A model of an electron in a quantum graph interacting with a two-level system Текст научной статьи по специальности «Математика»

NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2019, 10 (2), P. 131-140

A model of an electron in a quantum graph interacting with a two-level system

A. A. Boitsev, I. Y. Popov

ITMO University, Kronverkskiy, 49, St. Petersburg, 197101, Russia boitsevanton@gmail.com, popov1955@gmail.com

PACS 03.65.Nk, 73.23.Ad

DOI 10.17586/2220-8054-2019-10-2-131-140

A model of an electron in a quantum graph interacting with a two-level system is considered. The operator describing the model has the form of sum of tensor products. Self-adjoint extensions and a scattering matrix are written in terms of a boundary triplet, corresponding to the considered symmetric operator. Diagrams of reflection are calculated and numerical results are discussed .

Keywords: Aharonov-Bohm ring, nanostructure, quantum graph, scattering, solvable model.

Received: 1 March 2019 Revised: 30 March 2019

1. Introduction

Electron transport in nanostructures under the action of a magnetic field attracts great attention during last decades. The reason is observation of many interesting effects found applications in nanoelectronics. One of such intriguing problems is that of Aharonov-Bohm oscillations in quantum transport [1-3]. Several models were suggested to describe the phenomenon. One of the most effective in the field is the quantum graph model (see, e.g., [4-7]). An excellent review of the state of the art in quantum graph theory is presented in [8].

We are interested in the case when the system (quantum graph) with an Aharonov-Bohm ring interacts with another system having two energy levels. In this case, the operator of the whole system has the form of a sum of tensor products:

where self-adjoint operators A and T act in Hilbert spaces H and T, respectively. It is well known that such operators describe the interaction of two quantum systems. Extension technique for such operators is widely discussed in [9]. We introduce a model for an electron in a quantum graph interacting with a two-level system. Such an operator also preserves a tensor structure described above. The first operator A stands for the quantum graph and the second operator T, which is, actually, a 2 x 2 matrix, describes the two-level system.

In the following, we investigate the considered operator using boundary triplets approach and the results from [9]. Scattering matrix is obtained and the diagrams of the argument of the reflection coefficient (scattering phase) are constructed. The scattering has a resonance character.

2. Preliminaries 2.1. Linear relations

A linear relation © in H is a closed linear subspace of H © H. The set of all linear relations in H is denoted by C(H). We denote also by C(H) the set of all closed linear (not necessarily densely defined) operators in H. Identifying each operator T g C(H) with its graph gr (T), we regard C(H) as a subset of C(H).

The role of the set C(H) in extension theory becomes clear from Proposition 2.3. However, its role in the operator theory is substantially motivated by the following circumstances: in contrast to C (H), the set C(H) is

closed with respect to taking inverse and adjoint relations © 1 and ©*. The latter is given by: © 1 = {{g,f} : if,g}e ©} and

S := Ah ® I% + ® T%,

A linear relation © is called symmetric if © c ©* and self-adjoint if © = ©*.

2.2. Boundary triplets and proper extensions

Let us briefly recall some basic facts regarding boundary triplets. Let S be a densely defined closed symmetric operator with equal deficiency indices n±(S) := dim(N±j), Nz := ker (S* -z), z G C±, acting on some separable Hilbert space H.

Definition 2.1.

(i) A closed extension S of S is called proper if dom(S) c dom(S) c dom(S*).

(ii) Two proper extensions S', S are called disjoint if dom (S") n dom (S) = dom (S) and transversal if in addition dom (S') + dom (S) = dom (S*).

We denote by ExtS the set of all proper extensions of S completed by the non-proper extensions S and S* is denoted. For instance, any self-adjoint or maximal dissipative (accumulative) extension is proper.

Definition 2.2 ([10]). A triplet n = {H, r0, rj, where H is an auxiliary Hilbert space and r0, r : dom (S*) ^ H are linear mappings, is called a boundary triplet for S* if the "abstract Green's identity":

(S*f,g) - (f, S*g) = (rif, rog) - (rof,rig), f, g G dom (S*). (2)

is satisfied and the mapping r := (r0, r1)T : dom (S*) ^ H © H is surjective, i.e. ran (r) = H © H.

A boundary triplet n = {H, r0, r1} for S* always exists whenever n+(S) = n_(S). Note also that n±(S) = dim(H) and ker (r0) n ker (r1) = dom (S).

With any boundary triplet n, one associates two canonical self-adjoint extensions Sj := S* \ ker(r), j G {0,1}. Conversely, for any extension S0 = S* g ExtS there exists a (non-unique) boundary triplet n = {H, r0, ri} for S* such that S0 := S* \ ker (r0).

Using the concept of boundary triplets one can parameterize all proper extensions of A in the following way.

Proposition 2.3 ( [11,12]). Let n = {H, r0, r1} be a boundary triplet for S *. Then the mapping:

ExtS 3 S ^ rdom (S) = {(r0f, r1f)T : f G dom (S)} =: © G C(H) (3)

establishes a bijective correspondence between the sets ExtS and C(H). We write S = SQ if S corresponds to © by (3). Moreover, the following holds:

(i) SQ = Sq» , in particular, SQ = SQ if and only if ©* = ©.

(ii) SQ is symmetric (self-adjoint) if and only if © is symmetric (self-adjoint).

(iii) The extensions SQ and S0 are disjoint (transversal) if and only if there is a closed (bounded) operator B such that © = gr (B). In this case (3) takes the form:

Sq := Sgr(b) = S* \ ker (r - B^). (4)

In particular, Sj := S* \ ker (r) = Sq3 , j G {0,1}, where ©0 := ^^ j and ©1 := ^^ j = gr (O) where

O denotes the zero operator in H. Note also that C(H) contains the trivial linear relations {0} x {0} and H x H parameterizing the extensions S and S*, respectively, for any boundary triplet n.

2.3. Gamma field and Weyl function

It is well known that Weyl function is an important tool in the direct and inverse spectral theory of Sturm-Liouville operators. In [11,12] the concept of Weyl function was generalized to the case of an arbitrary symmetric operator S with n+(S) = n_(S) < to. Following [11] we briefly recall basic facts on Weyl functions and 7-fields associated with a boundary triplet n.

Definition 2.4 ( [11,12]). Let n = {H, r0, r1} be a boundary triplet for S* and S0 = S* \ ker (r0). The operator valued functions y(-) : p(S0) ^ [H, H] and M(•) : p(S0) ^ [H] defined by:

Y(z) := (r0 \ Nz)-1 M(z):=r17(z), z G KS0), (5)

are called the 7-field and the Weyl function, respectively, corresponding to the boundary triplet n.

Clearly, the Weyl function can equivalently be defined by:

M(z)r0fz =rfz, fz G Nz, z G p(S0). (6)

The Y-field y(-) and the Weyl function M(•) in (5) are well defined. Moreover, both y(-) and M(•) are holomorphic on p(S0) and the following relations:

Y(z) = (I + (z - Z)(So - z)-1)y(Z), z, Z G p(So), (7)

and

M(z) - M(Zr = (z - Z)y(Z)*y(z), Z, Z G p(So), (8)

hold. Identity (8) yields that M(•) is [H]-valued Nevanlinna function (M(•) g R[H]), i.e. M(•) is [H]-valued holomorphic function on C± satisfying:

M(z) = M(z)* and > 0, z G C+ u C_. (9)

It also follows from (8) that 0 g p(Im(M(z))) for all z G C±.

2.4. Krein-type formula for resolvents

Let n = {H, r0, r1} be a boundary triplet for S*, M(•) and y(-) the corresponding Weyl function and Y-field, respectively. For any proper (not necessarily self-adjoint) extension Se g ExtS with non-empty resolvent set p(Se) the following Krein-type formula holds (cf. [11,12]):

(SQ - z)-1 - (So - z)-1 = y(z)(© - M(z))-1Y*(z), z G p(So) n p(SQ). (10)

Formula (10) extends the known Krein formula for canonical resolvents to the case of any Se g ExtS with p(Se) = 0. Moreover, due to relations (3), (4) and (5) formula (10) is related to the boundary triplet n. We emphasize, that this relation makes it possible to apply the Krein-type formula (10) to boundary value problems (see, e.g., [13,14]).

2.5. Scattering

Let S be a densely defined closed symmetric operator with finite equal deficiency indices n±(S) and n = {H, r0, r1} is a boundary triplet for S*, Let S0 = S* \ kerr0 and Se is a self-adjoint extension corresponding to © G (7(H). As dimH is finite, by 10

(Sq - z)-1 - (So - z)-1, (11)

is a finite rank operator and the system {Se, S0} is a so-called complete scattering system, i.e. the wave operators:

W(Se,S0) = s - lim eitSee-itSoPoc(S0) (12)

exists and they are complete, i.e. their ranges coincide with the absolutely continuous subspace H0c(Se) of Se (see, e.g. [17], [15], [16]). By Poc(S0) we denote the orthogonal projection on absolutely continuous subspace Hac(S0) of S0. The scattering operator S(Se, S0) of a scattering system {Se, S0} is defined as:

S(Sq ,So) = W+(Se,So)*W-(Se,So). (13)

If we regard the scattering operator as an operator in Hoc(S0) then it becomes unitary and commutes with absolutely continuous part:

S0ac = So \ Hac(So) n dom (So). (14)

of S0 and thus it is unitarily equivalent to a multiplication operator induced by a family {Se(z)} of unitary operators in a spectral representation of S0c ( [17], Proposition 9.57). Tis family is called a scattering matrix of a scattering system S(Se, S0).

Since the dimension dimH is finite then the Weyl function M(•) corresponding to boundary triplet n = {H, r0, r1} is a matrix-valued Nevanlinna function. By Fatous theorem ( [18]), the limit:

M (A + i0) = Kmo M (A + ¿e) (15)

exists for almost all A g R. We denote the set of real point where the limit exists by £m. We will use the notation:

hm(a) = ran (M(A)), A G £m. (16)

By PM(a) we will denote the orthogonal projection on HM(a). We will also use the notation:

Ne(z) = (© - M(z))-1, z G C \ R, (17)

where © G (7(H) is a self-adjoint relation corresponding to Se. This function is well defined and the limit:

Ne(A + ¿0) = (© - M(A + ¿0))-1, (18)

exists almost for every A g R. This set we will denote as .

Theorem 2.5. ( [14]) Let S be a densely defined symmetric operator with finite deficiency indices in separable Hilbert space H, let n be a boundary triplet corresponding to S * with corresponding Weyl function M (•), Sq is a self-adjoint extension of S, S0 = S* f kerr0, © G C?(%), then in L2(R, dA, HM(A)) the scattering matrix of the complete scattering system {Sq, S0} is given by:

Sq(A) = /hm(a) + 2V^(M (A + i0))N (A + i0)VS(M (A + i0)), (19)

for A G SM n .

3. Model construction

3.1. An electron in quantum graph

Let us consider a Hilbert space Hi = L2(Cr U [-1,0]), r > 0, where:

Cr := {x G R2 : p(x, -1 - r) = r}, (20)

so that our Hilbert space is a union of a line segment [-1,1] and a ring with center at the point (-1 — r) g R of radius r. In a ring we consider an operator (x stands for the angle ^ in polar coordinates):

Arf := -(1 -f + f (21)

\r dx j

with domain dom AR = {f G W2'2[0, 2n] : f (0) = f (2 n) = 0}. Let us show that the operator is self-adjoint. Integrating by parts, we have:

2n 2n 2n 2n

1 -" 2i - f __1 i fll-J„. , 2i^ i ^2

(Arf, g) = - f (1 f " - -$f ' - f gdx = -\ f f ''gdx + - $ f f 'gdx - $2 f fgdx =

r2 r r2 r

0 0 0 0

-^ f '(2n)g(2n) - f '(0)g(0) + f (0)g'(0) - f (2n)g'(2n) + f "dxj +

(2n \ 2n

i (f (2n)g(2n) - f (0)g(0)) + y figdxl + $2 J fgdx =

00

1 2$

-(f'(2n)g(2n) - f'(0)g(0) + f (0)g'(0) - f (2n)g'(2n)) + —i (f (2n)g(2n) - f (0)g(0)) +

2n

This proves the statement.

f f( - 4g" + —ig' + $2g ]dx.

r2 r

d2

Now we introduce a self-adjoint operator AS := -^r in L2 [-1,0] with domain dom AS = {f G W2'2 [-1,0] :

dx2

f (-1) = f (0)=0}.

In compound system we consider an operator A1 acting in H1 as an operator AR on a circle and AS on a line segment. To make it symmetric, we restrict it on a set of functions with the conditions:

f2 ( 1 ) = fi(0) = /i(2n)

f<-D+(dX + «) *«» - (dX + «) /ic^c • (22)

where f1 G dom AR, f2 G dom AS. To find the deficiency elements of A1 we firstly solve the equation ARf = zf and come to an algebraic equation:

- f1A - i^ = z ^ 1A - = ±VZ (23)

rr

or A = i (r$ ± rVz). The deficiency elements (if we choose the branch SVz > 0) are eirx($±v/z). For the operator AS deficiency elements are e±iv/z. To find the deficiency elements of the considered operator A1, we solve the system of algebraic equations. We start by introducing:

fi := c3eirx($+^z) + C4eirx($-^z), /2 := ciei^z + C2e-i^, Ci G C (24)

and for simplicity introduce the notation a := e-i^, b := ei^z, / := e2nir($+^z), d := e2nir($-^z). Then, from the boundary conditions, we obtain:

cia + C2b = C3 + C4, C3 + C4 = C3/ + C4 d,

ci Vza - C2 Vzb + C3 (r($ + V)(1 - /) + - /)) + C4 (r($ - Vz)(1 - d) + $(1 - d)) = 0. Solving the system above, we obtain:

/1 = C4 eirx($+^ + C4eirx($-^z), (25)

C4 (d - / - 2rd + 2rd/ + 2r - 2r/\ ^^

/2 = 2al-W-^ + (26)

C4 (d - / + 2rd - 2rd/ - 2r + 2/ e-iyzx (26)

2b 1 - / So, the deficiency indices of the operator Ai are equal to 1, i.e. n±(Ai) = 1. The boundary triplet for the operator Ai is as follows:

HAl := C, ra1 := /(0), rf1 := -/'(0).

One immediately checks that the equation (2.2) is satisfied. For simplicity we put c4 = 2(1 - /) in (25) and (26). Then,

rf1 /2 = 1 (d - / - 2rd + 2rd/ + 2r - 2r/) + 1 (d - / + 2rd - 2rd/ - 2r + 2r/). (27)

Putting:

u := 1 (d - / - 2rd + 2rd/ + 2r - 2r/), (28)

a

v := 1 (d - / + 2rd - 2rd/ - 2r + 2r/), (29)

the Y-field yal (z) has the form (in accordance with definition 2.4):

YAl (z) := (uei^zx + ve-i^x) . (30)

u + v V /

Then the Weyl function (in accordance with definition 2.4) is:

Mai (z):=rai yai (z) = - i^z(u - V). (31)

u + v

To obtain the scattering matrix (19), we have to calculate the limit of the Weyl function (2.4) to the real axis from the upper half-plane. All the calculations with the final expressions are obtained in Appendix A.

3.2. Operator on a half-line

Let us consider an operator:

d2

A2 := --1-2 , (32)

d-2

with the domain dom A2 = Wo2 = {/ G W2'2(0, to) : /(0) = /'(0) = 0} in H2 = L2(0, to). It is symmetric and its deficiency indices are n±(A3) = 1. In accordance with [9], the Weyl function for this operator has the following form:

Ma3(z) := ¿Vz.

It is clear that:

sMa3 (A) = s (ivA) = 7A, A > 0

and 0 otherwise.

3.3. The direct sum of the operators

We consider the Hilbert space H = H1 © H2 where H1 and H2 are defined above. We define an operator A as the operator:

A := A1 © A2. (33)

The operator A is symmetric and has deficiency indices n±(A) = 2. It's Weyl function is obviously given by the expression:

M A(z) =

MAl (z) 0 0 M A2 (z)

г^(и — v)

0

u + v 0

(34)

3.4. Coupling to the two-level system

Now let us couple the considered operator to an operator:

10

T :=

,0 2,

acting on the Hilbert space T = C2. For this purpose we consider an operator

S = Ah ® IT + Is ® Tt.

In accordance with [9], we have:

MS(z) = MA(z — 1) ® J 0 + MA(z — 2) ® 0 1

(35)

(36)

(37)

(MAl (z — 1) 0 0

0 MAl (z — 2) 0

0 0 MA2 (z — 1)

\

\

(38)

0 0

12 (z — 1) 0

0 0 0 MA2 (z — 2)/

To find the scattering matrix, we need to calculate the limit of the Weyl function to the real axis which is, obviously:

MS (A) = MS (A + ¿0) =

(MAl (A — 1) 0 0

0 Mal (A — 2) 0

0 0 Ma2 (A — 1)

\

\

0

0

0

(39)

MA2 (A — 2)/

Now we need to calculate the imaginary part of the obtained limit. This gives us

^SM Ai (A — 1) 0

^SMS (A) =

V

^SM Ai (A — 2) 0 0

SM A2 (A — 1)

0 y'sM a2 (A — 2) j

(40)

3.5. Scattering matrix

Let us take the matrix of parameters © in the form (a, p G C):

/0 a 0 0\

a 0 p 0

© =

0 в 0 a \0 0 a 0/

(41)

Then:

Nq(A) = (© — Ms (A))-1 =

(—MAl (A — 1) a

a —MAl (A — 2)

0 в 00

00 в0

—MA2 (A — 1) a

a —MA2 (A — 2)/

1

(42)

0

We denote:

an = Mal (A - 1), a22 = Mal (A - 2), a33 = Ma(A - 1), a44 = Ma(A - 2).

Then, one has:

No =

1

aiia22a33a44 - ana441^|2 - aiia22|a|2 - a33a44|a|2 + |a|4

0 0\

/ (a44(-a22a33 + |fi|2) + |a|2a22 |a|2a - aa33a44

-ana33a44 + |a|2aii 0 0

-^ana44 0 0

V

|a|2a - aa33a44

-afia44

-a2fi

00 00

-afia44 —^ana44

-aiiafi

-a2fi -afiaii

+

00 \ \

0 0 -aiia22a44 + |a|2a44

\0 0 |a|2a - aiia22a -a33(aiia22 - |a|2) + aii|fi|2y

2

a| a - aiia22a 2

Assuming:

we, finally, obtain:

A :=

2i

aiia22a33a44 - aiia44|fi|2 - aiia22|a|2 - a33a44|a|2 + |a|4'

60(A) = I + A-

//9aii (a44(-a22a33 + |fi|2) + |a|2a2^ vSiiSâ^ (|a|2a - aa33a44) 0 0\

v/sâi

iSa22 (|a|2a - aa33a44)

V

-yj Saii 8«33 afia44

-yj Sa ii Sa44a2 fi

Sa22 (-aua33a44 + |a|2an) ^ V/Sa22 Sa33 fia ii a44 ^\ZSa22Sa44aii5fi

00 00 00

+

/0 0 - ^/Saii9;a33afia44

0 0 - v/Sa^^a^fiaiia44

0 0 Sa33 (-aua22a44 + |a|2a44)

- ^/SanSa44a2fi

- Y^Sa223;a44afiaii yj9;a339;a44 (|a|2a - ana22a)

\

\0 0 \JSa33Sa44 (|a|2a - ana22a) Sa44 (-a33(ana22 - |a|2) + aii|fi|2)/ Now, we have to take the projection onto the absolutely continuous part, we calculate:

® (1 0) j 60(A) I <8> (0 1H 60(A)

10 01

» Î • 1

+

and obtain:

2 + ASaii (a44(-a22a33 + |fi|2) + |a|2a2^ + ASa22 (-aiia33a44 + |a|2an) 0

- Ayj Saii 9;a33 afia44 - Ay/ Sa22Sa44aiiafi

0

+

(0 - Ay/ SaiiSa33 afia44 - A^/ Sa22Sa44afiaii

0 2 + ASa33 (-aua22a44 + |a|2a4^ + ASa44 (-a33(ana22 - |a|2) + an|fi|2)^

We are interested in the argument of the coefficient:

r := 2 + ASa33 (-aua22a44 + |a|2a4^ + ASa44 (-a33(ana22 - |a|2) + au|fi|2)

(43)

Fig. 1. Argument of the reflection coefficient argr as a function of energy A. Left column: a = p =1/3, values of $ vary from top to bottom through, consequently, 1/4,1/5,1/8,1/10; right column: $ = p = 1/3, a vary from top to bottom through, consequently, 1/3,1/5,1/8,1/12 (arbitrary units)

3.6. Results and discussion

The dependence of the scattering phase arg r on the energy is shown in Fig. 1. Jumps of the phase correspond to resonances which positions depend on the magnetic field. One can observe this dependence looking through the left column of pictures. Naturally, for a weaker magnetic field, one has resonances closer to the corresponding eigenvalues of the operator for the ring without the magnetic field. As for influence of parameter a, it is shown in the left column of pictures in Fig. 1. Parameter a is responsible for the connection between the ring and the segment. Note that the coupling condition between the ring and the segment (22) depends on the magnetic field $ (due to the "magnetic" derivatives the scattering phase changes). This explains the influence of a on the resonance position. As for p, which is responsible for the connection between the segment and the half-line, it does not influences on the resonance position essentially due to the absence of $ in the coupling condition. Correspondingly, we did not present the pictures for different values of p (their variations are not essential).

Acknowledgements

This work was partially financially supported by the Government of the Russian Federation (grant 08-08), by grant 16-11-10330 of Russian Science Foundation.

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Appendix A: Limit of the Weyl function MAl

Let us investigate Mal (A), A g R and calculate it's imaginary part. We put l = 2nr and calculate

u = -2 sin(lVX) sin(l$ + VA) - 4r cos(l VA) cos(VA + l$) + 2r cos(2l$ + VA) + 2r cos VA+ i ^2 sin(lVA) cos(l$ + VA) - 4r cos(lVA) sin(VA + l$) + 2r sin(2l$ + VA) + 2r sin VA)

and

v = -2 sin(lVA) sin(l$ - VA) + 4r cos(lVA) cos(VA - l$) - 2r cos(2l$ - VA) - 2r cos VA+ i (2sin(lVA) cos(l$ - VA) + 4r cos(lVA) sin(l$ - VA) - 2r sin(2l$ - VA) + 2r sin VA) .

Then,

u + v = -4sin(ZVA) sin(Z$) cos VA + 8r cos(ZVA) sin %/A sin(Z$) - 4r sin(2Z$) sin VA+ i ^4sin(ZVA) cos(Z$) cos VA - 8r cos(ZVA) sin %/A cos(Z$) + 4r cos(2Z$)sin VA + 4r sin Va).

If A < 0, then VA is purely complex, cos VA G R and sin VA are purely complex. Then for A > 0 the lines above give real and imaginary part of u + v, i.e.

K(u + v) =

-4 sin(ZVA) sin(Z$) cos VA + 8r cos(ZVA) sin VA sin(Z$) - 4r sin(2Z$) sin VA

S(u + v) =

4 sin(ZVA) cos(Z$) cos VA - 8r cos(ZVA) sin VA cos(Z$) + 4r cos(2Z$) sin VA + 4r sin VA and for A < 0

K(u + v) =

i (4 sin(ZVA) cos(Z$) cos VA - 8r cos(ZVA) sin VA cos(Z$) + 4r cos(2Z$) sin VA + 4r sin VA)

S(u + v) =

- ^-4sin(ZVA) sin(Z$) cos VA + 8r cos(ZVA) sin VAsin(Z$) - 4r sin(2Z$) sin VA In the same way we consider

u - v =

-4sin(ZVA) cos(Z$) sinVA - 8r cos(ZVA) cos VA cos(Z$) + 4r cos(2Z$) cos VA + 4r cos VA+ i ^4 sin(ZVA) sin(Z$) sin VA - 8r cos(ZVA) cos VA sin(Z$) + 4r sin(2Z$) cos VA^ . In this case for any A g R we have

K(u - v) =

-4 sin(ZVA) cos(Z$) sin VA - 8r cos(ZVA) cos VA cos(Z$) + 4r cos(2Z$) cos VA + 4r cos VA

S(u - v) =

4sin(ZVA) sin(Z$) sin VA - 8r cos(ZVA) cosVA sin(Z$) + 4r sin(2Z$) cos VA.

Then

Ma(a) = _¿vA(^(u - v) + iS(u - v)) (^(u + v) - iS(u + v))

( ) i K2(u + v) + S2(u + v) .

If A > 0, then

OJl^m- + vMu - v) + S(u + v)S(u - v)

sM (A) = -VA K2(u + v) + S2(u + v) ,

and if A < 0, then

c-il,fam_ /iTTS(u - vMu + v) - ^(u - v)S(u + v)

SM (A) = V|A| ^2(u + v) + S2(u + v) .