Expanding the class of conditionally exactly solvable potentials

PHYSICAL REVIEW A VOLUME 50, NUMBER 5 NOVEMBER 1994

Expanding the

class

of

conditionally

exactly

solvable

potentials

A.

de Souxa Dutra

Uniuersidade Estadual Paulista Ca-mpus de Guaratingueta-DFQ, Auenida Dr.Ariberto Pereira da Cunha, 333, Codigo de Enderegamento Postal 12500-000, Guaratingueta,

Sio

Paulo, Brazil

Frank Uwe Girlich

Fachbereich Physik, Universitat Leipzig, Augustusplatz 10,04109Leipzig, Germany

(Received 30December 1993)

Recently a class ofquantum-mechanical potentials was presented that is characterized by the fact that they are exactly solvable only when some oftheir parameters are fixed to a convenient value, so they

were christened as conditionally exactly solvable potentials. Here we intend toexpand this class by

in-troducing examples in two dimensions. As a byproduct ofour search, we found also another exactly

solvable potential.

PACSnumber{s): 03.65.Ca,03.65.Ge, 02.90.

+

p

Recently the existence

of

a whole class

of

exactly solv-able potentials in quantum mechanics was discovered

[1].

Furthermore it was observed that these potentials obey a

supersymmetric algebra

[2].

In fact the potentials dis-cussed in

Ref. [1]

were also discussed independently in a previous work

of

Stillinger

[3].

In that article the author did not perceive that such potentials were only the first representatives

of

an entire new class

of

potentials. What distinguishes these potentials from the usual exactly solv-able potentials isthe fact that at least one

of

their param-eters must be fixed

to

a specific value, because

if

you do

not do that the potentials no longer have exact solutions. On the other hand, they cannot be classified among the so called quasi exactly solvable potentials [4

—10],

because these have only afinite number

of

exactly solvable energy levels, and this happens only when the potential parame-ters obey some constraint equations, sothat only some

of

the parameters are free, and the others are fixed in terms

of

them. The continuous interest in obtaining and

classi-fying quantum-mechanical potentials is because

of

their importance as a basis for expanding more realistic cases, and also for use

of

their solutions to test numerical calcu-lations. Another possibility is that

of

trying to use them as a kind

of

theoretical laboratory, searching for new in-teresting physical features. Some

of

the examples that we will study below have, in fact, the interesting characteris-tic

of

representing a kind

of

semi-infinite charged string, because, ascan beseen in

Fig.

1,the potential isinfinitely

attractive along the

x

semiaxis. Here in this work we in-tend

to

expand the class

of

conditionally exactly solvable

(CES) potentials. This is going to be done through the presentation

of

two examples in two dimensions. As a byproduct

of

our search for

CES

potentials, we found another exactly solvable potential, whose solution will

also be presented.

The potentials that will be considered here are

and

Vn(x, y)

=

X A

arctan—

2/3

8

[arctan(y/x) ]

+

go

+C

[arctan(y

/x)

] (lb)

20

V p

where the function

arctan(y/x)

isdefined in the principal

branch.

It

can be observed that the first case above has an exact solution forany value

of

Go

if

8

=0.

Otherwise

the case

8%0

has an exact solution for

Go=

—

3irt _/32@,

and only for a restricted region

of

the space (x and y both bigger orboth less than zero).

The second potential has exact solutions only when

go=

—

5trt _/32@. In this case the solution is valid along all the x-y plane.

It

isnot difBcult toverify that these po-tentials are singular along the positive semiaxis

x,

as can

be seen from

Fig. 1.

Sothey represent akind

of

"charged

string.

"

V,

(x,y)

=

1

x

+y

B

arctan(y

/x

) [arctan(y

/x

}]'

2

Go

+

+C

[arctan(y

/x)

]

+D(x +y

)

FIG.

1. Plotofthe potential CES

II

with unitary parameters.

4370

BRIEF

REPORTS

d

₊

V,(u)

—

6'

'4(u)=0

t

2p du

where

(C+K —

iri /8p,)

with

K

being the separation constant and

_p

the mass

of

the particle. The subscript e in the potential stands for

its exact solvability. In the variable vwe have d

&

+

VCEs(u)

Eg(v—

)

=0,

2p dU

with VCEs(u) having two possibilities:

8

Go

VcEs(u)

=

+

+

U

(4a)

The solution for these potentials can be obtained through a suitable change

of

coordinates:

x

=u

cos(u),

y

=u

sin(u) .

After this change

of

variables, and their separation, the Schrodinger equation can be split into two others. In the variable u we have that

like multivaluedness

of

the wave function. However, one

can see that this transformation is typical

of

a stereo-graphic mapping, in this case

of R

onto

5

. This kind

of

compactification can be done provided that one has suit-able asymptotic conditions

[11].

In fact, this implies that one will have a finite wave function throughout variation

of

the variables, and particularly that at infinity it should vanish. This imposes the condition that the parameter D

be nonzero, because along the positive

x

semiaxis the

wave function will be singular at infinity

if

D

=0.

Fur-thermore, there is no periodicity

of

the wave function in

the variables

x

and y, aswe will see later.

So we see that the original potentials were, in some

cases, reduced to one-dimensional

CES

potentials, whose solutions were obtained previously. Now we proceed to

the solution

of

the above mentioned possibilities.

Case

I

for

8 =0

and Gu arbitrary The fi.rst example is

that

of

an exactly solvable potential. In this case the po-tential V,

(x,

y) ismapped into a harmonic oscillator with centrifugal barrier in the variable u, and a Coulomb plus

centrifugal barrier in the variable U. Sowe only need to

identify the corresponding parameters, substitute them in

the equation for the "energy,

"

which in the transformed equation is the parameter K,and solve it for the original eAergp

and

VCEs(u)II

=

Av2/3

+

+

(4b)

8 A D

(,,

'„„(a=0)=

4CD-

_A'

[2m+

I+a,

]'

(2n

+1),

1/2

The wave function can be obtained by returning

%(u,u)

=u

'

4(u)y(u)

tothe original variables.

At this point, we shall observe that the transformation used leads us to a mapping into an angle variable

u

=

arctan(y /x _{), which} can, in principle, create problems

I

where a&

—

=

(I+8pGO/fi

)'~ . It can be seen that the

above spectrum is a kind

of

mixing

of

the Coulomb and harmonic oscillator spectra. The normalized wave func-tions are

qI,B=0( ₎ 32pD

f2

1/8

21'(n

₊₁₎

1(a,

+n+1)

'_1/2 '_1/2

AD

$2 (X

+y2)

01 /2

arctan—

X

(2~)+1)/4 1/2

_—

SpK

$2

I(m

+1)

I

(a&+m+1)

arctan-y X

j

.1/2

arctan

—

y exp

pD

2%2 1/2

—

_8@K

(X2+y

)—

( 1/2

AD

₍

_z+

z)

J 1/2

—

_2pE

arctan—

$2 X where

p

2A

D

1/2

—

_(2n

_+1),

(2n

+1)

—

C,

2pA (2m

+

a,

+

I)iri

1/2 2

1 ₍ fi 2D

x=-

(7b)

4C

"'

_p

_p

and

L„'(x,

_y) is the generalized Laguerre polynomial.

It

is

remarkable that, as can be seen from Eq. (5), the domain

of

validity

of

the parameter C is defined through t

equation

in order to keep the energies real. So, as the left hand side

of

the above equation has its minimum value when

m

=0

(n arbitrary), we must have C

~

2@A

/(a

& +1)A' .

Apparently

if

C is less than this we should impose the condition that

where Int( ) stands for taking the first integer greater

50

BRIEF

REPORTS 4371

with

'_1/2

2A D

K

1

(2n+1)

—

C

.

(11)

p

As we are looking for

8'„,

we see that it is necessary to

solve athird order polynomial equation for

E,

and then use this solution inthe expression forthe energy

' _1/2

2A D

=+2&D

(K

+C)'

+

(2n

+1);

be verified through numerical calculation. Perhaps we

would need some kind

of

analytical continuation for the energy in such cases. Now, the next case.

Case

I

for

8%0

and

Go=

—

3A'

_/32p.

_{In this case we}

have our first example

of

a two-dimensional

CES

poten-tial. After repeating the previous procedure we get the following equation forthe energy eigenstates:

pB4

i'

(m+

—,

')K

+2pA K

+pAB

K

+

=0,

(10)

K

_m

=(R +D)' +(R

D—

)'

a1

3 (13a)

with

D—

:

+Q

+R,

Q:—

_(3az

—

a&

)/9,

R:—

(9a&a2

—

_27a

&

—

2a,

)/54, and

2@A

pAB

vari (m

+

—

')

fi (m

+

—

')

B4

8'

(m+

—,

')

(13b)

once more we have a limitation on the parameter C. The

solution

of

the equation for the parameter

E

isobtained following the procedure appearing in

Ref.

[1].

This gives

us

(12) The wave function is given by

'_1/2

('+')

$2

'_1/2

'

1/8

tg

(x,

y)

=

32p,

D

En+1)

2 +

m!I

(a+n+1)

1/2

AD

B

f2

(x

+y

)

H

P

&arctan(y/x)—

2E

XL:

B

Xexp

—

_z

(x

+y

)

—

&arctan(y/x)—

2E

2

a/2 '1/4

arctan-x

(14)

where we defined

' _1/2

—

_(2n

_+1),

2A2D

and

2R

D

p

1/2

(2n

+

1)

—

C,

(15a)

'_1/2

1/2

+4v'A

_(m

_+1/2)iri

₊

—

3 ₂

4

p

8pK

$2

'_1/4 with the solution

' _{1/2 1/2}

2

9iri A

₊

98

n, m

2

_p

4

and

H

(x,

y)isthe Hermite polynomial.

It

is easy

to

verify that the wave function, being pro-portional to

arctan(y/x),

vanishes along the positive

x

semiaxis. This should be expected due to the singularity

of

the potential in this region. Moreover, it is easy tosee from the wave function (14) that, as observed previously

when we presented the potential, one must restrict the re-gion

of

space such that

x

and yare both positive or both negative. Furthermore, the inverse tangent function is defined in the principal branch, as said previously.

Final-ly we treat the second case.

Case

II

(go=

—

5R

/72p).

Now we present the last ex-ample

of

a two-dimensional

CES

potential that will be discussed in this work.

If

we repeat once more the same procedure we get the following equation for the energy in this last case:

1/2 _1/2

+4CD

+

p

(2n

+

1)

.

(15b)

B~—

(16)

in order toinclude m

=0

in the solution. Besides, the pa-rameter

C

isalso restricted by

C)

4&A

(

+,

)

9A' _A

3 2p

1/2 1/2

(17) This time we need to do a careful analysis

of

the range

of

validity for the potential parameters. In the case

of

the parameter

B,

we have the restriction

BRIEF

REPORTS

Sowe see that the negative solution (positive in the above equation) introduces a limit on the number

of

bound

states. The positive case (negative above) is such that we

only need impose that

C

is bigger than the left hand side

of

Eq. (17) with m

=0.

After this brief analysis

of

the region

of

validity

of

the potential parameters, we present the corresponding wave

functions:

] /8

(x,

y)

=

32

D

tl, m

3r(

+1)

2

'm!I

(a

+n

+1)

2pa

$2

(x

+y

)

~/2 2/3 _]1/4

z~ arctan

—

P

XL„'

1/2

(x'+y

)

8

_~P

arctan—

2/3

pD

Xexp

2A

][/2

(x+y

)

——

2

arctan—

3' X

E

'

2A

where

p

2A

D

—

(2n

+1)

and

'_]/4

9@A 2A

(19)

This finishes the presentation

of

examples

of

CES

poten-tials, now in two-dimensional space. The principal feature presented by these potentials, besides belonging to the

CES

class

of

potentials, isthat astringlike singularity

of

the potential appears. This iscorroborated by the fact that the probability

of

finding the particle over this singu-larity iszero. This accounts for the interest in doing

fur-ther investigations

of

the characteristics

of

such

poten-I

tials.

It

would be interesting to study the behavior

of

their corresponding coherent states, and also see what happens with wave packets around the linear singularity

of

the potential.

Furthermore, in analogy with what has been done with

the exact potentials

[12,13],

and also with the quasiexact ones [8,

9],

it is our intention to look for the dynamical algebra behind the solvability

of

the

CES

potentials. This study is presently under development and we hope to

re-port on itin the near future.

One

of

the authors (A.

S.

D.

) thanks CNPq (Conselho

Nacional de Desenvolvimento Cientifico e Tecnologico)

of

Brazil for partial financial support and

FAP ESP

(Fundal'Ko de Amparo a Pesquisa no Estado de Sao Pau-lo)forproviding computer facilities.

[1]A.de Souza Dutra, Phys. Rev.A 47,R2435(1993).

[2]E.Papp, Phys. Lett. A17$,231(1993). [3]

F.

H.Stillinger,

J.

Math. Phys. 20,1891(1979).

[4]

G.

P.Flessas, Phys. Lett. 72A, 289(1979);78A, 19(1980); 81A,17(1981);

J.

Phys. A14, L209 (1981).

[5]P.G.L.Leach,

J.

Math. Phys. 25, 974 (1984); Physica D 17,331(1985).

[6] A.deSouza Dutra, Phys. Lett.A 142,319 (1988).

[7]R.S.Kaushal, Phys. Lett. A142, 57(1989).

[8]M.A.Shiffman, Iut.

J.

Mod. Phys. A 4, 2897(1989).

[9] A.de Souza Dutra and H.Boschi Filho, Phys. Rev. A 44, 4721(1991).

[10]L. D.Salem and R.Montemayor, Phys. Rev. A 43, 1169 (1991).

[11]

T.

Eguchi, P. B.Gilkey, and A.

J.

Hanson, Phys. Rep. 66, 213 (1980).

[12] A. O.Barut, A. Inomata, and

R.

Wilson,

J.

Phys. A 20, 4075(1987);20,4083(1987}.

[13]H.Boschi-Filho and A.N. Vaidya, Ann. Phys. (NY) 212,