- Email: [email protected]
Matrix superpotential linear in variable parameter
Commun Nonlinear Sci Numer Simulat 17 (2012) 1522–1528
Contents lists available at SciVerse ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Matrix superpotential linear in variable parameter Yuri Karadzhov Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka Street, 01601 Kyiv-4, Ukraine
a r t i c l e
i n f o
Article history: Received 3 August 2011 Received in revised form 19 September 2011 Accepted 20 September 2011 Available online 28 September 2011
a b s t r a c t The paper presents the classification of matrix valued superpotentials corresponding to shape invariant systems of Schrödinger equations. All inequivalent irreducible matrix superpotentials realized by matrices of arbitrary dimension with linear dependence on variable parameter are presented explicitly. Ó 2011 Elsevier B.V. All rights reserved.
Keywords: Schrödinger equation Exact solutions Darboux transformation Shape-invariance SSQM
1. Introduction Supersymmetric quantum mechanic presents a powerful and elegant tool for obtaining explicit solutions of quantum mechanics problems described by Schrödinger equations [1]. Invented by Gendenstein [2], property of discrete reparametrization of potentials, known as shape-invariance, helps to determine whether eigenvalues of Hamiltonians can be calculated by algebraic methods. Though shape-invariant potentials do not exhaust the full class of potentials of solvable Schrödinger equations [13], it was interesting to find new integrable models. Previously several attempts of describing the class of shape-invariant potentials were made. In paper of Cooper et al. [3] a wide class of scalar shape-invariant potentials was described. An attractive example of a matrix problem which admits a shape invariant supersymmetric formulation was discovered by Pron’ko and Stroganov [4], who studied a motion of a neutral non-relativistic fermion which interacts anomalously with the magnetic field generated by a thin current carrying wire. The supersymmetric approach to the Pron’ko–Stroganov problem was applied in papers [5–7]. Particular cases of matrix potentials were discussed in papers [8–10,14]. Matrix superpotentials appear also in some supersymmetric systems related to the crystalline structures in Gross–Neveu model [17,18]. Two-dimensional matrix superpotentials, including shape invariant, were studied in papers [15,16]. In paper [11] Fukui considers a certain class of shape invariant potentials, which however was ad hoc restricted to 2 2 matrices with superpotentials linearly dependent on the variable parameter. Thus, in contrast to the class of scalar potentials, the class of known matrix potentials is presented by important but rather particular examples. The remaining part of the class is still undiscovered and requires further research. It seems to be interesting to widen the class of shape-invariant matrix potentials, because in such a way we will be able to describe new exactly-integrable systems of Schrödinger equations.
E-mail address: [email protected] 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.09.025
Y. Karadzhov / Commun Nonlinear Sci Numer Simulat 17 (2012) 1522–1528
1523
A systematic study of the problem was carried out in our recent paper [12] where we presented a complete description of irreducible matrix potentials, which include a term linear in variable parameter, and is proportional to the unit matrix. In the present paper the classification is continued and Fukui’s results are generalized. All shape-invariant matrix potentials of an arbitrary dimension with superpotentials linearly dependent on the variable parameter were found. 2. Shape-invariant potentials Let us start with a spectral problem
Hk w ¼ Ek w;
ð1Þ
where Hk is a Hamiltonian with a matrix potential, Ek and w are its eigenvalues and eigenfunctions correspondingly, moreover w is n-component spinor. In the Schrödinger equation Hamiltonian has the form
Hk ¼
@2 þ V k ðxÞ; @x2
ð2Þ
where Vk(x) is a n-dimensional matrix potential depending on variable x and parameter k. Suppose that Hamiltonian accepts factorization
Hk ¼ ayk ak ;
ð3Þ
then its superpartner is defined as follows
Hþk ¼ ak ayk :
ð4Þ
The most common representation of operators ak and ayk has the form
ak ¼ Ak ðxÞ
@ þ Bk ðxÞ; @x
ayk ¼
@ y A ðxÞ þ Byk ðxÞ; @x k
ð5Þ
where Ak(x), Bk(x) are matrices depending on x and Ayk ðxÞ; Byk ðxÞ are hermitian conjugate to them. Substituting this representation into Eq. (3) we obtain the equation
Hk ¼ Ayk Ak
0 @ 0 @2 þ Byk Ak Ayk Bk Ayk Ak þ Byk Bk Ayk Bk ; @x @x2
ð6Þ
which is supposed to be Schrödinger equation of the form (2). It leads to the following conditions
Ayk Ak ¼ I;
0 Byk Ak Ayk Bk Ayk Ak ¼ 0; 0 Byk Bk Ayk Bk ¼ V k :
ð7Þ
In terms of new variable W k ðxÞ ¼ Ayk ðxÞBk ðxÞ this condition take the form
W yk ¼ W k ;
ð8Þ
V k ¼ W 2k W 0k : The same result can be obtained analogously with simpler representation of operators ak and ayk
ak ¼
@ þ W k ðxÞ; @x
ayk ¼
@ þ W k ðxÞ: @x
ð9Þ
Wk(x) is called a matrix superpotential. As Wk(x) is hermitian, then the corresponding potential and its superpartner V þ k ðxÞ, i.e.
Vk ¼
@W k þ W 2k ; @x
V þk ¼
@W k þ W 2k @x
ð10Þ
are hermitian too. The goal is to find such superpotentials which generate shape-invariant Hamiltonians
Hþk ¼ HF k þ C k ;
ð11Þ
where Fk, Ck are scalar functions of k. In terms of superpotential, the last condition has the form
W 2k þ W 0k ¼ W 2F k W 0F k þ C k :
ð12Þ
It is sufficient to search for irreducible superpotentials, which means matrix Wk cannot be transformed to block-diagonal form with a constant unitary transformation.
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Y. Karadzhov / Commun Nonlinear Sci Numer Simulat 17 (2012) 1522–1528
Note that if Fk = k then Eq. (12) transforms into
W 0k ¼
1 Ck 2
ð13Þ
and it follows that
Wk ¼
1 C k x þ Xk ; 2
ð14Þ
where Xk is hermitian matrix depending on k which can be diagonalized. So in this case superpotential is completely reducible to the one-dimensional shifted oscillators. In case Fk – k it makes sense to restrict ourselves to unit shifts
F k ¼ k þ 1;
ð15Þ
see discussion section for details. In the following sections the classification of superpotentials linearly dependent on parameter k is presented. 3. The determining equations Let us consider the superpotential of the form
W k ¼ kQ þ P;
ð16Þ
where P and Q are n n hermitian matrices dependent on x. Suppose that Q is not proportional to the unit matrix because in this case the superpotential is reducible. Substituting (16) into (12) and taking into account Eq. (15) we obtain the equation
ð2k þ 1ÞðQ 0 Q 2 Þ fQ ; Pg þ 2P0 ¼ C k ; 0
ð17Þ
0
where Q ¼ @Q ; P ¼ @P and {Q, P} = QP + PQ is an anticommutator of matrices Q and P. After variable separation the last equa@x @x tion transforms into a system of determining equations
Q 0 ¼ Q 2 þ m; 1 P0 ¼ fQ ; Pg l; 2 C k ¼ ð2k þ 1Þm 2l;
ð18Þ ð19Þ ð20Þ
where l; m 2 R are arbitrary constants. For scalar values unit matrix I is omitted throughout the paper and written, for instance, as l instead of lI. 4. Solving the determining equations In this section a solution for determining system of arbitrary dimension n is presented. Let us start with Eq. (18). Introducing new variable
M ¼ Q u;
ð21Þ
where scalar function u is defined in the following way
8 > < k tanðkx þ cÞ; u ¼ k tanhðkx þ cÞ; > : 1 xþc ;
m ¼ k2 > 0 m ¼ k2 < 0 m¼0
ð22Þ
and c 2 R is chosen so that M is not a singular matrix, we can transform Eq. (18) into
M1 M 0 M1 ¼ I þ 2uM 1 :
ð23Þ
Note that
ðM 1 Þ0 ¼ M1 M 0 M1 : So if we set N = M1 then we obtain the following linear equation:
N0 ¼ I 2uN:
ð24Þ
Its general solution has the form
N ¼ qðxÞI þ hðxÞC;
ð25Þ
Y. Karadzhov / Commun Nonlinear Sci Numer Simulat 17 (2012) 1522–1528
1525
where q(x) and h(x) are scalar real valued functions and C is an arbitrary constant matrix,
81 m ¼ k2 > 0 > < 2k sinð2ðkx þ cÞÞ; q ¼ 2k1 sinhð2ðkx þ cÞÞ; m ¼ k2 < 0 > : x þ c; m¼0 8 2 2 ðkx þ c Þ; m ¼ k >0 cos > < 2 h ¼ cosh ðkx þ cÞ; m ¼ k2 < 0 > : m¼0 ðx þ cÞ2 ;
ð26Þ
Q is hermitian, so N is hermitian too. Moreover, if unitary matrix U is such that UNU is a diagonal matrix, then UQU is also diagonal. As q and h are scalar functions, then diagonalization transformation U should not depend on x. So Q can be diagonalized
Q ¼ diagfq1 ; . . . ; qn g
ð27Þ
and Eq. (18) transforms into system
q0i ¼ q2i þ m;
i ¼ 1 . . . n;
ð28Þ
which has following solutions
qi ¼ k tanðkx þ ci Þ;
i ¼ 1 . . . n;
m ¼ k2 > 0;
2
k tanhðkx þ ci Þ; i ¼ 1 . . . m 6 qi ¼ 4 k cothðkx þ ci Þ; i ¼ m þ 1 . . . l ;
qi ¼
xþ1c ; i ¼ 1 . . . m i
0;
ð29Þ
i ¼ l þ 1...n
k; "
m ¼ k2 < 0;
i ¼ m þ 1...n
;
m ¼ 0;
where ci 2 R; i ¼ 1 . . . n are integration constants. In cases m < 0 and m = 0 matrix Q consists of blocks of size m, l m + 1, n l + 1 and m, n m + 1 correspondingly. Some of them may or may not appear, i.e. have zero size. So we define matrix Q up to constant unitary transformation U. Now let us consider linear Eq. (19) for P = {pij} which can be solved element-wise: If m = k2 > 0
l
tanðkx þ ci Þ þ uii secðkx þ ci Þ; i ¼ 1 . . . n k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pij ¼ uij secðkx þ ci Þ secðkx þ cj Þ; i ¼ 1 . . . n; j ¼ 1 . . . n
pii ¼
ð30Þ
If m = k2 < 0
2
lk tanhðkx þ ci Þ þ uii sec hðkx þ ci Þ; i ¼ 1 . . . m 6 l pii ¼ 6 4 k cothðkx þ ci Þ þ uii cschðkx þ ci Þ; i ¼ m þ 1 . . . l i ¼ l þ 1...n lk þ uii expðkxÞ; 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u sec hðkx þ ci Þ sec hðkx þ cj Þ; i ¼ 1 . . . m; j ¼ 1 . . . m 6 ij 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 6 uij sec hðkx þ ci Þcschðkx þ cj Þ; i ¼ 1 . . . m; j ¼ m þ 1 . . . l 6 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 6 uij sechðkx þ ci Þ expðkxÞ; i ¼ 1 . . . m; j ¼ l þ 1 . . . n 6 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pij ¼ 6 6 uij cschðkx þ ci Þcschðkx þ cj Þ; i ¼ m þ 1 . . . l; j ¼ m þ 1 . . . l 6 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 6 uij cschðkx þ ci Þ expðkxÞ; i ¼ m þ 1 . . . l; j ¼ l þ 1 . . . n 6 6 6 uij expðkxÞ; ðHÞ i ¼ l þ 1 . . . n; j ¼ l þ 1 . . . n 4 uij ; ðÞ i ¼ l þ 1 . . . n; j ¼ l þ 1 . . . n Use formula (w) if corresponding diagonal entries qi and qj have the same sign and formula (⁄) otherwise.
ð31Þ
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Y. Karadzhov / Commun Nonlinear Sci Numer Simulat 17 (2012) 1522–1528
If m = 0
"u
lx ii 2 ðxþ2ci Þ
xþci
pii ¼ 2
;
i ¼ 1...m
lx þ uii ; i ¼ m þ 1 . . . n
uij pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; i ¼ 1 . . . m; j ¼ 1 . . . m 6 ðxþci Þðxþcj Þ 6 uij pij ¼ 6 pffiffiffiffiffiffiffi ; i ¼ 1 . . . m; j ¼ m þ 1 . . . n 4 xþci uij ; i ¼ m þ 1 . . . n; j ¼ m þ 1 . . . n
ð32Þ
where uji ¼ uij 2 C are integration constants. Numbers m and l in the above intervals correspond to the numbers defined in (29). Matrix P can be further simplified with unitary transformation, that reduces the number of non-zero entries in similar blocks. Obtained superpotentials are irreducible if there are enough non-zero entries uji. 5. Solving spectral problem Consider the Schrödinger equation
b k w ¼ ðHk þ ck Þw ¼ Ek w; H
ð33Þ
where ck vanishes with constant multiplied by unit matrix in the Hamiltonian Hk. It follows from shape-invariant condition (12) that
C k ¼ ckþ1 ck :
ð34Þ
Since all considered Hamiltonians are shape-invariant, Eq. (33) can be solved using the standard SSQM technique. An algorithm for constructing exact solutions of supersymmetric shape-invariant Schrödinger equations can be found in [3]. Ground state w0k ðxÞ is proportional to the square-integrable solution of the first order equation
ak w0k ðxÞ
@ þ W k w0k ðxÞ ¼ 0: @x
ð35Þ
Function w0k solves Eq. (33) with eigenvalues
E0k ¼ ck :
ð36Þ
Solution which corresponds to the nth excited state wnk ðxÞ can be represented as
wnk ðxÞ ¼ ayk aykþ1 aykþn1 w0kþn ðxÞ
ð37Þ
and must be a square-integrable function of x. The corresponding eigenvalue is
Enk ¼ ckþn :
ð38Þ
It follows from Eqs. (20), (34), (36), (38) that
Enk ¼ E0k þ 2nl ðn2 þ 2knÞm:
ð39Þ
At the end of the section, an example of exactly integrable problem is presented. Let Q and P, defined by formulas (29), (32), be
Q¼
1x 0 0
0
!
lx
P¼
;
2
1 2x puffiffix
puffiffix
lx
! ð40Þ
;
where l, u are real constants, l > 0. Note that, in contrast with (32), the constant l is changed to l to simplify notation. Then corresponding Schrödinger equation looks like
@2 2 @x
/ n
0 þ@
4k2 1 4x2
2 2
2
þ l 4x þ ux lk pffiffi 3ul x uk pffiffiffiffi 2 x3
uffiffiffi kffi p x3
pffiffi 1 x
3ul2
2
l2 x2 þ ux
A
/ n
¼ Ek
/ n
ð41Þ
0 and ck is equal to l. / The square integrable ground state solution w0 ¼ has the form n0
/0 ¼ C 1 /1 þ C 2 /2 ; n0 ¼ C 1 n1 þ C 2 n2
ð42Þ
Y. Karadzhov / Commun Nonlinear Sci Numer Simulat 17 (2012) 1522–1528
1527
for non integer k > 0, where
pffiffiffiffi lx2 4u2 l 1 /1 ¼ xkþ2 e 2 HB k; 0; 2 k; pffiffiffiffi ; x ; l 2 pffiffiffiffi p ffiffiffiffi pffiffiffiffi l kþ1 l2x2 0 4u 2 l l kþ2 l2x2 4u2 l x e HB k; 0; 2 k; pffiffiffiffi ; x e HB k; 0; 2 k; pffiffiffiffi ; x þ x n1 ¼ l 2 2u l 2 2u
ð43Þ
pffiffiffiffi l x2 4u2 l 1 /2 ¼ x2 e 2 HB k; 0; 2 k; pffiffiffiffi ; x ; l 2 pffiffiffiffi pffiffiffiffi pffiffiffiffi l l2x2 0 4u2 l 2k lx2 lx2 4u 2 l xe HB k; 0; 2 k; pffiffiffiffi ; e 2 HB k; 0; 2 k; pffiffiffiffi ; x þ x : n2 ¼ l 2 2u l 2 2u
ð44Þ
and
Here HB denotes the Heun Biconfluent function. For integer k solutions (43) and (44) are identical and second linearly independent solution of Eq. (35) should be used instead of (44). It is however cumbersome and is omitted in the paper. Note that constants C1 and C2 should be chosen so, that
Z
1
ðw0 Þy w0 dx ¼ 1:
ð45Þ
1
The spectrum of the problem is described by formula
En ¼ ð2n þ 1Þl:
ð46Þ
6. Discussion In this section the possibility of transforming unknown function Fk in the shape-invariance condition (11) to unit shift is discussed. During the computation it was assumed that Fk = k + 1, let us show it is a reasonable restriction. Under invertible transformation of variables
k ! aðkÞ
ð47Þ
the function Fk = F(k) changes by a similar transformation
FðkÞ ! aF a1 ðkÞ ¼ aðFða1 ðkÞÞÞ:
ð48Þ
Searching for such transformation, that would change function F(k) to unit shift, we get the equation
Fða1 ðkÞÞ ¼ a1 ðk þ 1Þ:
ð49Þ
The above equation is known as Abel functional equation. The results concerning the solution of this equation were obtained in papers [19–22]. Let X be R or Rþ . It is proved that (C1) if F : X ! R is an injective function such that for every compact set K X there exists p 2 N such that 8n; m 2 N0 ; jn mj P p :
F n ðKÞ \ F m ðKÞ ¼ ; then there exists a solution to the Abel functional equation. So if the above condition is fulfilled, Fk can be transformed into unit shift. 7. Conclusion Restricting the class of superpotentials to a simple form (16), it was possible to find new exactly solvable systems of Schrödinger equations. As corresponding Hamiltonians satisfy shape-invariant condition (12), the systems can be solved using a standard SSQM technique. An illustrative example of such a system was presented. Related superpotentials (29)–(32) are matrices of arbitrary dimension n that can be further simplified with unitary transformation. The Hamiltonian factorization and the shape-invariance condition itself were discussed. It was shown that an alternative representation (5) of operators ak and ayk is equivalent to a standard one (9). An unknown function Fk that appears in forminvariance condition significantly relies on the superpotential’s form. If it equals to identical function Fk = k, the corresponding superpotential is direct sum of shifted one-dimensional oscillators. In other cases, it is reasonable to consider Fk to be unit shift, as it was shown in the discussion section. In spite of this, the class of obtained potentials is strongly restricted by the superpotential’s form. In future works, a more general form of superpotential will be discussed.
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Y. Karadzhov / Commun Nonlinear Sci Numer Simulat 17 (2012) 1522–1528
Acknowledgment The author thanks to Prof. Anatoly Nikitin for useful discussions and valuable comments. References [1] Witten E. Nucl Phys B 1981;513:185; Witten E. Nucl Phys B 1982;253:202. [2] Gendenshtein L. JETP Lett 1983;38:356. [3] Cooper F, Khare A, Sukhatme U. Phys Rep 1995;251:267. [4] Pron’ko GP, Stroganov YG. Sov Phys JETP 1977;45:1075. [5] Voronin AI. Phys Rev A 1991;43:29. [6] Hau LV, Golovchenko GA, Burns MM. Phys Rev Lett 1995;74:3138. [7] Ferraro E, Messina N, Nikitin AG. Phys Rev A 2010;81:042108. [8] Andrianov AA, Ioffe MV. Phys Lett B 1991;255:543. [9] Andrianov AA, Ioffe MV, Spiridonov VP, Vinet L. Phys Lett B 1991;272:297. [10] Andrianov AA, Cannata F, Ioffe MV, Nishnianidze DN. J Phys A: Math Gen 1997;30:5037. [11] Fukui T. Phys Lett A 1993;178:1. [12] Nikitin AG, Karadzhov Yuri. J Phys A: Math Theor 2011;44:305204. 21. [13] Ui H. Prog Theor Phys 1984;72:192. [14] de Lima Rodrigues R, Bezerra VB, Vaidyac AN. Phys Lett A 2001;287:45. [15] Tkachuk VM, Roy P. Phys Lett A 1999;263:245–9. [16] Tkachuk VM, Roy P. J Phys A 2000;33:4159–67. [17] Correa F, Dunne GV, Plyushchay MS. Ann Phys 2009;324:2522–47. [18] Correa F, Jakubsky’ V, Nieto Luis-Miguel, Plyushchay MS. Phys Rev Lett 2008;101:030403. [19] Korkine A. Bull Sci Math 1882(2):228–42. [20] Belitskii G, Lyubich Yu. Stud Mat 1998;127:81–9. [21] Belitskii G, Lyubich Yu. Stud Mat 1999;134(2). [22] Laitochová J. Nonlinear Anal: Hybrid Syst 2007;1(1):95–102.
Contents lists available at SciVerse ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Matrix superpotential linear in variable parameter Yuri Karadzhov Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka Street, 01601 Kyiv-4, Ukraine
a r t i c l e
i n f o
Article history: Received 3 August 2011 Received in revised form 19 September 2011 Accepted 20 September 2011 Available online 28 September 2011
a b s t r a c t The paper presents the classification of matrix valued superpotentials corresponding to shape invariant systems of Schrödinger equations. All inequivalent irreducible matrix superpotentials realized by matrices of arbitrary dimension with linear dependence on variable parameter are presented explicitly. Ó 2011 Elsevier B.V. All rights reserved.
Keywords: Schrödinger equation Exact solutions Darboux transformation Shape-invariance SSQM
1. Introduction Supersymmetric quantum mechanic presents a powerful and elegant tool for obtaining explicit solutions of quantum mechanics problems described by Schrödinger equations [1]. Invented by Gendenstein [2], property of discrete reparametrization of potentials, known as shape-invariance, helps to determine whether eigenvalues of Hamiltonians can be calculated by algebraic methods. Though shape-invariant potentials do not exhaust the full class of potentials of solvable Schrödinger equations [13], it was interesting to find new integrable models. Previously several attempts of describing the class of shape-invariant potentials were made. In paper of Cooper et al. [3] a wide class of scalar shape-invariant potentials was described. An attractive example of a matrix problem which admits a shape invariant supersymmetric formulation was discovered by Pron’ko and Stroganov [4], who studied a motion of a neutral non-relativistic fermion which interacts anomalously with the magnetic field generated by a thin current carrying wire. The supersymmetric approach to the Pron’ko–Stroganov problem was applied in papers [5–7]. Particular cases of matrix potentials were discussed in papers [8–10,14]. Matrix superpotentials appear also in some supersymmetric systems related to the crystalline structures in Gross–Neveu model [17,18]. Two-dimensional matrix superpotentials, including shape invariant, were studied in papers [15,16]. In paper [11] Fukui considers a certain class of shape invariant potentials, which however was ad hoc restricted to 2 2 matrices with superpotentials linearly dependent on the variable parameter. Thus, in contrast to the class of scalar potentials, the class of known matrix potentials is presented by important but rather particular examples. The remaining part of the class is still undiscovered and requires further research. It seems to be interesting to widen the class of shape-invariant matrix potentials, because in such a way we will be able to describe new exactly-integrable systems of Schrödinger equations.
E-mail address: [email protected] 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.09.025
Y. Karadzhov / Commun Nonlinear Sci Numer Simulat 17 (2012) 1522–1528
1523
A systematic study of the problem was carried out in our recent paper [12] where we presented a complete description of irreducible matrix potentials, which include a term linear in variable parameter, and is proportional to the unit matrix. In the present paper the classification is continued and Fukui’s results are generalized. All shape-invariant matrix potentials of an arbitrary dimension with superpotentials linearly dependent on the variable parameter were found. 2. Shape-invariant potentials Let us start with a spectral problem
Hk w ¼ Ek w;
ð1Þ
where Hk is a Hamiltonian with a matrix potential, Ek and w are its eigenvalues and eigenfunctions correspondingly, moreover w is n-component spinor. In the Schrödinger equation Hamiltonian has the form
Hk ¼
@2 þ V k ðxÞ; @x2
ð2Þ
where Vk(x) is a n-dimensional matrix potential depending on variable x and parameter k. Suppose that Hamiltonian accepts factorization
Hk ¼ ayk ak ;
ð3Þ
then its superpartner is defined as follows
Hþk ¼ ak ayk :
ð4Þ
The most common representation of operators ak and ayk has the form
ak ¼ Ak ðxÞ
@ þ Bk ðxÞ; @x
ayk ¼
@ y A ðxÞ þ Byk ðxÞ; @x k
ð5Þ
where Ak(x), Bk(x) are matrices depending on x and Ayk ðxÞ; Byk ðxÞ are hermitian conjugate to them. Substituting this representation into Eq. (3) we obtain the equation
Hk ¼ Ayk Ak
0 @ 0 @2 þ Byk Ak Ayk Bk Ayk Ak þ Byk Bk Ayk Bk ; @x @x2
ð6Þ
which is supposed to be Schrödinger equation of the form (2). It leads to the following conditions
Ayk Ak ¼ I;
0 Byk Ak Ayk Bk Ayk Ak ¼ 0; 0 Byk Bk Ayk Bk ¼ V k :
ð7Þ
In terms of new variable W k ðxÞ ¼ Ayk ðxÞBk ðxÞ this condition take the form
W yk ¼ W k ;
ð8Þ
V k ¼ W 2k W 0k : The same result can be obtained analogously with simpler representation of operators ak and ayk
ak ¼
@ þ W k ðxÞ; @x
ayk ¼
@ þ W k ðxÞ: @x
ð9Þ
Wk(x) is called a matrix superpotential. As Wk(x) is hermitian, then the corresponding potential and its superpartner V þ k ðxÞ, i.e.
Vk ¼
@W k þ W 2k ; @x
V þk ¼
@W k þ W 2k @x
ð10Þ
are hermitian too. The goal is to find such superpotentials which generate shape-invariant Hamiltonians
Hþk ¼ HF k þ C k ;
ð11Þ
where Fk, Ck are scalar functions of k. In terms of superpotential, the last condition has the form
W 2k þ W 0k ¼ W 2F k W 0F k þ C k :
ð12Þ
It is sufficient to search for irreducible superpotentials, which means matrix Wk cannot be transformed to block-diagonal form with a constant unitary transformation.
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Y. Karadzhov / Commun Nonlinear Sci Numer Simulat 17 (2012) 1522–1528
Note that if Fk = k then Eq. (12) transforms into
W 0k ¼
1 Ck 2
ð13Þ
and it follows that
Wk ¼
1 C k x þ Xk ; 2
ð14Þ
where Xk is hermitian matrix depending on k which can be diagonalized. So in this case superpotential is completely reducible to the one-dimensional shifted oscillators. In case Fk – k it makes sense to restrict ourselves to unit shifts
F k ¼ k þ 1;
ð15Þ
see discussion section for details. In the following sections the classification of superpotentials linearly dependent on parameter k is presented. 3. The determining equations Let us consider the superpotential of the form
W k ¼ kQ þ P;
ð16Þ
where P and Q are n n hermitian matrices dependent on x. Suppose that Q is not proportional to the unit matrix because in this case the superpotential is reducible. Substituting (16) into (12) and taking into account Eq. (15) we obtain the equation
ð2k þ 1ÞðQ 0 Q 2 Þ fQ ; Pg þ 2P0 ¼ C k ; 0
ð17Þ
0
where Q ¼ @Q ; P ¼ @P and {Q, P} = QP + PQ is an anticommutator of matrices Q and P. After variable separation the last equa@x @x tion transforms into a system of determining equations
Q 0 ¼ Q 2 þ m; 1 P0 ¼ fQ ; Pg l; 2 C k ¼ ð2k þ 1Þm 2l;
ð18Þ ð19Þ ð20Þ
where l; m 2 R are arbitrary constants. For scalar values unit matrix I is omitted throughout the paper and written, for instance, as l instead of lI. 4. Solving the determining equations In this section a solution for determining system of arbitrary dimension n is presented. Let us start with Eq. (18). Introducing new variable
M ¼ Q u;
ð21Þ
where scalar function u is defined in the following way
8 > < k tanðkx þ cÞ; u ¼ k tanhðkx þ cÞ; > : 1 xþc ;
m ¼ k2 > 0 m ¼ k2 < 0 m¼0
ð22Þ
and c 2 R is chosen so that M is not a singular matrix, we can transform Eq. (18) into
M1 M 0 M1 ¼ I þ 2uM 1 :
ð23Þ
Note that
ðM 1 Þ0 ¼ M1 M 0 M1 : So if we set N = M1 then we obtain the following linear equation:
N0 ¼ I 2uN:
ð24Þ
Its general solution has the form
N ¼ qðxÞI þ hðxÞC;
ð25Þ
Y. Karadzhov / Commun Nonlinear Sci Numer Simulat 17 (2012) 1522–1528
1525
where q(x) and h(x) are scalar real valued functions and C is an arbitrary constant matrix,
81 m ¼ k2 > 0 > < 2k sinð2ðkx þ cÞÞ; q ¼ 2k1 sinhð2ðkx þ cÞÞ; m ¼ k2 < 0 > : x þ c; m¼0 8 2 2 ðkx þ c Þ; m ¼ k >0 cos > < 2 h ¼ cosh ðkx þ cÞ; m ¼ k2 < 0 > : m¼0 ðx þ cÞ2 ;
ð26Þ
Q is hermitian, so N is hermitian too. Moreover, if unitary matrix U is such that UNU is a diagonal matrix, then UQU is also diagonal. As q and h are scalar functions, then diagonalization transformation U should not depend on x. So Q can be diagonalized
Q ¼ diagfq1 ; . . . ; qn g
ð27Þ
and Eq. (18) transforms into system
q0i ¼ q2i þ m;
i ¼ 1 . . . n;
ð28Þ
which has following solutions
qi ¼ k tanðkx þ ci Þ;
i ¼ 1 . . . n;
m ¼ k2 > 0;
2
k tanhðkx þ ci Þ; i ¼ 1 . . . m 6 qi ¼ 4 k cothðkx þ ci Þ; i ¼ m þ 1 . . . l ;
qi ¼
xþ1c ; i ¼ 1 . . . m i
0;
ð29Þ
i ¼ l þ 1...n
k; "
m ¼ k2 < 0;
i ¼ m þ 1...n
;
m ¼ 0;
where ci 2 R; i ¼ 1 . . . n are integration constants. In cases m < 0 and m = 0 matrix Q consists of blocks of size m, l m + 1, n l + 1 and m, n m + 1 correspondingly. Some of them may or may not appear, i.e. have zero size. So we define matrix Q up to constant unitary transformation U. Now let us consider linear Eq. (19) for P = {pij} which can be solved element-wise: If m = k2 > 0
l
tanðkx þ ci Þ þ uii secðkx þ ci Þ; i ¼ 1 . . . n k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pij ¼ uij secðkx þ ci Þ secðkx þ cj Þ; i ¼ 1 . . . n; j ¼ 1 . . . n
pii ¼
ð30Þ
If m = k2 < 0
2
lk tanhðkx þ ci Þ þ uii sec hðkx þ ci Þ; i ¼ 1 . . . m 6 l pii ¼ 6 4 k cothðkx þ ci Þ þ uii cschðkx þ ci Þ; i ¼ m þ 1 . . . l i ¼ l þ 1...n lk þ uii expðkxÞ; 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u sec hðkx þ ci Þ sec hðkx þ cj Þ; i ¼ 1 . . . m; j ¼ 1 . . . m 6 ij 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 6 uij sec hðkx þ ci Þcschðkx þ cj Þ; i ¼ 1 . . . m; j ¼ m þ 1 . . . l 6 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 6 uij sechðkx þ ci Þ expðkxÞ; i ¼ 1 . . . m; j ¼ l þ 1 . . . n 6 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pij ¼ 6 6 uij cschðkx þ ci Þcschðkx þ cj Þ; i ¼ m þ 1 . . . l; j ¼ m þ 1 . . . l 6 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 6 uij cschðkx þ ci Þ expðkxÞ; i ¼ m þ 1 . . . l; j ¼ l þ 1 . . . n 6 6 6 uij expðkxÞ; ðHÞ i ¼ l þ 1 . . . n; j ¼ l þ 1 . . . n 4 uij ; ðÞ i ¼ l þ 1 . . . n; j ¼ l þ 1 . . . n Use formula (w) if corresponding diagonal entries qi and qj have the same sign and formula (⁄) otherwise.
ð31Þ
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Y. Karadzhov / Commun Nonlinear Sci Numer Simulat 17 (2012) 1522–1528
If m = 0
"u
lx ii 2 ðxþ2ci Þ
xþci
pii ¼ 2
;
i ¼ 1...m
lx þ uii ; i ¼ m þ 1 . . . n
uij pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; i ¼ 1 . . . m; j ¼ 1 . . . m 6 ðxþci Þðxþcj Þ 6 uij pij ¼ 6 pffiffiffiffiffiffiffi ; i ¼ 1 . . . m; j ¼ m þ 1 . . . n 4 xþci uij ; i ¼ m þ 1 . . . n; j ¼ m þ 1 . . . n
ð32Þ
where uji ¼ uij 2 C are integration constants. Numbers m and l in the above intervals correspond to the numbers defined in (29). Matrix P can be further simplified with unitary transformation, that reduces the number of non-zero entries in similar blocks. Obtained superpotentials are irreducible if there are enough non-zero entries uji. 5. Solving spectral problem Consider the Schrödinger equation
b k w ¼ ðHk þ ck Þw ¼ Ek w; H
ð33Þ
where ck vanishes with constant multiplied by unit matrix in the Hamiltonian Hk. It follows from shape-invariant condition (12) that
C k ¼ ckþ1 ck :
ð34Þ
Since all considered Hamiltonians are shape-invariant, Eq. (33) can be solved using the standard SSQM technique. An algorithm for constructing exact solutions of supersymmetric shape-invariant Schrödinger equations can be found in [3]. Ground state w0k ðxÞ is proportional to the square-integrable solution of the first order equation
ak w0k ðxÞ
@ þ W k w0k ðxÞ ¼ 0: @x
ð35Þ
Function w0k solves Eq. (33) with eigenvalues
E0k ¼ ck :
ð36Þ
Solution which corresponds to the nth excited state wnk ðxÞ can be represented as
wnk ðxÞ ¼ ayk aykþ1 aykþn1 w0kþn ðxÞ
ð37Þ
and must be a square-integrable function of x. The corresponding eigenvalue is
Enk ¼ ckþn :
ð38Þ
It follows from Eqs. (20), (34), (36), (38) that
Enk ¼ E0k þ 2nl ðn2 þ 2knÞm:
ð39Þ
At the end of the section, an example of exactly integrable problem is presented. Let Q and P, defined by formulas (29), (32), be
Q¼
1x 0 0
0
!
lx
P¼
;
2
1 2x puffiffix
puffiffix
lx
! ð40Þ
;
where l, u are real constants, l > 0. Note that, in contrast with (32), the constant l is changed to l to simplify notation. Then corresponding Schrödinger equation looks like
@2 2 @x
/ n
0 þ@
4k2 1 4x2
2 2
2
þ l 4x þ ux lk pffiffi 3ul x uk pffiffiffiffi 2 x3
uffiffiffi kffi p x3
pffiffi 1 x
3ul2
2
l2 x2 þ ux
A
/ n
¼ Ek
/ n
ð41Þ
0 and ck is equal to l. / The square integrable ground state solution w0 ¼ has the form n0
/0 ¼ C 1 /1 þ C 2 /2 ; n0 ¼ C 1 n1 þ C 2 n2
ð42Þ
Y. Karadzhov / Commun Nonlinear Sci Numer Simulat 17 (2012) 1522–1528
1527
for non integer k > 0, where
pffiffiffiffi lx2 4u2 l 1 /1 ¼ xkþ2 e 2 HB k; 0; 2 k; pffiffiffiffi ; x ; l 2 pffiffiffiffi p ffiffiffiffi pffiffiffiffi l kþ1 l2x2 0 4u 2 l l kþ2 l2x2 4u2 l x e HB k; 0; 2 k; pffiffiffiffi ; x e HB k; 0; 2 k; pffiffiffiffi ; x þ x n1 ¼ l 2 2u l 2 2u
ð43Þ
pffiffiffiffi l x2 4u2 l 1 /2 ¼ x2 e 2 HB k; 0; 2 k; pffiffiffiffi ; x ; l 2 pffiffiffiffi pffiffiffiffi pffiffiffiffi l l2x2 0 4u2 l 2k lx2 lx2 4u 2 l xe HB k; 0; 2 k; pffiffiffiffi ; e 2 HB k; 0; 2 k; pffiffiffiffi ; x þ x : n2 ¼ l 2 2u l 2 2u
ð44Þ
and
Here HB denotes the Heun Biconfluent function. For integer k solutions (43) and (44) are identical and second linearly independent solution of Eq. (35) should be used instead of (44). It is however cumbersome and is omitted in the paper. Note that constants C1 and C2 should be chosen so, that
Z
1
ðw0 Þy w0 dx ¼ 1:
ð45Þ
1
The spectrum of the problem is described by formula
En ¼ ð2n þ 1Þl:
ð46Þ
6. Discussion In this section the possibility of transforming unknown function Fk in the shape-invariance condition (11) to unit shift is discussed. During the computation it was assumed that Fk = k + 1, let us show it is a reasonable restriction. Under invertible transformation of variables
k ! aðkÞ
ð47Þ
the function Fk = F(k) changes by a similar transformation
FðkÞ ! aF a1 ðkÞ ¼ aðFða1 ðkÞÞÞ:
ð48Þ
Searching for such transformation, that would change function F(k) to unit shift, we get the equation
Fða1 ðkÞÞ ¼ a1 ðk þ 1Þ:
ð49Þ
The above equation is known as Abel functional equation. The results concerning the solution of this equation were obtained in papers [19–22]. Let X be R or Rþ . It is proved that (C1) if F : X ! R is an injective function such that for every compact set K X there exists p 2 N such that 8n; m 2 N0 ; jn mj P p :
F n ðKÞ \ F m ðKÞ ¼ ; then there exists a solution to the Abel functional equation. So if the above condition is fulfilled, Fk can be transformed into unit shift. 7. Conclusion Restricting the class of superpotentials to a simple form (16), it was possible to find new exactly solvable systems of Schrödinger equations. As corresponding Hamiltonians satisfy shape-invariant condition (12), the systems can be solved using a standard SSQM technique. An illustrative example of such a system was presented. Related superpotentials (29)–(32) are matrices of arbitrary dimension n that can be further simplified with unitary transformation. The Hamiltonian factorization and the shape-invariance condition itself were discussed. It was shown that an alternative representation (5) of operators ak and ayk is equivalent to a standard one (9). An unknown function Fk that appears in forminvariance condition significantly relies on the superpotential’s form. If it equals to identical function Fk = k, the corresponding superpotential is direct sum of shifted one-dimensional oscillators. In other cases, it is reasonable to consider Fk to be unit shift, as it was shown in the discussion section. In spite of this, the class of obtained potentials is strongly restricted by the superpotential’s form. In future works, a more general form of superpotential will be discussed.
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Acknowledgment The author thanks to Prof. Anatoly Nikitin for useful discussions and valuable comments. References [1] Witten E. Nucl Phys B 1981;513:185; Witten E. Nucl Phys B 1982;253:202. [2] Gendenshtein L. JETP Lett 1983;38:356. [3] Cooper F, Khare A, Sukhatme U. Phys Rep 1995;251:267. [4] Pron’ko GP, Stroganov YG. Sov Phys JETP 1977;45:1075. [5] Voronin AI. Phys Rev A 1991;43:29. [6] Hau LV, Golovchenko GA, Burns MM. Phys Rev Lett 1995;74:3138. [7] Ferraro E, Messina N, Nikitin AG. Phys Rev A 2010;81:042108. [8] Andrianov AA, Ioffe MV. Phys Lett B 1991;255:543. [9] Andrianov AA, Ioffe MV, Spiridonov VP, Vinet L. Phys Lett B 1991;272:297. [10] Andrianov AA, Cannata F, Ioffe MV, Nishnianidze DN. J Phys A: Math Gen 1997;30:5037. [11] Fukui T. Phys Lett A 1993;178:1. [12] Nikitin AG, Karadzhov Yuri. J Phys A: Math Theor 2011;44:305204. 21. [13] Ui H. Prog Theor Phys 1984;72:192. [14] de Lima Rodrigues R, Bezerra VB, Vaidyac AN. Phys Lett A 2001;287:45. [15] Tkachuk VM, Roy P. Phys Lett A 1999;263:245–9. [16] Tkachuk VM, Roy P. J Phys A 2000;33:4159–67. [17] Correa F, Dunne GV, Plyushchay MS. Ann Phys 2009;324:2522–47. [18] Correa F, Jakubsky’ V, Nieto Luis-Miguel, Plyushchay MS. Phys Rev Lett 2008;101:030403. [19] Korkine A. Bull Sci Math 1882(2):228–42. [20] Belitskii G, Lyubich Yu. Stud Mat 1998;127:81–9. [21] Belitskii G, Lyubich Yu. Stud Mat 1999;134(2). [22] Laitochová J. Nonlinear Anal: Hybrid Syst 2007;1(1):95–102.