H+ and H− being the partner Hamiltonians in Hs, we can easily isolate the corresponding partner potentialsV±from (2.27). Actually these potentials may be expressed as
V±(x) =1 2
W2(x)∓W(x) (2.29) with W(x) =ωx. We shall refer to the functionW(x) as the super- potential. The representations (2.29) were introduced by Witten [8]
to explore the conditions under which SUSYmay be spontaneously broken.
The general structure of V±(x) in (2.29) is indicative of the pos- sibility that we can replace the coordinatexin (2.27) by an arbitrary
functionW(x). Indeed the forms (2.29) ofV± reside in the following general expression of the supersymmetric Hamiltonian
Hs= 1 2
p2+W2•+1
2σ3W (2.30)
W(x) is normally taken to be a real, continuously differentiable func- tion in. However, should we run into a singularW(x), the necessity of imposing additional conditions on the wave functions in the given space becomes important [10].
Corresponding toHs, the associated supercharges can be written in analogy with (2.21) as
Q = √1 2
0 W +ip
0 0
Q+ = √1 2
0 0 W −ip 0
(2.31) As in (2.22), here tooQand Q+ may be combined to obtain
Hs=Q, Q+ (2.32)
Furthermore,Hs commutes with both Qand Q+ [Q, Hs] = 0
Q+, Hs
= 0 (2.33)
Relations (2.30) - (2.33) provide a general nonrelativistic basis from which it follows that Hs satisfies all the criterion of a formal supersymmetric Hamiltonian. It is obvious that these relations allow us to touch upon a wide variety of physical systems [12-53] including approximate formulations [54-63].
In the presence of the superpotential W(x), the bosonic opera- tors band b+ go over to more generalized forms, namely
√2ωb→A = W(x) + d
√ dx
2ωb+→A+ = W(x)− d
dx (2.34)
In terms ofA and A+ the Hamiltonian Hs reads 2Hs= 1
2
A, A+•+1
2σ3A, A+ (2.35)
Expressed in a matrix structure Hs is diagonal Hs ≡ diag (H−, H+)
= 1
2 diag AA+, A+A (2.36) Note that Hs as in (2.30) is just a manifestation of (2.34). In the literature it is customery to refer to H+ and H− as “bosonic” and
“fermionic” hands of Hs, respectively.
The components H±, however, are deceptively nonlinear since any one of them, sayH−, can always be brought to a linear form by the transformation W =u/u. Thus for a suitable u, W(x) may be determined which in turn sheds light on the structure of the other component.
It is worth noting that both H± may be handled together by taking recourse to the change of variables W = gu/u where, g, which may be positive or negative, is an arbitrary parameter. We see that H± acquire the forms
2H±=− d2
dx2 +g2±g u u
2
∓g u
u
(2.37) It is clear that the parameter g effects an interchange between the
“bosonic” and “fermionic” sectors : g → −g, H+ ↔ H−. To show how this procedure works in practice we take for illustration [64] the superpotential conforming to supersymmetric Liouville system [24]
described by the superpotential W(x) = √a2gexp ax2 , g and a are parameters. Then u is given by u(x) = exp2√
2exp ax2 /a2. The Hamiltonian H+ satisfies −dxd22 +W2−Wψ+ = 2E+ψ+. Transforming y = 4a√22gexp ax2 , the Schroedinger equation forH+
becomes d2
dy2ψ++1 y
d dyψ+
1 2g −1
4
ψ++8E+
a2y2ψ+= 0 (2.38) The Schroedinger equation for H− can be at once ascertained from (2.38) by replacingg→ −g which means transformingy→ −y. The relevant eigenfunctions turn out to be given by confluent hypogeo- metric function.
The construction of the SUSYQM scheme presented in (2.30) - (2.33) remains incomplete until we have made a connection to the Schroedinger HamiltonianH. This is what we’ll do now.
Pursuing the analogy with the harmonic oscillator problem, specif- ically (2.27a), we adopt for V the form V = 12 W2−W+λ in- Wwhich the constantλcan be adjusted to coincide with the ground- state energyE0 ohH+. In other words we write
V(x)−E0 = 1 2
W2−W (2.39)
indicating thatV andV+can differ only by the amount of the ground- state energy valueE0 of H.
IfW0(x) is a particular solution, the general solution of (2.39) is given by
W(x) =W0(x) + exp [2xW0(τ)dτ]
β−xexp [2yW0(τ)dτ]dy, β∈R (2.40) On the other hand, the Schroedinger equation
−1 2
d2
dx2 +V(x)−E0
ψ0= 0 (2.41)
subject to (2.39) has the solution ψ0(x) = Aexp
− xW(τ)dτ
+Bexp
− xW(τ)dτ x
exp
2 yW(τ)dτ
dy (2.42)
whereA, B,∈R and assuming ψ(x) ∈L2(−∞,∞). If (2.40) is sub- stituted in (2.42), the wave function is the same [65] whether a par- ticular W0(x) or a general solution to (2.39) is used in (2.42).
In N = 2 SUSYQM, in place of the supercharges Qand Q+ de- fined in (2.31), we can also reformulate the algebra (2.32) - (2.35) by introducing a set of hermitean operatorsQ1 and Q2 being expressed as Q= (Q1+iQ2)/2, Q+= (Q1−iQ2)/2 (2.43)
While (2.32) is converted to Hs=Q21 =Q22 that is
{Qi, Qj}= 2δijHs (2.44)
(2.33) becomes
[Qi, Hs] = 0, i= 1,2 (2.45) In terms of the superpotential W(x), Q1 and Q2 read
Q1 = √1 22
σ1W −σ2√p m
Q2 = √1 22
σ1√pm +σ2W
(2.46) On account of (2.45), Q1 and Q2 are constants of motion: ˙Q1 = 0 and ˙Q2 = 0.
From (2.44) we learn that the energy of an arbitrary state is strictly nonnegative. This is because [66]
Eψ = < ψ|Hs|ψ >
= < ψ|Q+1Q1|ψ >
= < φ|φ >≥0 (2.47) where |φ >= Q1|ψ >, and we have used in the second step the representation (2.44) of Hs.
For an exact SUSY
Q1|0> = 0
Q2|0> = 0 (2.48)
So |φ > = 0 would mean existence of degenerate vacuum states|0>
and|0>related by a supercharge signalling a spontaneous symmetry breaking.
It is to be stressed that the vanishing vacuum energy is a typ- ical feature of unbroken SUSYmodels. For the harmonic oscillator whose Hamiltonian is given by (2.3) we can say that HB remains invariant under the interchange of the operatorsband b+. However, the same does not hold for its vacuum which satisfies b|0 >. In the case of unbroken SUSYboth the Hamiltonian Hs and the vacuum are invariant with respect to the interchange Q↔Q+.