interwining of exactly solvable generalized schr ö dinger equations e. velicheva, a. suzko jinr...
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Interwining of Exactly Solvable Generalized
Schrödinger Equations
E. Velicheva, A. SuzkoJINR
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
Светлой памяти ученого, учителя и коллеги Сузько Алины Алексеевны посвящается…
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
Content
1. Introduction2. The first-order Darboux transformations for the
generalized Schrödinger equations3. Chain of Darboux transformations4. Application
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
)x()x()),x(vH( 0
)x()x(x f12
1
dx
d
2
1nn
22
2
)x(q)x(v),x(v
)x()x()x(v)x(dx
d
)x(m
12
2
Generalized Schrödinger Equations:1) with position-dependent effective mass
2) with energy-dependent potentials
linearly energy-dependent potentials
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
)1)(x()x(q)x()x(v)x(dx
d
)x(m
1
dx
d
)2()x(q
)x(v
dx
d
)x(m
1
dx
d
q(x)
1-H ,H
)3()x(q
)x(v~
dx
d
)x(m
1
dx
d
q(x)
1-H
~ ,
~~H~
The first-order Darboux transformations for the generalized Schrödinger equations
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
Conventional intertwining technique for Schrödinger equation
)x(vH xx
)x(v~H~
xx
)4(LH~
LH
L ~
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
Intertwining relations and intertwining operator for generalized Schrödinger equation
dx
d)x(B)x(AL
)x(m)x(q)x(B
B
AK ,0K
m
1
q
1
dx
d
q
v
dx
dKK
qm
1
dx
d 2
K
m
1
q
1
dx
d
q
v
dx
dKK
qm
1 2Riccati equation:
qm
KA
in the terms of K, q, m, v and take
to account
m
q
dx
d
q
1
dx
d
m
q
qm
K
dx
d
m
q2vv~
LH~
LH
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
,U
UK
U
U
dx
d
qm
1K
dx
d
qm
1L
U is the function of the transformation and solutionof Riccati equation and also auxiliary equation of Schrödinger equation
m
q
dx
d
q
1
dx
d
m
q
U
U
qm
1
dx
d
m
q2vv~
U
U
dx
d
qm
1L
~
Intertwining relations and intertwining operator the generalized Schrödinger equation
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
Solutions at the energy of transformation
0LU
U
1
q
mUL Solution at the energy of transformation:
is the energy of transformation and
and are linearly independent solutions of Riccati equation U U
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
Chain of Darboux transformations...LLL 12,LL1
,Kdx
d
qm
1L 22
,K
1
12
,U
UK
1
11
,K
dx
d
qm
1L 11
,KK1 ,UU1
21211 UKdx
d
qm
1UL
2U
1U is solution of the equation (2) at energy 1
is solution of the equation (2) at energy 2
1 is solution of the equation (3)
)2()x(q
)x(v
dx
d
)x(m
1
dx
d
q(x)
1-H ,H
)3()x(q
)x(v~
dx
d
)x(m
1
dx
d
q(x)
1-H
~ ,
~~H~
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
Chain of Darboux transformations
)5()x(q
)x(v
dx
d
)x(m
1
dx
d
q(x)
1-H ,H 2
2222
,Kdx
d
qm
1L 12122
m
q
dx
d
q
1
dx
d
m
q
qm
K
dx
d
m
q2vv 2
12
,L 11 ~
1
can be taken as a new transformation function for to generate a new potential and solution for the equation
12v 2
H~
Thus, the action of the 2-th-order operator on solution of the equation (2) leads to solution of the equation (5)
2
122 LLL
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
,L...LLL 11nn
)5( LHLH n
,Kdx
d
qm
1L nn
m
q
dx
d
q
1
dx
d
m
qn
qm
K
dx
d
m
q2vv n
1nn
1nnn L
1n
1nnK
Chain of n-order of first-order Darboux transformations
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
It can be shown that the result action of a chain of first-orderDarboux transformations is equivalent to every nth-order Darboux transformation and can be expressed in terms of solutions of an initial equation, with no use of the solutions to intermediate equations.
11nn1nnn L...LLL
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
21212,1 UUUUW
21 KKK
)6(m
q
dx
d
q
1
dx
d
m
q
qm
K
dx
d
m
q2vv 2
12
The second-order Darboux transformations for generalized Schrödinger equations
1
1
U
UK
,U
W
qm
1ln
dx
dK
1
2,1
1
12
1
11 U
UK
qm
Wln
dx
dK 2,1
)7(m
q
dx
d
q
1
dx
d
m
q2
qm
W
dx
d
W
1
dx
d
m
q2vv 2,1
2,12
211
2,11 UK
dx
d
qm
1
U
W
qm
1
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
m
q
dx
d
q
1
dx
d
m
q
qm
K
dx
d
m
q2vv 2
12
The second-order Darboux transformations for generalized Schrödinger equations
m
q
dx
d
q
1
dx
d
m
q2
qm
W
dx
d
W
1
dx
d
m
q2vv 2,1
2,12
1
2,11 U
W
qm
1
m
q
dx
d
q
1
dx
d
m
q
U
U
qm
1
dx
d
m
q2vv1
m
q
dx
d
q
1
dx
d
m
q2
qm
W
dx
d
W
1
dx
d
m
q2
U
U
qm
1
dx
d
m
q2
U
U
qm
1
dx
d
m
q2-v
m
q
dx
d
q
1
dx
d
m
qln
dx
d
qm
1
dx
d
m
q2
U
U
qm
1
dx
d
m
q2v
m
q
dx
d
q
1
dx
d
m
q
qm
K
dx
d
m
q2vv
2,1
2,11
1
1
1
11
1
212
,K1
12
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
The second-order Darboux transformations for generalized Schrödinger equations
1
,11 U
W
qm
1
)8(1
2,1
,2,1122 W
W
qmL
,LLL 122
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
2,1
3,2,13122 W
W
qm
1ULL
,K2
23
Next step
33 K
dx
d
qm
1L
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
)9(11
2 ..2,1
..2,1
m
q
dx
d
qdx
d
m
qn
qm
W
dx
d
Wdx
d
m
qvv n
nn
It is shown that the n-order Darboux transformations are equivalent to the resulting action of a chain of first-order Darboux transformations.
The transformation potential in the case of n-order Darboux transformation over the initial
potential
Thus, we can express the transformed potentials in terms of the initial potentials v, q, the effective mass m and the family of auxiliary solutions , (j=1,2,,...n ) of the initial equation.
nv
jU
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
)10)(()()()()()(
1xxqxxvx
dx
d
xmdx
d
ApplicationThe 1-st example
)11()cos()sin(
)( 11
xk
kxC
xk
kxCx
,x/1m ,xq )x4/(1)x(v
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
,x
)xcosh(C
1
11
xtanhx2
1K~
111
)12(cosh
2
4
1tanh12
4
1)(
12
21
1211 x
x
xxx
xxv
)13(tanh2
1~)( 1111
x
xdx
dK
dx
dx
,2 xcoshxC
1
q
mU
1
1
1
The 1-st example
The solution at the transformation energy
,211
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
The simple example of exactly solvable problem for the generalized equation with a real potential which is singular at zero. The potentials have one bound state at different energies.
1v
The 1-st example
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
The 1-st example: potentials and solutions of the with two bound states
,x
)xcosh(C
1
11
,x
)xsinh(C
2
22
,K~
K~
K~
21 ,K~
1
11
,K~
1
12
,K
~
qm
12121
,qm
Wln
dx
dK~ 12
211
222
m
q
dx
d
q
1
dx
d
m
q2
qm
K~
dx
d
m
q2vv2
)14(ln24
92,12
2
2 Wdx
d
xv
)15(sinhsinhcoshcosh, 21121221
2
212,1 xxxxx
CWW
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
)16(ln
cosh ,12,1
,11
12
Wdx
WdW
dx
d
xC
x
)17(sinsinhcoscosh 1111
1,1 kxxkxxk
xk
CCW
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
Potentials for the generalized Schrödinger equation having three bound states
)18(ln24
133,2,12
2
3 Wdx
d
xv
)19(
sinhcoshcosh -
coshsinhcosh
sinhsinhsinh
sinhsinhsinh
coshsinhcosh
sinhcoshcosh
321212
321213
321221
321232
321223
321232
2/3321
3
3,2,1
xxx
xxx
xxx
xxx
xxx
xxx
x
CW
,x
)xcosh(C
1
11
,x
)xsinh(C
2
22
x
)xcosh(C
3
33
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
21 41 ,41
,162 ,162
,41 ,253 321
1v
2vat energies of transformation
Potentials :at the energy of transformation
3v at energies of transformation
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
The 2-nd example
)20)(()()()()()(
1xxqxxvx
dx
d
xmdx
d
,x/m 22 ,1q 0)x(v
)x()x(dx
d
)x(m
1
dx
d
),21(lncoslnsin)( 21 xcxcx
x 2
22
4
41
k
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
Potentials with two bound states
1,2i ,4
412
2i
2
i
),xlncosh(
x 11
),xlnsinh(x 22
m
1
dx
d
m
2
m
W
dx
d
W
1
dx
d
m
2vv
2
22,1
2,12
)22(ln2
2,12
Wx
dx
dxv
xxxxx
W lnsinhlnsinhlncoshlncosh 1212122
2
2,1
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
Asymmetric double-well potentials for generalized Schrödinger equations with
effective mass21
,0.21 1->
2-> ,0.21 0.62 75.32
,0.51 0.62 0.62 ,5.51
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
),xlncosh(x 33
),xlnsinh(x 22
),xlncosh(x 11
1,2,3i ,4
412
2i
2
i
)23(ln2
3,2,13
Wx
dx
dxv
Third-order Darboux transformation. Potentials with two three states
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
Asymmetric triple-well potentials for generalized Schrödinger equations with effective mass
,0.11
,0.11
,75.32
,75.32
,0.21 ,75.42 ,75.32 ,0.21 1->
2->
321
25.63
0.53
0.63 0.53
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
ConclusionI. By application of the intertwining operator technique to generalizedSchrödinger equation with position dependent mass and with energydependent potentials, Darboux transformations of an arbitrary orderhave been constructed. II. It has been shown that nth-order Darboux transformation is equivalent
to the resulting action of a chain of first-order Darboux transformations. III. On concrete examples it has been shown how to apply the Darbouxtransformation technique for modeling quantum well potentials with the
given spectrum for investigation of low-dimensional structures in nanoelectronics.
IV. Hamiltonians with different number of levels havebeen constructed and the influence of the distance between levels onthe shape of constructed potentials has been investigated, inparticular, asymmetric double well and triple well potentials havebeen created.
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
Thank you for attention
Gomel, Belarus The XIII-th International School-Conference “The Actual Problems of Microworld Physics 27.07-7.08.2015
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