Shape invariance of second orderin one-dimensional quantum Mechanics Текст научной статьи по специальности «Математика»

2013 ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА Сер. 4. Вып. 1

ФИЗИКА

УДК 539.1(05)

M. S. Bardavelidze, D. N. Nishnianidze

SHAPE INVARIANCE OF SECOND ORDER IN ONE-DIMENSIONAL QUANTUM MECHANICS

Introduction. The notion of shape invariance [1] is one of the new very fruitful ideas appeared in the framework of supersymmetrical quantum mechanics [2] (see review papers [3-6]). It allowed to reproduce in a purely algebraic way all known one-dimensional exactly solvable potentials, i. e. potentials with analytical expressions of all energy eigenvalues and corresponding wave functions [7]. Even for these potentials, shape invariance gave a new look onto familiar problems. Also, by using the supersymmetrical intertwining relations it provides some new exactly solvable models [8, 9].

Recently [10, 11], it was proved that the problem of finding of all potentials with the so called additive shape invariance is equivalent to solution of the one-dimensional Euler equation, which originally expresses momentum conservation for inviscid fluid flow. This equation was solved, and its solution provides all familiar exactly solvable potentials. Some new potentials were also found allowing for the superpotential to depend on Planck constant explicitly.

The Hamiltonian

depending on a space variable x and a constant parameter a is said to be shape invariant if two conditions are fulfilled: 1) it satisfies supersymmetrical intertwining relations:

with some intertwining operators q of first order in derivatives, and 2) two potentials in (2) are related as follows:

V2(x; ao) + g(ao) = Vi(x; ai) + g(ai),

where g(a) does not depend on coordinate x, and the parameters a0 and a1 are related. The relation between two Hamiltonians just means that they have the same shape, and

Malkhaz Simon Bardavelidze — teacher, Akaki Tsereteli State University, Kutaisi, Georgia; e-mail: m.bardaveli@gmail.com

David Nodar Nishnianidze — Professor, Akaki Tsereteli State University, Kutaisi, Georgia; e-mail: cutaisi@yahoo.com

© M. S. Bardavelidze, D. N. Nishnianidze, 2013

Ht(x; a) = -Ь2д2 + Vt (x, a),

(1)

Hi(x; a)q(x) = q(x)H2(x; a),

(2)

slightly differ due to a change of parameter of the model. In the case under discussion below, similarly to [10, 11], the different values of constant a will be:

a„+i = an + h.

This form of change of parameter values is dictated by the "supersymmetrical" expression for potentials Vi,2 through superpotential W(x, a) with explicit dependence on Planck constant:

V1>2(x, a) = W2(x, a) ^ hdxW(x, a).

Usually in the literature, h is invisible, since it is taken unity. It is easy to check that all well known exactly solvable (shape invariant) potentials just obey these properties.

Intertwining of second order. The useful and popular generalization of conventional supersymmetrical quantum mechanics — polynomial SUSY QM — is based on the intertwining relations with supercharge operators q of second order in derivatives [12-14]. The analogous generalizations were also used successfully [15-19] in two-dimensional SUSY QM providing some new exactly solvable models. In the present paper, we shall generalize the investigation of [10, 11] analyzing the additive shape invariance of second order, i. e. with intertwining operator:

q = h2d2 - 2hf (x,a)d + b(x, a), (3)

where the rule for appearance of h is very simple: each derivative in x gives a power of h (see also the recent preprint [20]).

Substitution of (1) and (3) into intertwining relations (2) after equating coefficients in front of powers of derivative leads to three nonlinear differential equations for functions

Vi,V2 ,f,b:

V - Vi = 4hf '(x, a), (4)

h2f ''(x, a) - hb'(x, a) - (V1 - V2)f (x, a) = hV', (5)

hb''(x, a) + 4b(x, a)f '(x, a) + hY' - 2f (x, a)V2' = 0, (6)

where prime means derivative over x. Actually, just this system of equations was solved in [13], though with hidden Planck constant: h =1. Now, with reproduced again multipliers h, the general solution is:

2 ^ , h2 f f''(x,a) f'2(x,a)\ d(a)

VlAx, a) = T2 */<(*, a) + a) + y ^^ - ^¿-L j - ^ + a (a). (7)

It is convenient to rewrite potentials in the form:

Vi,2 = vi,2(x, a) + a(a),

where a is shape invariant parameter, and a(a) does not depend on x.

Conditions of shape invariance. The condition of shape invariance reads:

vi (a + h, x) — V2 (x, a) = -(a(a + K) — a(a)), (8)

and substitution of v(x, a) as in (7) allows to expand it in powers of K. It is useful to introduce new functions:

1 ( f"(x,a) f'2(x, a) \ . d(a) .

' >=cp(x,a), j = p(x, a). (9)

2\ f (x,a) 2f 2(x,a)J ' ' f 2(x,a)

so that (8) can be presented in the form:

oE (df(x,a+h) df(x,a)\ 2 2

~ I-dx----dx— 1 (x, a + H) — f (x, a) -

— fi2 (^(x, a + fi) — ^(x, a)) — (p(x, a + fi) — p(x, a)) = —(a(a + fi) — a(a)). Expanding in Taylor series in fi, one obtains:

_ 2 hi + V Hk dk+1f(a,x)\ + 2hf(a,x)df(a,x) +

\ dx k! dxdak / da

\ k=l J

^ /2/(a, x) 0"/(a, x) ^ 1 dfc/(«, x) d"-fc/(a, x) \ ^ y n\ dan f^klin-ky. dak dan-k J

h2 y^ hn <9™cp(a, x) y^ ^ dnp(a, x) _ ~ (a)

+ n\ dan n\ dan ~ k\

n=1 n=1 k=l

The coefficients of different powers in fi have to vanish. For the first one (fi1) this means:

ndf(x,a) rf ,df(x,a) . 1 d p(x,a) a'(a)

- /(X'a)—dT~ + 2 ~da = —' (10)

and taking into account (9), it takes the form:

4/3(x, " 2/4(x, + f(x, a)d'(a) -

- 2d(a)' a) - /3(x, a)a'(a) = 0. (11) da

The zero value of next coefficient (of fi2) leads to:

0d2f(x,a) d2 f (x, a) fdf(x,a)\2 1 d2p(x,a)_ a"(a) - J a)-^---— +

dadx ' da2 \ da J 2 da2 2 '

and one can check that this condition is not independent: it follows from (10). Other terms give the conditions:

2 dk+1f(a,x) 2 f(a,x)dk+1f(a,x) 1 dk+1p(a,x)

~ k! dxdak + (k+ 1)! dak+l (k + 1)! dak+l +

A 1 dif{a,x)dk+1~if{a,x) 1 dk~1y{a,x)_ a fc+1(a)

+ iirz, i _ ,'V Jhii flnk+l-i

!(k + 1 — i)! da* dak+1-i (k — 1)! dak-1 (k +1)!

i=i

for k ^ 2. After some straightforward algebraic manipulations with (10), these conditions can be written in a compact form of the system of differential equations:

2(k - 1) dk+1f(x, a) dk-l(ç{x, a) _ k(k + l) dxdak da^1 ~ '

which, in turn, is equivalent to:

<9cp (x, a) 1 d3f{x,a) d5f(x,a) <93cp (x, a)

da 3 dxda3 ' dxda4 ' da3 The result can be represented in a form:

f (a, x) = U(a) + a3f3(x) + a2f2(x) + afi(x) + fo(x),

cp(a, x) = a2/g(x) + | a/^(x) + cp0(x), (12)

where functions fi(x), tyi(x) must be found. It is clear that if any fi in (12) is constant, it can be taken zero.

After substitution of (12) into (9), the latter takes the form of functional-differential equation (different functions depend on different arguments). It has the following four variants of solutions:

U(a) = 0, f(x,a) = afi(x);

U (a) = 2 ca2, fix, a) = 2abx -|— x2;

c

£/ (a) = —, fix, a) = a (— x2 + e ) + 5x;

a 4c

c

U(a) = —, /(x, a) = abx + e. a

Since three expressions for f (x) above have the forms of polynomials in x, they can not satisfy the equation (11). Thus we have to study only the case of f (x,a) = afi(x). Its substitution into (11) gives a new functional-differential equation:

2d(a) - ad'(a) + a3a'(a)f2(x) + 2a4(f4(x) - 2f (x)f2(x)) = 0.

Fortunately, its solution is known [21, Eq. (31)]:

2d(a) - ad'(a) = -2Aia4, a3a'(a) = 2a4A2, f4(x) - 2f (x)f2(x) = Ai - A2f?(x),

and therefore:

d(a) = Aia4 + Bia2, a(a) = A2a2 + B2;

2f[(x) = fHx) -+ A2, (13)

where A1,A2 are constants.

The shape invariant potential (7) takes now the form:

, ч 2 ^ h2 ( f\'(x) f\2(x) \ A 1a2 + Bi „ 2

Vi,2(x, a = T2fia/ x + a2/2 x + - - " ,2Д + Ma2 + B2,

2 \fi(x) 2/ 2(x^ f2(x)

where the functions f1 (x) are the solutions of (13).

Explicit solutions. The last problem we have to solve concerns the differential equation (13). It can be easily solved in elementary functions if A1 = 0 :

fi = V^

■ aexp(-y/-A2 x/2) - йехр(л/-^2 x/2)

аехр(-л/-^2 x/2) + йехр(л/-^2 x/2) 210

These functions fi(x) correspond to potentials Vi,2, which are already known in the framework of first order shape invariance: Scarf and Poschl—Teller potentials. When both parameters Ai = A2 =0 vanish, one obtains the singular oscillator potential.

For other values of parameters, it is convenient to rewrite (13) in a polynomial form in terms of new function y(x) = 1/fi(x):

2y'(x) = Aiy4 - A2y2 - 1. The solution is easy to write but only as the expression for inverse function x = x(y):

dy x

Aiy4 - A2y2 - 1 2'

(14)

In order to analyze an opportunity to find the direct expression of (14) in the form y = y(x), which could be inserted into potentials Vi,2, three different regions of constants Ai,A2 must be considered:

1) A2 > -4Ai; 2) A2 < -4Ai; 3) A2 = -4Ai.

The chance to express y as a function of x exists only in the case 1) with Ai < 0, when both roots of polynomial in the denominator of (14) are of the same sign. If in addition A2 > 0, the integral (14) is expressed in terms of elementary trigonometric functions, and we will not consider this case below. For A2 < 0, it is convenient, instead of constants Ai,A2, to introduce new ones:

Ai = -&2&2, A2 = -(b2 + b2), b2 > 0, bi > b2.

Then from (14) we obtain:

const • exp bl • (15)

The constant in the r.h.s. can be made unity by a suitable shift of x, and in terms of function fi Eq.(15) is equivalent to:

/■-M""VA+M_±„ m-^Y (16)

y-i/bi bi/2 i/b2 b2/2

V + l/bi y - i/b2

fi + bj Vfi - b2) V b2

The explicit expressions for fi as a function of x can be derived from (16) only in two cases: bi/b2 = 2 and bi/b2 = 3. Nevertheless, one can extract the asymptotic behaviour of fi(x) at the origin and at infinity without explicit solution:

fi(xo) = 0 or fi(xo) = ±c only for xo = 0; (17)

fi(-c) ^ bi or fi(-c) ^ -b2; fi(+c) ^ -bi or fi(+ro) ^ b2.

Therefore, the potentials are finite at both infinities x ^ ±c», and in particular,

BB

Vlj2(±00) ->■ B2 - -g^ or Vlj2(±00) B2- -g^-.

It follows from (17), that potentials may be singular only in the origin x = 0. Since the behaviour of f1(x) in this point can be defined directly from (13):

№) 2

x^0 x

the potentials have singularities of the form:

h2

' x^o 4x2 . . 16a(a T h) + 3h2

' x^o 4x2

Let us discuss in more details the case of b1 = 2b2 = 8p. Eq.(16) is a cubic algebraic equation for f1(x). For the sign minus at the r.h.s. of (16), this equation has the only real solution:

Mx) = 8|3cosh2(3[3x) tanh1^3(3|3x) / + ^4/3^ _ 4(3tanh(6(3x). cosh(6px) v /

For the sign plus at the r.h.s. of (16), this equation has three real solutions, which can be written as:

A + A A — A

/i(x) = A + A-4|3coth(6|3x), /i(x) =--—2—^-^coth^Px),

where

A = 4(3(sinh(6|3x) - 02.

coth(6^x) 1 /co N sinh2(6^x)

These solutions can be written in trigonometric form:

/i(x) = 4|3coth(6|3x) ( 2cos | - ; (18)

/i(x) = —4|3coth(6|3x) ( 2cos^y^ + ; (19)

/i (x) = -4(3 coth(6(3x) ( 2 cos ^^ + 1 j , (20)

where

6 6 1 cos — = tanh(6(3x), sin — —

2 v 1 h 2 cosh(6|3x)'

For x = 0, the solutions (18), (19) are nonzero constants:

f(\ + 813

and therefore, as it was noticed above, the problem is nonsingular for such solutions. 212

For b\ = 3b2 = 3y, Eq.(16) is algebraic equation of fourth order:

ft - 87^44 ft + 18Y2/i - 27y4 = 0, e(x) = ±exp(4Yx). (21)

1 — £(X )

The explicit expressions for solutions of (21) are rather cumbersome. However, it is solvable by Ferrari method [22], since one of solutions of corresponding resolvent can be found easily. For example, for e(x) = — exp(4yx), this solution is:

2 12 Y2

y = 6Y2 + ■ r

cosh2/3(2yx)'

and solution of Eq. (21) is reduced to solution of quadratic equation. References

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Статья поступила в редакцию 3 сентября 2012 г.