One of the simplest non-Hermitian Hamiltonians, first proposed by Schwartz in 1960, that may possess a spectral singularity is analysed from the point of view of the non-Hermitian generalization of quantum mechanics. It is shown that the η operator, being a second-order differential operator, has supersymmetric structure. Asymptotic behaviour of the eigenfunctions of a Hermitian Hamiltonian equivalent to the given non-Hermitian one is found. As a result, the corresponding scattering matrix and cross section are given explicitly. It is demonstrated that the possible presence of a spectral singularity in the spectrum of the non-Hermitian Hamiltonian may be detected as a resonance in the scattering cross section of its Hermitian counterpart. Nevertheless, just at the singular point, the equivalent Hermitian Hamiltonian becomes undetermined.
Recently, there has been a growing interest in non-Hermitian Hamiltonians possessing a real spectrum and spectral singularities [1–15]. Probably, this is due to the remark that they may produce a resonance-like effect in some optical experiments and may find an optical realization as a certain type of lasing effect that occurs at the threshold gain [6–9].
It is well known that, for any self-adjoint (or essentially self-adjoint) scattering Hamiltonian, continuous spectrum eigenfunctions can be expressed in terms of the Jost solution f(k,x) and Jost function F(k), which, in the simplest case, is the Jost solution taken at x=0, F(k)=f(k,0) . One of the characteristic features of any self-adjoint scattering Hamiltonian is that its Jost function never vanishes if k is a spectral point, F(k)≠0 for all k>0 .
An essential feature of the spectral singularity k0 is that this point belongs to the continuous part of the spectrum of a non-Hermitian Hamiltonian H and the corresponding Jost function vanishes at this point, F(k0)=0. As a result, there is no way to construct a Hermitian1 operator h related to H by an equivalence transformation . In other words, there is no way to redefine the inner product in the spirit of the study by Bender et al.  with respect to which H would become Hermitian.
Probably, just for this reason, some authors have claimed that for a Hamiltonian possessing the spectral singularity no resolution of the identity operator is possible [1–5]. In the study of Andrianov et al. , the completeness of biorthogonal sets of eigenfunctions of non-Hermitian Hamiltonians possessing spectral singularities was carefully analysed. The results obtained are illustrated by a number of concrete examples. In particular, the authors  showed that the contribution of the spectral singularity to the resolution of the identity operator depends on the class of functions used for physical states. Further progress was made in Samsonov , where a special regularization procedure for the resolution of the identity operator was proposed. Note that in Guseinov  a concise analysis of the general concept of the spectral singularity of non-Hermitian Hamiltonians is given.
Till now Hamiltonians possessing spectral singularities have been studied mainly as a possible source of new properties of optical media [1–9]. Probably, this is because of the fact that their association with quantum mechanical observables is involved. Nevertheless, as shown in Andrianov et al. , spectral singularities ‘are physical’ since ‘they contribute to transmission and reflection coefficients of a non-Hermitian Hamiltonian dramatically enhancing their values’.
On the other hand, any non-Hermitian diagonalizable Hamiltonian H with real and purely discrete spectrum possesses a Hermitian counterpart h=h† that is related to H by a similarity transformation . Such a transformation does not exist if the Hamiltonian is not diagonalizable. (Such a Hamiltonian cannot be reduced to a diagonal form by changing the basis. The interested reader can find a discussion about quantum mechanics with non-diagonalizable Hamiltonians in Sokolov et al.  and Andrianov et al. .)
For scattering Hamiltonians, additional obstruction appears for the existence of a similarity transformation between H and h. Such a transformation does not exist if the spectral singularity is present in the continuous spectrum of H. In many cases H depends on a parameter, H=H(d), and the spectral singularity appears at d=d0. If d≠d0 the continuous spectrum of H is regular. If there exists an invertible and positive definite operator η such that ηH(d)=H†(d)η, where H†(d) is the Hermitian conjugate of H(d), then one can construct the Hermitian counterpart h(d) of the operator H(d). Below, using a very simple example, I show that the possible presence of a spectral singularity in H(d), for d≈d0, may be detected indirectly as a resonance in the scattering cross section for h(d).
In this paper, I present a careful analysis of Schwartz’s example of a non-Hermitian Hamiltonian H with the possible presence of a spectral singularity. This is one of the simplest examples because the Hamiltonian contains only the kinetic energy, and its non-Hermitian character is hidden in a boundary condition at x=0. In the next section, I define this Hamiltonian and give a definition of the spectral singularity. In §3, I introduce a Hermitian Hamiltonian h together with an η operator that intertwines H and H†. In §4, I reveal the supersymmetric (SUSY) nature of the η operator and introduce its superpartner . In §5, I construct an integro-differential operator that being applied to the function gives eigenfunctions Φk(x) of h. In §6, I calculate the asymptotic behaviour of Φk(x) and scattering matrix for h. In §7, I show that, if H has the spectral singularity, operator h becomes undetermined. In §8, I review briefly our main findings and draw some conclusions.
2. Non-Hermitian Hamiltonian H
Following Schwartz , consider a non-Hermitian operator (Hamiltonian) 2.1 with the domain 2.2 where d and b are real numbers. It is a simple exercise to find its eigenfunctions φk(x), 2.3 which for d<0 form a bi-orthonormal set in , 2.4 with the completeness condition of the form Here and in what follows I denote the identity operator by I and the asterisk means the complex conjugate.
I would like to emphasize that if d>0 then the Hamiltonian H has a discrete level , the possibility that I would like to avoid, and, therefore, in what follows I assume d<0.
There are several equivalent definitions of spectral singularities . The one that is suitable for my purpose uses the kernel R(x,ξ,λ) of the resolvent Rλ of H, A point λ0 belonging to the continuous part of the spectrum of H satisfies where the limit should be taken along any path belonging to the resolvent set of H. The function R(x,ξ,λ)  is constructed with the help of two linearly independent solutions φ(x,λ) and e(ξ,k) of the differential equation 2.5 as follows: 2.6 Here W(λ) is the Wronskian of the functions φ(x,λ) and e(ξ,k),
In particular, if the function φ(x,λ) is such that and e(x,k) is the Jost solution of equation (2.5) defined by its asymptotic behaviour then the Wronskian W(λ) coincides with the Jost function for the Hamiltonian H, Because the resolvent becomes infinite at any point where W(λ)=0, i.e. in the current case at λ=−(d+ib)2, the Hamiltonian H has a spectral singularity at point λ=b2, i.e. at d=0. Just at this point, as mentioned in §1, the Jost function for H vanishes. As shown in Samsonov , in this case the corresponding continuous spectrum eigenfunction has zero binorm and the resolution of the identity operator needs a special regularization procedure.
3. Equivalent Hermitian operator h
To establish an equivalence between the non-Hermitian operator H and a Hermitian operator h, I will use ideas formulated in Scholtz et al.  for quasi-Hermitian Hamiltonians and further developed in Mostafazadeh  for pseudo-Hermitian Hamiltonians. First, one has to find a Hermitian positive definite and invertible operator η such that 3.1 In the present case the adjoint operator H† is defined by the same differential expression (2.1) with the domain 3.2 It is not difficult to check that a second-order differential operator, 3.3 satisfies equation (3.1). Evidently, equation (3.1) defines η up to a transformation η→A†ηA, with any invertible A such that [A,H]=0 . It is convenient to use the form (3.3) of the η operator.
If η were bounded, its domain would be the whole Hilbert space, and no problems would occur in acting by both the left- and the right-hand sides of (3.1) on functions belonging to DH. Unfortunately, this is not the case here because operator (3.3) is unbounded and should have its own domain in . It is reasonable to assume that the domain of η coincides with that of H, 3.4 As I show below, this assumption is justified by the property that the operator η defined in this way is self-adjoint as well as positive definite and invertible on Dη. It is not difficult to find its eigenfunctions and eigenvalues, 3.5 where 3.6 From here I conclude that η in (3.3) is positive definite. Moreover, because its spectrum is bounded below by d2≠0, the operator η−1 is bounded in and can be continued from any initial domain to the whole .
I note also that functions (3.6) form a complete and orthonormal set in , 3.7 This property follows from the fact that η is self-adjoint with respect to the usual inner product in . Indeed, as usual, assuming that ψ1∈Dη and integrating by parts twice the term with the second derivative and once the term with the first derivative yields To justify the last equality, consider the integrated term at x=0: The first line here follows from the property that ψ1∈Dη=DH given in (2.2) and in the last line I have used ψ2∈Dη=DH.
In the next step, I have to check that η in (3.3) is invertible on Dη. For that, compute the kernel space of the differential expression (3.3). This is a two-dimensional linear space with the basis vectors Evidently, f+(x) does not satisfy the boundary condition given in equation (2.2) while f−(x) does satisfy this condition. Therefore, for d<0, when , we have f−(x)∉Dη and, hence, η in (3.3) is invertible on Dη. Thus, as already mentioned, in what follows I assume that d<0 except for §7 where I consider the case d=0.
From intertwining relation (3.1), it follows that the operator defined as 3.8 is Hermitian, h=h†, and at the same time is related to H by equivalence transformation (3.8). According to the first equality (3.8), if ψ1∈Dh then φ1=η−1/2ψ1 should belong to DH, φ1∈DH. Note that because both η−1 and η−1/2 are bounded, the function φ1 is well defined. Hence, I can define as a set of functions ψ1=η1/2φ1 when φ1 runs through , where will be specified below. The function ψ1 here is also well defined because . Similarly, according to the second equality in (3.8), I can define as a set of functions ψ2=η−1/2φ2 when φ2 runs through . It is not difficult to see that, for any φ1(x) satisfying the boundary condition (2.2), the function 3.9 satisfies the boundary condition (3.2). Note that, because η is a second-order differential expression and φ2(x)∈DH†, the function φ1(x) should be smoother than is required by equation (2.2), namely it should be such that where φ(iv)(x) is the fourth derivative of φ(x). Thus, we have Moreover, from (3.9) it follows that This means that I can put with Furthermore, because η has an empty kernel on DH, the set Dh is dense in and it can be taken as an initial domain for h where it is Hermitian, i.e. This property follows from the following chain of equalities:
4. Supersymmetric partner of the η operator
Like any positive definite second-order differential operator, η admits a factorization by first-order operators L and L†, 4.1 thus revealing its SUSY nature. The corresponding SUSY algebra is based on the above factorization properties and intertwining relations  4.2 where 4.3 Note that intertwining relations (4.2) are nothing but identities: Although the operator , which is a SUSY partner of η, is defined by the same differential expression as operator η in (3.3), its domain is different from Dη in (3.4). This, in particular, follows from intertwining relations (4.2). Indeed, according to these relations, operator L† transforms eigenfunctions of η to eigenfunctions of , and operator L realizes an inverse mapping. Taking into account factorization properties (4.1) and (4.3), we find 4.4 Factor (k2+d2)−1/2 guarantees the normalization of these functions, Thus, using (4.4) and (3.6) one finds the eigenfunctions of , 4.5 Note that these functions satisfy the Dirichlet boundary condition at x=0. One can check that the operator defined on the domain by the differential expression (3.3) is self-adjoint.
Evidently, the functions (4.5) are d-independent and form an orthonormal and complete (in the sense of distributions) basis in , 4.6
Another remarkable property of intertwining operators (4.1), which will be needed below, is the value of the composition 4.7
5. Eigenfunctions of h
First, note that the eigenfunctions φk of H (2.3) may be obtained by applying operator L* (4.7) to the functions 5.1 which yields Therefore, the eigenfunctions Φk of h, obtained by operating with the metric operator η1/2 on the eigenfunctions of H, may also be expressed in terms of the functions ψk: 5.2 Here, the factor [k2+(d+ib)2]−1/2 guarantees the normalization of the functions on the Dirac delta function. Also note that L*ψk(x)∉Dη1/2, but this should not cause trouble because all continuous spectrum eigenfunctions are here generalized eigenfunctions of the corresponding operators and should be understood in the sense of distributions.
It will be useful to express Φk(x) as the result of the action on the functions (4.5) by an integro-differential operator. To this end, first insert the identity operator (3.7) between η1/2 and L in (5.2) and use equation (3.5) to obtain 5.4 and then in the obtained expression replace Ψk(x) according to (4.4): 5.5 Here, the operator L has been moved from the left side in the inner product to the right side, where it becomes adjoint L†, the action of the superposition L†L* has been replaced by its explicit expression (4.7) and formula (5.3) has been used.
An advantage of using the functions in (5.5) with respect to using Ψk in (5.4) is that the integral operator in (5.5) is the positive and Hermitian square root of a resolvent operator and, therefore, it is bounded in whereas the integral operator in (5.4) is unbounded.
6. Asymptotic behaviour of functions Φk(x): scattering matrix and cross section for h
Note that, according to formula (5.5), the eigenfunctions Φk(x) of the Hermitian operator h are expressed in terms of elementary functions (4.5) and (5.1). Nevertheless, no simple explicit expression for these functions exists. Below I calculate their asymptotic behaviour as .
Inserting formulae (4.7) and (5.3) into (5.5) yields 6.1 where Expanding the product of sine functions into the difference of cosine functions reduces the above integral to the one published in Gradshteyn & Ryzhik  (see formula no. 3.754.2), Here K0(z) is the standard modified Bessel function . Accordingly, the integral from (6.1) has two contributions, 6.2 where the first term I1(x) contains the function K0(d|y−x|), and the second term I2(x) is expressed in terms of the function K0(d(y+x)), With the change of the integration variable in the last integral, d(y+x)=t, and letting x tend to infinity, we see that 6.3 Making a similar replacement in the first integral, d(y−x)=t, and also letting , we obtain a non-zero result where With the help of standard trigonometric formulae, we reduce the product of sine and exponential into a sum of four terms, two of which are even and two others are odd with respect to the replacement t→−t. Because of the symmetric integration limits, the odd terms vanish and the integration limits in the integrals with the even terms can be reduced to the semiaxis . As a result, both these terms reduce to the standard integral (see , formula no. 6.671.14) so that 6.4 Now using equations (6.1)–(6.4), we finally obtain where From here the phase shift is found to be 6.5 and the S-matrix is 6.6 This result agrees perfectly with the general formula for the S-matrix obtained in Samsonov .
Note that the scattering matrix 6.7 leads to a Breit–Wigner resonance formula  which in the energy scale reads 6.8 with Γ=4bd and E0=b2−d2. Now assume that |d| is small enough so that b2>d2. Near the resonance, E≈E0 and k≈b, so that equation (6.8) reduces to the celebrated Breit–Wigner formula . From here I conclude that the S-matrix (6.6) is the square root of the Breit–Wigner S-matrix SBW given in (6.7).
The phase shift δ (6.5) corresponding to S (6.6) is one half of δBW, . It leads to a cross section with a square root branch point , 6.9 I choose here that sign of the square root which corresponds to positive definite operator ρ=η1/2 . It is not difficult to see that σ(0)=4πd2/(b2+d2)2>0, , σ(k)>0 for and dσR(0)/dk>0 for . These results mean that, for any fixed value of b and small enough value of |d|, the function σ(k) in (6.9) has a maximum and, therefore, exhibits a resonance behaviour. This is just a consequence of the fact that the Hamiltonian H is, in a sense, close to that which has a spectral singularity.
7. Spectral singularity, d=0
As was discussed in §2, a spectral singularity appears in H only when d=0. Note that the functions Ψk (3.6) as well as the operator η have no singularity at d=0: 7.1 Thus, the operator η is well defined in at the spectral singularity of H, and its eigenfunctions form an orthonormal and complete set in , 7.2 It is apparent that operator ρ, being the positive and Hermitian square root of η, 7.3 is also well defined for d=0. This means that, using this operator, one is able to construct a physical Hilbert space, but this does not mean that the non-Hermitian Hamiltonian H will be mapped to a Hermitian Hamiltonian by a similarity transformation. To illustrate this impossibility, I will calculate the eigenfunctions Φk(x) of h at d=0.
Let us denote 7.4 Then the integro-differential operator (5.5), applied to function ψk(y), yields 7.5 The integral in (7.4) with functions given in (4.5) is standard (see , formula no. 3.741.1) so that the kernel A(x,y) reads 7.6 Further integration in (7.5) with ψk given in (5.1) can also be carried out explicitly if one uses formulae 4.382.1 and 4.382.2 from Gradshteyn & Ryzhik . Finally, after some tedious calculations, assuming, for instance, b>0, one gets where Note that the last term here is undetermined for k=b. I thus conclude that the point k=b cannot belong to the continuous spectrum, nor can it belong to a discrete spectrum. Therefore, an operator that has such eigenfunctions cannot be Hermitian in . This conclusion is also supported by the fact that for k=b the imaginary part of the integral in (7.5) is divergent.
This result is not surprising. Indeed, at d=0 the eigenfunction of H corresponding to k=±b is proportional to . Although η remains invertible on DH, it is not invertible when applied to generalized eigenfunctions of H.
In this paper, I have analysed one of the simplest non-Hermitian Hamiltonians H, which at a specific value of a parameter may possesses a spectral singularity in its continuous spectrum, first proposed by Schwartz . It contains only kinetic energy, but the functions from its domain of definition satisfy a complex boundary condition at x=0. I have shown that the η operator (η=ρ2) is a second-order differential operator with constant coefficients and revealed its SUSY nature. This approach permitted me to express the eigenfunctions Φk(x) of h, where h is Hermitian and related to H by a similarity transformation, in terms of a bounded integral operator defined in the Hilbert space . With the help of this bounded operator, I succeeded in finding asymptotic behaviour of the functions Φk(x) and calculating the scattering matrix and cross section for h. Finally, I have shown that, at the point in the parameter space where H has a spectral singularity, the Hermitian operator h becomes undetermined. Thus, using this specific example I demonstrated that a non-Hermitian Hamiltonian possessing a spectral singularity cannot be mapped to a Hermitian Hamiltonian by any similarity transformation. Nevertheless, the possible presence of a spectral singularity in H may be detected as a resonance in the scattering cross section in h.
One contribution of 17 to a Theme Issue ‘𝒫𝒯 quantum mechanics’.
↵1 An operator A in a Hilbert space is said to be self-adjoint if A=A† where A† is Hermitian adjoint to A. This definition assumes that DA=DA†. A densely defined operator B in a Hilbert space is said to be symmetric if 〈ψ|Bφ〉=〈Bψ|φ〉 for all ψ,φ∈DB. In this paper, I do not differentiate between symmetric and Hermitian operators.
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