Interwining of Exactly Solvable Generalized Schr ö dinger Equations E. Velicheva, A. Suzko JINR Gomel, Belarus The XIII-th International School-Conference

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


Светлой памяти ученого, учителя и коллеги Сузько Алины Алексеевны посвящается…


Content

1. Introduction2. The first-order Darboux transformations for the

generalized Schrödinger equations3. Chain of Darboux transformations4. Application


)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


)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


Conventional intertwining technique for Schrödinger equation

)x(vH xx

)x(v~H~

xx

)4(LH~

LH

L ~


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


,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


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


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~


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


,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


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


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


m

q

dx

d

q

1

dx

d

m

q

qm

K

dx

d

m

q2vv 2

12


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



1

,11 U

W

qm

1

)8(1

2,1

,2,1122 W

W

qmL

,LLL 122


2,1

3,2,13122 W

W

qm

1ULL

,K2

23

Next step

33 K

dx

d

qm

1L


)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


)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


,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


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


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


)16(ln

cosh ,12,1

,11

12

Wdx

WdW

dx

d

xC

x

)17(sinsinhcoscosh 1111

1,1 kxxkxxk

xk

CCW


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


21 41 ,41

,162 ,162

,41 ,253 321

1v

2vat energies of transformation

Potentials :at the energy of transformation

3v at energies of transformation


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


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


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


),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


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


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.


Thank you for attention


Documents

Interwining of Exactly Solvable Generalized Schr ö dinger Equations E. Velicheva, A. Suzko JINR Gomel, Belarus The XIII-th International School-Conference