Metrics and isospectral partners for the most generic cubic -symmetric non-Hermitian Hamiltonian
We investigate properties of the most general -symmetric non-Hermitian Hamiltonian of cubic order in the annihilation and creation operators as a ten parameter family. For various choices of the parameters we systematically construct an exact expression for a metric operator and an isospectral Hermitian counterpart in the same similarity class by exploiting the isomorphism between operator and Moyal products. We elaborate on the subtleties of this approach. For special choices of the ten parameters the Hamiltonian reduces to various models previously studied, such as to the complex cubic potential, the so-called Swanson Hamiltonian or the transformed version of the from below unbounded quartic -potential. In addition, it also reduces to various models not considered in the present context, namely the single site lattice Reggeon model and a transformed version of the massive sextic -potential, which plays an important role as a toy model to identify theories with vanishing cosmological constant.
Non-Hermitian Hamiltonians are usually interpreted as effective Hamiltonians associated with dissipative systems when they possess a complex eigenvalue spectrum. However, from time to time also non-Hermitian Hamiltonians whose spectra were believed to be real have emerged sporadically in the literature, e.g. the lattice version of Reggeon field theory [1, 2]. Restricting this model to a single site leads to a potential very similar to the complex cubic potential . Somewhat later it was found  for the latter model that it possess a real spectrum on the real line. More recently the surprising discovery was made  that in fact the entire infinite family of non-Hermitian Hamiltonians involving the complex potentials for possess a real spectrum, when its domain is appropriately continued to the complex plane.
Thereafter it was understood [4, 5] that the reality of the spectra can be explained by an unbroken -symmetry, that is invariance of the Hamiltonian and its eigenfunctions under a simultaneous parity transformation and time reversal . In case only the Hamiltonian is -symmetric the eigenvalues occur in complex conjugate pairs. In fact, the -operator is a specific example of an anti-linear operator for which such spectral properties have been established in a generic manner a long time ago by Wigner . However, in practical terms one is usually not in a position to know all eigenfunctions for a given non-Hermitian Hamiltoinian and therefore one has to resort to other methods to establish the reality of the spectrum. Since Hermitian Hamiltonians are guaranteed to have real spectra, one obvious method is to search for Hermitian counterparts in the same similarity class as the non-Hermitian one. This means one seeks similarity transformations of the form
Non-Hermitian Hamiltonians respecting the property (1) are referred to as pseudo-Hermitian . Besides these spectral properties it is also understood how to formulate a consistent quantum mechanical description for such non-Hermitian Hamiltonian systems [8, 9, 5] by demanding the -operator to be Hermitian and positive-definite, such that it can be interpreted as a metric to define the -inner product. A special case of this is the -inner product , which results by taking with . For some recent reviews on pseudo Hermitian Hamiltonians see [10, 11, 12, 13, 14].
Since the metric-operator is of central importance many attempts have been made to construct it when given only a non-Hermitian Hamiltonian. However, so far one has only succeeded to compute exact expressions for the metric and isospectral partners in very few cases. Of course when the entire spectrum is known this task is straightforward, even though one might not always succeed to carry out the sum over all eigenfunctions. However, this is a very special setting as even in the most simple cases one usually does not have all the eigenfunctions at ones disposal and one has to resort to more pragmatic techniques, such as for instance perturbation theory [15, 16, 17, 18]. Rather than solving equations for operators, the entire problem simplifies considerably if one converts it into differential equations using Moyal products [19, 20, 21] or other types of techniques . Here we wish to pursue the former method for the most generic -symmetric non-Hermitian Hamiltonian of cubic order in the creation and annihilation operators.
We refer models for which the metric can be constructed exactly as solvable pseudo-Hermitian (SPH) systems.
Our manuscript is organised as follows: In section 2 we introduce the model we wish to investigate in this manuscript, formulating it in terms of creation and annihilation operators and equivalently in terms of space and momentum operators. We comment on the reduction of the model to models previously studied. In section 3 we discuss in detail the method we are going to employ to solve the equations (1), namely to exploit the isomorphism between products of operator valued functions and Moyal products of scalar functions. In section 4 we construct systematically various exact solutions for the metric operator and the Hermitian counterpart to . As special cases of these general considerations we focus in section 5 and 6 on the single site lattice Reggeon model and the massive -potential. In section 7 we provide a simple proof of the reality for the -potentials and some of its generalizations. We state our conclusions in section 8.
2 A master Hamiltonian of cubic order
The subject of our investigation is the most general -symmetric Hamiltonian, which is maximally cubic in creation and annihilation operators , respectively,
The Hamiltonian is a ten-parameter family with . It is clear that this Hamiltonian is -symmetric by employing the usual identification and with the operators in -space and . The effect of a simultaneous parity transformation and time reversal , on the creation and annihilation operators is , . Without loss of generality we may set the parameter to one in the following as it is simply an overall energy scale.
In terms of the operators and the separation into a Hermitian and non-Hermitian part is somewhat more transparent and we may introduce in addition a coupling constant in order to be able to treat the imaginary part as perturbation of a Hermitian operator. In terms of these operators the most general expression is, as to be expected, yet again a ten-parameter family
Table 1: Special reductions of the Hamiltonian The map is defined in equation (73) and are coupling constants of the models. We abbreviated and .
We have symmetrized in terms which contain and by introducing anticommutators, i.e. . This allows us to separate off conveniently the real and imaginary parts of by defining
with and . In addition, the symmetrized version (3) will lead to very simple expressions when we convert products of operator valued functions into expressions involving scalar functions multiplied via Moyal products. For our definition of the Moyal product it implies that the parameters do not need to be re-defined. Depending on the context, one (2) or the other (3) formulation is more advantageous. Whereas the usage of creation and annihilation operators is more proned for an algebraic generalization, see e.g. , the formulation in terms of operators and is more suitable for a treatment with Moyal brackets. The relation between the two versions is easily computed from the aforementioned identifications between the , and ,via the relations and with being a -matrix. Below we will impose some constraints on the coefficients and it is therefore useful to have an explicit expression for at our disposal in order to see how these constraints affect the expression for the Hamiltonian in (2). We compute the matrix
and the inverse
which also exists for since .
The Hamiltonian encompasses many models and for specific choices of some of the it reduces to various well studied examples, such as the simple massive ix-potential  or its massless version, the so-called Swanson Hamiltonian [25, 26, 27, 21, 28], the complex cubic potential together with his massive version  and also the transformed version of the -potential . As we will show below, in addition it includes several interesting new models, such as the single site lattice version of Reggeon field theory , which is a thirty year old model but has not been considered in the current context and the transformed version of the - potential, which serves as a toy model to identify theories with vanishing cosmological constant . The latter models have not been solved so far with regard to their metric operators and isospectral partners. Besides these models, also includes many new models not considered so far, some of which are even SPH.
To enable easy reference we summarize the various choices in table 1.
Most SPH-models which have been constructed so far are rather trivial, such as the massive -potential or the so-called Swanson Hamiltonian. The latter model can be obtained simply from the standard harmonic oscillator by means of a Bogolyubov transformation and a subsequent similarity transformation, which is bilinear in and . Beyond these maximally quadratic models, the complex cubic potential was the first model which has been studied in more detail. Unfortunately so far it can only be treated perturbatively. The transformed version of the from below unbounded -z-potential is the first SPH-model containing at least one cubic term. Here we enlarge this class of models. As a special case we shall also investigate the single site lattice version of Reggeon field theory  in more detail. Before treating these specific models let us investigate first the Hamiltonian in a very generic manner.
Our objective is to solve equation (1) and find an exact expression for the positive-definite metric operator , subsequently to solve for the similarity transformation and construct Hermitian isospectral partner Hamiltonians.
3 Pseudo-Hermitian Hamiltonians from Moyal products
Taking solely a non-Hermitian Hamiltonian as a starting point, there is of course not a one-to-one correspondence to one specific Hermitian Hamiltonian counterpart. The conjugation relation in (1) admits obviously a whole family of solutions. In order to construct these solutions we will not use commutation relations involving operators, but instead we will exploit the isomorphism between operator valued function in and and scalar functions multiplied by Moyal products in monomial of scalars and . We associate to two arbitrary operator valued functions and two scalar functions , such that
The Moyal product is a distributive and associative map obeying the same Hermiticity properties as the operator valued functions on the right hand side of (7), that is . Following standard arguments we provide now explicit representations for the and . We may formally Fourier expand an arbitrary operator valued functions and scalar functions as
respectively. In terms of this representation the multiplication of two operator valued functions yields
which follows using the identities and . Is now straightforward to verify that the definition of the Moyal product (8) guarantees that the isomorphism (7) holds, since yields formally the same expression as (10) with replaced by .
The Hermiticity property is important for our purposes. We find that
This is easily seen by computing using the representation (9). Then this function is Hermitian if and only if the kernel satisfies , which in turn implies that is real. Positive definiteness of an operator valued function is guaranteed if the logarithm of the operator is Hermitian, that is we need to ensure that is real. Furthermore, it is easy to see that is -symmetric if and only if .
As an instructive example we consider for which we compute the corresponding kernel as . From this it is easy to see that is -symmetric, since satisfies .
In the present context of studying non-Hermitian Hamiltonians this technique of exploiting the isomorphism between Moyal products and operator products has been exploited by Scholtz and Geyer [19, 20], who reproduced some previously known results and also in , where new solutions were constructed. In [19, 20] a more asymmetrical definition than (8) of the Moyal product was employed, i.e. . In comparison with (8) this definition leads to some rather unappealing properties: i) the loss of the useful and natural Hermiticity relation, i.e. , ii) the right hand side of the isomorphism in (11) is replaced by the less transparent expression and iii) in  it was shown that the definition leads to more complicated differential equations than the definition . The representation for the operator valued functions , which satisfies the properties resulting from the definition differs from (9) by replacing in the Fourier expansion.
3.2 Construction of the metric operator and isospectral partners
We briefly recapitulate the main steps of the procedure [19, 20, 21] of how to find for a given non-Hermitian Hamiltonian a metric operator , a similarity transformation and an Hermitian counterpart using Moyal products. First of all we need to solve the right hand side of the isomorphism
for the “scalar metric function” . Taking as a starting point the non-Hermitian Hamiltonian , we have to transform this expression into a scalar function by replacing all occurring operator products with Moyal products. We can use this expression to evaluate the right hand side of the isomorphism of (12), which is a differential equation for whose order is governed by the highest powers of and in . Subsequently we may replace the function by the metric operator using the isomorphism (7) now in reverse from the right to the left. Thereafter we solve the differential equation for . Inverting this expression we obtain , such that we are equipped to compute directly the scalar function associated to the Hermitian counterpart by evaluating
Finally we have to convert the function into the operator valued function and the “Hermitian scalar function” into the Hamiltonian counterpart , once more by solving (7) from the right to the left.
So far we did not comment on whether the metric is a meaningful Hermitian and positive operator. According to the isomorphism (11) we simply have to verify that , and are real functions in order to establish that the corresponding operator valued functions , and are Hermitian. We may establish positive definiteness of these operators by verifying that their logarithms are real.
3.3 Ambiguities in the solution
Obviously when having a non-Hermitian Hamiltonian as the sole starting point there is not a unique Hermitian counterpart in the same similarity class associated to the adjoint action of one unique operator . Consequently also the metric operator is not unique. The latter was pointed out for instance in  and exemplified in detail for the concrete example of the so-called Swanson Hamiltonian in . In fact, it is trivial to see that any two non-equivalent metric operators, say and , can be used to construct a non-unitary symmetry operator for the non-Hermitian Hamiltonian
We may solve (14) and express one metric in terms of the other as
Thus we encounter here an infinite amount of new solutions. Likewise this ambiguity can be related to the non-equivalent Hermitian counterparts
with symmetry operators and . When and we obviously also have and . The expression for the symmetry operator for was also identified in .
There are various ways to select a unique solution. One possibility  is to specify one more observable in the non-Hermitian system. However, this argument is very impractical as one does not know a priori which variables constitute observables.
4 SPH-models of cubic order
Let us study by converting it first into a scalar function . Most terms are non problematic and we can simply substitute , but according to our definition of the Moyal bracket (8) we have to replace , , etc. Replacing all operator products in this way we convert the Hamiltonian in (3) into the scalar function
for the “metric scalar function” . There are various simplifications one can make at this stage. First of all we could assume that either or is an observable in the non-Hermitian system, such that does not depend on or , respectively. As pointed out before it is not clear at this stage if any of these choices is consistent. However, any particular choice or will be vindicated if (4) can be solved subsequently for or , respectively. Here we will assume that admits a perturbative expansion. Making a very generic exponential -symmetric ansatz, which is real and cubic in its argument for , we construct systematically all exact solutions of this form. Substituting the ansatz into the differential equation (4) and reading off the coefficients in front of each monomial in and yields at each order in ten equations. by solving these equations we find five qualitatively different types of exact solutions characterized by vanishing coefficients and some additional constraints. We will now present these solutions.
4.1 Non-vanishing -term
4.1.1 Constraints 1
We consider the full Hamiltonian in (17) and impose as the only constraint that the -term does not vanish, i.e. . For this situation we can solve the differential equation (4) exactly to all orders in perturbation theory for
where we imposed the additional constraints
Since is real it follows from (11) that the corresponding metric operator is Hermitian. Next we solve for . Up to order we find
The corresponding Hermitian counterpart corresponding to this solution is computed by means of (13) to
Notice that since we demanded to be non-vanishing these solutions can not be reduced to any of the well studied models presented in table 1, but represent new types of solutions. We may simplify the above Hamiltonians by setting various s to zero.
Since only depends on in this case, we can compute exactly . The Hermitian counterpart results to
4.1.2 Constraints 2
In the construction of the previous solution some coefficients had to satisfy a quadratic equations in the parameters to guarantee the vanishing of the perturbative expansion. The other solution for this equation leads to the constraints , such that the non-Hermitian Hamiltonian simplifies. If we now impose the additional constraints
we can solve the differential equation (4) exactly. For
we compute the exact scalar metric function to
Clearly is a Hermitian and positive definite operator, which follows from the facts that and are real, respectively. Notice the fact that the Hamiltonian (27) does not follow as a specialization of (19), since the constraints (26) do not result as a particular case of (20). The Hermitian Hamiltonian counterpart corresponding to (27) is computed with by means of (13) to
Once again we may simplify the above Hamiltonians by setting various s to zero or other special values, except for the case for which the constraints (26) reduce the non-Hermitian part of the Hamiltonian (27) to zero.
Thus this case requires a separate consideration:
4.2 Non-vanishing -term and vanishing -term
Let us therefore embark on the treatment of the complementary case to the previous subsection, namely and . For these constraints we can solve the differential equation (4) exactly for the Hamiltonian
when we impose one additional constraint
The “metric scalar function” results to
Once again is a Hermitian and positive definite operator, which follows again from the facts that and are real. Since only depends on , we can simply take the square root to compute . Then the corresponding Hermitian counterpart is computed by means of (13) to
In fact we can implement the constraint (31) directly in the solution. The function