. Sarafanov,
Asymptotic Theoryof Resonant Tunneling in 3D Quantum Waveguides of Variable Cross-Section , SIAMJ. Appl. Math., 70(2009), no. 5, pp. 1542–1566.[2]
J. T. Londergan, J. P. Carini, and D. P. Murdock , Binding and Scattering inTwo-Dimensional Systems: Application to Quantum Wires, Waveguides and PhotonicCrystals , Springer-Verlag, Berlin, 1999.[3] L.M.Baskin, P.Neittaanm¨aki, B.A.Plamenenevskii, and A.A.Pozharskii, On elec-tron transport in 3D quantum waveguides of variable cross-section, Nanotechnology,17(2006), pp. 19-23.[4] V.G.Maz’ya, S.A.Nazarov, and B.A.Plamenevskii,