PACS: 03.65.-w; 03.65.Db; 03.65.Ge2

1. Introduction

As is well known, the problem of a quantum particle moving on a real line with a point interaction (or a singular perturbation) at a single point, can be treated in two equivalent modes: (*i*) by considering an alternative free system without the singular potential (*i.e*., *V*(*x*) = 0) and excluding the singular point, in which case the interaction is encoded in proper boundary conditions, and (*ii*) by explicitly considering the singular interaction by means of a local singular potential. See *e.g*. Ref. 1 and references therein.

The principal aim of this paper is to study and analyze some representative examples of nonrelativistic (Schrödinger) point interactions, *i.e*., boundary conditions and singular potentials, and their corresponding bound states. In this introduction, we extract these examples from a general family of boundary conditions for the system described in the case (*i*), and from a general singular potential written in terms of the Dirac delta and derivatives *d*/*dx* for the system described in (*ii*). The introduction of the present paper is an abridged (and also complementary) version of Ref. 1, *i.e*., it is a survey of point interactions with examples. In Sec. 2, we obtain and discuss the bound states for all these examples. The conclusions are given in Sec. 3. In the Appendix A we study some general aspects related with the eigenvalues and eigenvectors of the Hamiltonian operator corresponding to the case (*i*). Finally, in the Appendix B we explicitly solve the Schrödinger equation for a potential that is the first derivative of the Dirac delta, but we do not use the same definition of *δ*'(*x*) that was used throughout the article.

1.1. Case (*i*): point interactions as boundary conditions

In this case, one considers the line (ℝ) with the origin (*x* = 0) excluded (a hole or a single defect). The Hamiltonian operator is,

Where *x* = ℝ - {0} ≡ Ω. The operator *ψ* such that *ψ* ∈ 𝓗 ≡ 𝓛^{2}(Ω) (*i.e*., ‖*ψ*‖ < ∞ in Ω, with the usual definitions of the norm and the scalar product, *ψ* must satisfy some of the following general boundary conditions:

The parameter *λ* is inserted for dimensional reasons and the 2 x 2 matrix ^{2}. We use the notation *ψ'*. We write the matrix

Where *ϕ* ∈ [0, π], and quantities *m _{A}
* ∈ ℝ (A=0, 1, 2, 3.) satisfy
(

*m*

_{0})

^{2}+ (

*m*

_{1})

^{2}+ (

*m*

_{2})

^{2}+ (

*m*

_{3})

^{2}= 1.

Another 4-parameter family of boundary conditions can algebraically be obtained from Eq. (2) ^{1}:

where the matrix

Note that *S*_{11} and *S*_{22} are real, and *m*_{1} + sin(*ϕ*) = 0 in (4); nevertheless, if we have a boundary condition where *m*_{1} + sin(*ϕ*), the singularity in Eq. (5) could be conveniently avoided, and the respective boundary condition could thus emerge from Eq. (4) ^{1}.

The following boundary conditions are included in Eqs. (2) and (4). Some of the names that identify these boundary conditions are obvious but others will be justified by studying their respective singular potentials:

which is obtained by setting: *m*_{0} - cos(*ϕ*),
*m*_{1} + sin(*ϕ*) and
*m*_{2} = *m*_{3} = 0. Note
that, by making *ϕ* = π/2(⇒ *m*_{0}/
*m*_{1} = 0) in Eq. (6), we obtain the periodic
boundary condition, *ψ*(0+) = *ψ*(0-) and
*ψ*'(0+) = *ψ*'(0-).

which is obtained by setting: *m*_{0} = *m*_{2}
= 0 ⇒ ((1 - *m*_{3})/*m*_{1}) =
*m*_{1}/(1 + *m*_{3}),
cos(*ϕ*) = 0 and sin(*ϕ*) = 1 ⇒
*ϕ* = π/2.

which is obtained by making: *m*_{0} = *m*_{3}
= 0 ⇒ (*m*_{1})^{2} +
(*m*_{2})^{2} = 1, cos(*ϕ*) =
0 and sin(*ϕ*) = 1 ⇒ *ϕ* = π/2. Note that, by
making *m*_{1} = +1 and *m*_{2} =
0 in Eq. (8), we obtain the periodic boundary condition (*ψ*(0+)
= *ψ*(0-) and *ψ'*(0+) = *ψ'*(0-)).
Likewise, by making *m*_{1} = -1 and
*m*_{2} = 0, we obtain the antiperiodic boundary
condition, *ψ*(0+)= -*ψ*(0-) and
*ψ'*(0+) = -*ψ'*(0-).

which is obtained by setting: *m*_{0} = cos(*ϕ*),
*m*_{1} = sin(ϕ) and *m*_{2} =
*m*_{3} = 0. As in the example of the boundary
condition (a), the case *ϕ* = π/2 (⇒
*m*_{0}/*m*_{1} = 0) leads to
the periodic boundary condition.

which is obtained by setting: *m*_{0} = +1,
*m*_{2} = *m*_{3} = 0 (⇒
*m*_{1} = 0) and *ϕ* = π.

which is obtained by setting: *m*_{0} = +1, *m*_{2} = *m*_{3} = 0 (⇒ *m*_{1} = 0) and *ϕ* = 0.

It is worth mentioning that, boundary condition (e) is obtained from boundary condition (a)
by noticing that -2*m*_{0}/*m*_{1}
= +2cot(*ϕ*) = -∞ (because *ϕ* → π-), thus,
*ψ*(0+) = *ψ*(0-) and *ψ'*(0+)
= ((-∞) × *ψ*(0-)) + *ψ'*(0-) ⇒
*ψ*(0-) = 0, and therefore *ψ*(0+) =
*ψ*(0-) = 0. Likewise, boundary condition (f) is obtained
from boundary condition (d) by noticing that
-2*m*_{0}/*m*_{1} =
-2cot(*ϕ*) = -∞ (because *ϕ* → 0+), so
*ψ'*(0+) = *ψ'*(0-) and *ψ*(0+)
= *ψ*(0-) + ((-∞) × *ψ'*(0-)) ⇒
*ψ'*(0-) = 0, and therefore *ψ'*(0+) =
*ψ'*(0-) = 0.

1.2. Case (*ii*): point interactions as singular potentials

In this case, one considers the line (ℝ) with a singular potential at the origin (*x* = 0). The Hamiltonian operator is,

Where *x* ∈ ℝ. A plausible formal expression for a general singular potential *d*/*dx* is the following:

where *g _{B}
* ∈ ℝ (

*B*= 1,2,3,4.)

^{1}

^{,}

^{4}. In this paper, the derivative of the Dirac delta is written as

*δ'*(

*x*) ≡

*dδ*/

*dx*, that is, with the prime on the delta. The operator

^{1}. It has also been proved that every

The singular potential *ψ*(0) = ⟨*δ*, *ψ*⟩ and *ψ'*(0) = -⟨*δ'*, *ψ*⟩. In essence, the latter formulas can be obtained by using the (symbolic) sifting property for the Dirac delta:

and

because *δ'*(*x*)*ψ*(*x*) = (*d*/*dx*)(*δ*(*x*)*ψ*(*x*)) - *δ*(*x*)*ψ'*(*x*) = *δ'*(*x*)*ψ*(0) - *δ*(*x*)*ψ'*(0)

(the common delta function properties

and

were also used above). Because functions *ψ*(*x*) and *ψ'*(*x*) are not generally continuous at *x* = 0, *ψ*(0) and *ψ'*(0) may be written as the average at the discontinuity (this is certainly only a plausible choice for discontinuous test functions):

(see Ref. 6 for a discussion about situations in which the latter definitions do not hold). Thus, one can also write expression (13) as follows:

where ⟨*F*, *ψ*⟩ (with *F* = *δ* or *δ'*) also denotes the action *F*[*ψ*] of the distribution (or linear functional) *F* on the test function *ψ*. Note that, if one defines the quantities, *t*_{11} ≡ *g*_{4}, then these coefficients {*t _{pq}*} define a 2 x 2 hermitian matrix

^{5}.

Due to the presence of *δ*(*x*) and *δ'*(*x*) in *n*-th derivative of a delta function potential in the following way ^{7}: integrating *ϵ* to +*ϵ* and taking the limit *ϵ* → 0 gives the following first boundary condition:

where *α* ≡ 2*m*/ℏ^{2}. Similarly, integrating *x* = -*L* (with *L* > 0) to *x*, then once more from -*ϵ* to +*ϵ* and taking the limit *ϵ* → 0 again, one obtains a second boundary condition:

where the relations

(Θ(*x*) is the Heaviside function: Θ(*x* < 0) = 0 and Θ(*x* > 0) = 1) and

should be used. Note that Eqs. (18) and (19) precisely constitute the family of boundary conditions (4), where, in this case, the matrix

By comparing the matrix

Thus, if we use Eqs. (21)-(24), we can relate boundary conditions included in (4) with potentials dependent of deltas included in (13) (or (17)). The following potentials correspond respectively to the examples of boundary conditions that were introduced above:

which is obtained by setting: *m*_{0} = -cos(*ϕ*),
*m*_{1} = sin(*ϕ*) and
*m*_{2} = *m*_{3} = 0, thus,
(from relations (21)-(24)) *g*_{1} =
2cot(*ϕ*)/*αλ* and
*g*_{2} = *g*_{3} =
*g*_{4} = 0. Therefore, (from Eq. (13)) we obtain the
result given in Eq. (25). Note that, by making *ϕ* = π/2, we
obtain *g*_{1} = 0, and therefore *ϕ* → *π*-, we obtain
*g*_{1} → -∞ (this is the case (e), which is
presented below).

which is obtained by setting: *m*_{0} = *m*_{2}
= 0 ⇒ ((1 - *m*_{3})/*m*_{1}) =
*m*_{1}/(1+ *m*_{3}),
cos(*ϕ*) = 0 and sin(*ϕ*) = 1 ⇒
*ϕ* = π/2, thus, (from relations (21)-(24))
*g*_{2} = 2*m*_{3}/α(1 +
*m*_{1}) and *g*_{1} =
*g*_{3} = *g*_{4} = 0.
Therefore, (from Eq. (13)) we obtain the result given in Eq. (26). Note that, by
making *m*_{1} = 1, we
obtain *g*_{2} = 0, and therefore

which is obtained by setting: *m*_{0} = *m*_{3}
= 0 ⇒ (*m*_{1})^{2} +
(*m*_{2})^{2} = 1, cos(*ϕ*) =
0 and sin(*ϕ*) = 1 ⇒ *ϕ* = π/2 , thus, (from
relations (21)-(24)) *g*_{3} =
-2*m*_{2}/*α*(1 +
*m*_{1}) and *g*_{1} =
*g*_{2} = *g*_{4} =0.
Therefore, (from Eq. (13)) we obtain the result given in Eq. (27). It is worth
noting that, by making *m*_{1} = -1 and
*m*_{2} = 0, we obtain *g*_{3}
= 0/0. However, in this case we can write *g*_{3} =
-(2/*α*) [(2/*m*_{2}) -
(*m*_{2}/2) +
*O*((*m*_{2})^{3})], which
implies that *g*_{3} → -∞ when
*m*_{2} → 0 (⇒ *m*_{1} → -1).
Precisely, the latter case corresponds to the antiperiodic boundary condition
(see the paragraph that follows Eq. (8)). Likewise, if
*m*_{1} = +1 and *m*_{2} = 0,
we obtain *g*_{3} = 0). Incidentally, the
Hamiltonian operator (12) with the potential (27) can also be written as ^{5}^{,}
^{8}.

which is obtained by setting: *m*_{0} = cos(*ϕ*),
*m*_{1} = sin(*ϕ*) and
*m*_{2} = *m*_{3} = 0, thus,
(from relations (21)-(24)) *g*_{4} =
2*λ*cot(*ϕ*)/*α* and
*g*_{1} = *g*_{2} =
*g*_{3} = 0. Hence, (from Eq. (13)) we obtain the
result given in Eq. (28). Note that, by making *ϕ* = π/2, we
obtain *g*_{4} = 0, and therefore *ϕ* → 0+, we obtain
*g*_{4} → +∞ (this is the case (f), which is
presented below). It is worth noting that, the general singular potential ^{1}.

which is obtained by making: *m*_{0} = +1,
*m*_{2} = *m*_{3} = 0 and
*ϕ* = π, thus, (from relations (21)-(24))
*g*_{1} = -4/*αλm*_{1} and
*g*_{2} = *g*_{3} =
*g*_{4} = 0. Also, *m*_{1} = 0
and therefore *g*_{1} = -∞ (in fact,
*m*_{1} → 0+ ⇒ *g*_{1} → -∞,
and *m*_{1} → 0- ⇒ *g*_{1} → +∞).
Therefore, (from Eq. (13)) we obtain the result given in Eq. (29). Note that the
Dirichlet potential is the Dirac delta potential with infinite strength, and it
can (heuristically) be written in the form

which is obtained by setting: *m*_{0} = +1, *m*_{2} = *m*_{3} = 0 and *ϕ* = 0, thus, (from relations (21)-(24)) *g*_{4} = 4λ/*αm*_{1} and *g*_{1} = *g*_{2} = *g*_{3} = 0. Also, *m*_{1} = 0 and therefore *g*_{4} = +∞ (in fact, *m*_{1} → 0+ ⇒ *g*_{1} → +∞, and *m*_{1} → 0- ⇒ *g*_{1} → -∞). Therefore, (from Eq. (13)) we obtain the result given in Eq. (30). Note that the Neumann potential is the “delta-prime” interaction potential with infinite strength.

2. Bound States

In this section, we present the (normalized) bound state eigenfunctions and their respective energy eigenvalues corresponding to the examples of point interactions that were introduced above.

(a) For the Hamiltonian with the Dirac delta potential (25), *E* < 0:

Where *g*_{1} < 0. This eigenfunction satisfies the boundary condition (6): *ψ*(0+) = *ψ*(0-) ≡ *ψ*(0) and *λψ'*(0+) - *λψ'*(0-) = 2cot(*ϕ*)*ψ*(0), where *g*_{1} = 2cot(*ϕ*)/*αλ*. A nice discussion of the Dirac delta potential, which includes the scattering states, can be found in the book by Griffiths ^{9}. For studies on the completeness of the eigenfunctions in this problem, see Refs. 10 and 11.

(b) For the Hamiltonian with the potential first derivative of the Dirac delta (26), *ψ*(*x*) = 0) with zero energy *E* = 0, *i.e*., there is no a nontrivial square integrable solution that satisfies the boundary condition (7): *ψ*(0+) = ((1 + *m*_{3})/*m*_{1} )*ψ*(0-) and *ψ'*(0+) = (*m*_{1}/(1 + *m*_{3}))*ψ'*(0-), where *g*_{2} = 2*m*_{3}/*α*(1 + *m*_{1}) and (*m*_{1})^{2} + (*m*_{3})^{2} = 1. For a concise discussion of this potential, which includes the scattering states, we recommend Ref. 12. For a study that considers the potential -*aδ*(*x*) + *bδ'*(*x*), see Ref. 13. It should be noted that different definitions of the derivative of the delta interaction exist in the literature; see *e.g*. Refs. 6 and 14 and other references quoted therein. Finally, another article that presents a very particular study that involves the same potential *δ'*(*x*) used by us throughout the article can be found in Ref. 15. In the Appendix B, we treat precisely with a different but very natural definition of this potential. However, we do not get a nontrivial bound state in this case either.

(c) For the Hamiltonian with the quasi-periodic potential (27), *E* = 0, where *ψ*(0+) = (*m*_{1} - i*m*_{2})*ψ*(0-) along with *ψ'*(0+) = (*m*_{1} - i*m*_{2})*ψ'*(0-) is the corresponding boundary condition (formula (8)), and *g*_{3} = -2*m*_{2}/*α*(1 + *m*_{1}) with (*m*_{1})^{2} + (*m*_{2})^{2} = 1. We have not found a complete discussion of the scattering states for this potential (with *m*_{1} ≠ 0 and *m*_{2} ≠ 0) in the literature. However, see Refs. 5 and 8 where various aspects related to the boundary condition associated with this potential are discussed.

(d) For the Hamiltonian with the “delta-prime” interaction potential (28), *E* < 0:

where *g*_{4} > 0 and sgn(*x*) is the sign function (sgn(*x* > 0) = +1 and sgn(*x* < 0) = -1). This eigenfunction satisfies the boundary condition (9): *ψ*(0+)-*ψ*(0-) = -2cot(*ϕ*)*λψ'*(0) and *ψ'*(0+) = *ψ'*(0-) ≡ *ψ'*(0), where *g*_{4} = 2*λ*cot(*ϕ*)/*α*. Scattering states arising from this boundary condition were obtained, for example, in Ref. 16 and the most important spectral properties associated with the free Hamiltonian for this boundary condition (as well as with others) were analyzed in ^{3}. In Ref. 17, it was shown that the boundary condition defining this interaction arises precisely from the potential (28).

(e) Because the Dirichlet potential (29) is obtained from the Dirac delta potential (25) by setting the limit to *g*_{1} → -∞, the eigenfunction and the respective energy eigenvalue for the Hamiltonian with the Dirichlet potential can be obtained from (31) by taking the same limit. Thus, we obtain the following formal results:

where we have used the following representation of the Dirac delta ^{18}: *ψ*(*x*) looks like a highly localized state with infinite energy, in fact, it is essentially the square root of the Dirac delta. Despite these results, it is easy to show that the scalar product of *ψ*(*x*) with a square integrable function, *f* ∈ 𝓗 ≡ 𝓛^{2}(ℝ), vanishes. The latter result implies that the distribution (or linear functional) associated with *ψ*(*x*),

is precisely zero. In fact,

In the last step we used the representation of the Dirac delta that was used to derive Eq. (33), and also the property

Thus, we conclude that the eigenfunction is really trivial, *i.e*., *ψ*(*x*) = 0 everywhere on ℝ, and it satisfies the boundary condition (10): *ψ*(0+) = *ψ*(0-) = 0 (of course, to the system corresponding to the case (*i*) where the origin is excluded). A similar result to that given in Eq. (34) emerges in the problem of the one-dimensional hydrogen atom. In that case the state *ψ*(*x*) corresponds to the (nonexistent) ground state of infinite binding energy ^{19}^{,} ^{20}.

(f) Because the Neumann potential (30) is obtained from the “delta-prime” interaction potential (28) by setting the limit to *g*_{4} → ∞, the eigenfunction and the respective energy eigenvalue for the Hamiltonian with the Neumann potential can be obtained from (32) by taking the same limit. Thus, we obtain the following results:

This is the trivial bound state with zero energy, and it obviously satisfies the boundary condition (11): *ψ'*(0+) = *ψ'*(0-) = 0.

3. Conclusion

We have presented and examined the bound states for a number of representative examples of (Schrödinger) point interactions, *i.e*., boundary conditions and singular potentials, that were introduced, related and also discussed, throughout the article. As we have seen, the (attractive) Dirac delta function potential provides an even-parity bound state; this is a well-known fact. If this potential has infinite strength it becomes the Dirichlet potential, and therefore the state must satisfy the Dirichlet boundary condition. Thus, the bound state becomes trivial in this latter case. Likewise, the labelled as “delta-prime” interaction potential (this is not the first derivative of the Dirac delta potential) also provides a bound state (an odd-parity state). If this potential has infinite strength it becomes the Neumann potential, *i.e*., the state must satisfy the Neumann boundary condition. However, this state is equal to zero. On the other hand, in our model, the potential first derivative of the Dirac delta function does not provide a nontrivial bound state. If we change the definition of *δ'*(*x*) for a more natural, we do not obtain a nontrivial bound state either. It is worth mentioning that this new potential is also a legitimate point interaction because it corresponds to a boundary condition included in the domain of the (self-adjoint) Hamiltonian