Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians
For a large class of time-dependent non-Hermitain Hamiltonians expressed in terms linear and bilinear combinations of the generators for an Euclidean Lie-algebra respecting different types of PT-symmetries, we find explicit solutions to the time-dependent Dyson equation. A specific Hermitian model with explicit time-dependence is analyzed further and shown to be quasi-exactly solvable. Technically we constructed the Lewis-Riesenfeld invariants making use of the metric picture, which is an equivalent alternative to the Schrödinger, Heisenberg and interaction picture containing the time-dependence in the metric operator that relates the time-dependent Hermitian Hamiltonian to a static non-Hermitian Hamiltonian.
Quasi-exactly solvable (QES) quantum systems are characterized by the feature that only part of their infinite energy spectrum and corresponding eigenfunctions can be calculated analytically. Systematic studies of such type of systems have been carried out by casting them into the form of Lie algebraic quantities [1, 2] and making use of the property that the eigenfunctions of the corresponding Hamiltonian systems form a flag which coincides with the finite dimensional representation space of the associated Lie algebras. QES systems that can be cast into such a form are usually referred to as QES models of Lie algebraic type [3, 4]. The relevant underlying algebras are either of -type, with their compact and non-compact real forms and , respectively , or of Euclidean Lie algebras type [6, 7, 8]. The latter class was found to be particularly useful when dealing with certain types of non-Hermitian systems.
While many QES models have been studied in stationary settings, little is known for time-dependent systems. So far a time-dependence has only been introduced into the eigenfunctions in form of a dynamical phase [9, 10]. However, no QES systems with explicitly time-dependent Hamiltonians have been considered up to now. The main purpose of this article is to demonstrate how they can be dealt with and to initiate further studies of such type of systems. We provide the analytical solutions to a QES Hamiltonian quantum system with explicit time-dependence. As a concrete example we consider QES systems of -Lie algebraic type. Technically we make use of the metric picture [11, 12], which is an alternative to the Schrödinger, Heisenberg and interaction picture. It will allow us to solve a Hermitian time-dependent Hamiltonian system by solving first a static non-Hermitian system as an auxiliary problem with a time-dependence in the metric operator.
The Hermitian Hamiltonian systems we study here are of the general form
where the time-dependent coefficient functions , , are real and , and denote the three generators that span the Euclidean-algebra . They obey the commutation relations
Considering here only Hermitian representations with , and , the Hamiltonian in equation (1) is clearly Hermitian. Standard representation are for instance the trigonometric representation , and or a two-dimensional representation , or with , , , denoting Heisenberg canonical variables with non-vanishing commutators . We have set here and mostly in what follows to .
We briefly recall from [11, 12] what is meant by the metric picture. It is well known that the Schrödinger and the Heisenberg picture are equivalent with the former containing the time-dependence entirely in the states and the latter entirely in the operators. -symmetric/quasi-Hermitian systems [13, 14, 15] allow for yet another equivalent variant in which the time-dependence is contained entirely in the metric operator. In order to see that we first need to solve the time-dependent Dyson relation [16, 17, 18, 19, 20, 11, 12, 21, 22, 23] which in general reads
involving a time-dependent non-Hermitian Hamiltonian and the Dyson operator related to the metric operator as . For our purposes we will eventually take the Hamiltonian to be time-independent , with satisfying the time-dependent Schrödinger equation and the time-independent Schrödinger equation with energy eigenvalue . The corresponding wavefunctions are related as .
Before we solve a concrete system in a quasi-exactly solvable fashion we consider first the fully time-dependent Dyson relation with time-dependent non-Hermitian Hamiltonian and investigate which type of Hamiltonians can be related to the Hermitian Hamiltonian in (1). We will see that in some cases we are even forced to take or part of it to be time-independent. As not many explicit solutions to the time-dependent Dyson relation are known, this will be a valuable result in itself.
Our manuscript is organized as follows: In section 2 we explore various types of -symmetries that leave the Euclidean -algebra invariant and investigate time-dependent non-Hermitian Hamiltonians in terms -algebraic generators that respect these symmetries. We find new solutions to the time-dependent Dyson relation for those type of Hamiltonians by computing the corresponding Hermitian Hamiltonians and the Dyson map. In section 3 we provide analytical solutions for a concrete model respecting a particular -symmetry. We compute the eigenstates of the Lewis-Riesenfeld invariants and the time-dependent Hermitian Hamiltonian in a quasi-exactly solvable fashion. A three-level system is presented in more detail. Our conclusions are stated in section 4.
2 Solutions to the time-dependent Dyson equation for -Hamiltonians
A key property in the study and classification of Hamiltonian systems related to the -algebra are the antilinear symmetries  that leave the algebra (2) invariant. Given the general context of -symmetric/quasi-Hermitian systems we call these symmetries As discussed in more detail in [25, 26], there are many options which all give rise to models with qualitatively quite distinct features. It is easy to see that each of the following antilinear maps leave all the commutation relations (2) invariant
Next we seek non-Hermitian Hamiltonians that respect either of these symmetries. Focussing here on time-dependent Hamiltonians consisting entirely of linear and bilinear combinations of -generators they can all be cast into the general form
Demanding that , the symmetries are implemented by taking the coefficient functions to be either real, purely imaginary or relate different functions to each other by conjugation. For the different symmetries in (4) we are forced to take
Except for very specific combinations of the coefficient functions, the Hamiltonians are non-Hermitian in general.
We now solve the time-dependent Dyson relation (3) for by mapping different -symmetric versions of to a Hermitian Hamiltonian of the form (1). For the time-dependent Dyson map we make an Ansatz in terms of all the -generators
At this point we allow , keeping in mind that does not have to be Hermitian. We exclude here unitary operators, i.e. , as in that case just becomes a gauge transformation. The adjoint action of this operator on the -generators is computed by using the standard Baker-Campbell-Haussdorff formula
The gauge-like term in (3) acquires the form
As common, we abbreviate here time-derivatives by overdots. For the computation of the time-dependent energy operator , see below, we also require the term
Using (8)-(10) we calculate next the adjoint action of on and add the expression in (11). Demanding that the result is Hermitian will constrain the time-dependent functions , , and . We need to treat each -symmetry separately.
2.1 Time-dependent -invariant Hamiltonians
For convenience we take the coefficient function to be time-independent. For the -invariant Hamiltonian with coefficient functions as specified in (6) we have to be aware that for the Hamiltonian becomes Hermitian. Substituting the general form for into (3), using (8)-(10), (11), reading off the coefficients in front of the generators and demanding that the right hand side becomes Hermitian enforces to take the functions in (7). The resulting Hermitian Hamiltonian is
with 7 constraining relations
Thus from the original 12 free parameters, i.e. the 9 coefficient functions and the 3 functions in the Dyson map, we can still freely choose 5. In comparison with the other -symmetries, this is the most constrained case. We also note that this system is the only one in which all three functions in the Dyson map are constrained when we take the coefficient functions as primary quantities.
2.2 Time-dependent -invariant Hamiltonians
The Hamiltonian becomes Hermitian for , , , but is non-Hermitian otherwise. Preceding as in the previous section the implementation of (3) enforces to take and in (7), which makes the Dyson map -symmetric. The Hermitian Hamiltonian is computed to
with 5 constraining relations
We note that we have less constraints as in the previous section, but some of the coefficient functions can no longer be taken to be time-dependent and one even has to vanish. One of the three functions in the Dyson map, e.g. , can be freely chosen. Compared to the other cases this is the only one for which has the same -symmetry as the corresponding non-Hermitian Hamiltonian when taking the constraints on into account.
2.3 Time-dependent -invariant Hamiltonians
The Hamiltonian becomes Hermitian for and . Using the same arguments as above, we are forced to take and in (7). The Hermitian Hamiltonian is computed to
with 5 constraining relations
Once again one of the coefficient functions has to be time-independent and one of the three functions in the Dyson map can be chosen freely.
2.4 Time-dependent -invariant Hamiltonians
The Hamiltonian becomes Hermitian for and . By the same reasoning as above we have to take and in (7). The Hermitian Hamiltonian results to to
with 5 constraining relations
This case is similar to the previous one with one of the coefficient functions forced to be time-independent and one of the three functions in the Dyson map being freely choosable.
2.5 Time-dependent -invariant Hamiltonians
The Hamiltonian becomes Hermitian for and . Here we have to take and in (7). The Hermitian Hamiltonian is computed to
with only 4 constraining relations
In comparison with the other symmetries, this is the least constraint case. From the three functions in the Dyson map only one is constraint and the others can be chosen freely. However, one of the coefficient functions needs to be time-independent.
3 Time-dependent quasi-exactly solvable systems
We will now specify one particular model and show how it can be quasi-exactly solved in the metric picture. Since the symmetry appears to be somewhat special, in the sense that it is the only case for which the Dyson map respects the same symmetry as the Hamiltonian, we consider a particular non-Hermitian -symmetric time-independent Hamiltonian of the form
Given the constraining equations (16), we could in principle take , to be time dependent, but to enforce the metric picture we take here all four coefficients , , and to be time-independent real constants. According to the analysis in section 2.2, the time-dependent Dyson map
with , maps the time-independent non-Hermitian Hamiltonian to the time-dependent Hermitian Hamiltonian
Here we are free to chose the time-dependent function . As previously pointed out for non-Hermitian systems with time-dependent metric, one needs to distinguish between the Hamiltonian, that is a non-observable operator, and the observable energy operator. This feature remains also true when the non-Hermitian Hamiltonian is time-independent, but the metric is dependent on time. In reverse, it simply means that when one identifies the non-Hermitian Hamiltonian with the energy operator one has made the choice for the metric to be time-independent. With as specified in (24), the energy operator is computed with the help of (12) to
We note that is also -symmetric when we include into the symmetry transformation. In order to demonstrate that this system is quasi-exactly solvable we specify the constants in the Hamiltonian (23) further to , , , so that it becomes
This Hamiltonian can be obtained from one discussed in  by transforming , in the trigonometric representation. The constants in are chosen so that it exhibits an interesting double scaling limit when assuming that . In the trigonometric representation this limiting Hamiltonian is the Mathieu Hamiltonian.
The Hermitian Hamiltonian (3) simplifies in this case to
where we denoted the Casimir operator by and abbreviated . In the aforementioned double scaling limit we obtain a time-dependent Hamiltonian of the form .
3.1 Quasi-exactly solvable Lewis-Riesenfeld invariants
The most efficient way to solve the time-dependent Dyson equation (3) is to use the Lewis-Riesenfeld approach  and compute at first the respective time-dependent invariants and for the Hamiltonian and , see [28, 29, 22], by solving the equations
Unlike the corresponding Hamiltonians that have to obey (3), the invariants are related by a similarity transformation
Computing the eigenstates of the invariants
the solutions to the time-dependent Schrödinger equations for , are simply related by a phase factor to the eigenstates of the invariants , . It is easy to to derive that the two phase factors have to be identical . They can be determined from
Taking now to be time-independent, we may assume with being some constant. The Lewis-Riesenfeld then just becomes a dynamical phase factor
such that .
3.1.1 The quasi-exactly solvable symmetry operator
We make a general Ansatz for the invariant of of the form
with unknown constants . The invariant for the time-independent system is of course just a symmetry and we only need to compute the commutator of with to determine the coefficients in (35). We find the most general symmetry or invariant to be
where in the last equation we have taken . Since the last term only produces an overall shift in the spectrum we set for convenience.
Next we compute the eigensystem for by solving (32). Assuming the two linear independent eigenfunctions to be of the general forms
with constants where denotes the Pochhammer symbol. The ground state is taken to be -symmetric. The constants are chosen conveniently to ensure the simplicity of the polynomials , in the eigenvalues . We then find that the functions and satisfy the eigenvalue equation provided the coefficient functions and obey the three-term recurrence relations
The well-known and crucial feature responsible for a system to be quasi-exactly solvable is the occurrence of the three-term recurrence relations and that they can be forced to terminate at certain values of . This is indeed the case and for our relations (40), (42) and can be achieved for some specific values or , respectively. To see this we take and note that the polynomials and factorize for , as
where the first -polynomials are
Since according to (45) the polynomials and are factor in all and for and , respectively, all higher order polynomial vanish when setting . These latter constraints are the quantization conditions for . Thus setting at the different levels , we find the real eigenvalues
with , and from we find the real eigenvalues