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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 716109, 8 pages
Finding of the Metric Operator for a Quasi-Hermitian Model
Department of Mathematics, Karabuk University, 78050 Karabuk, Turkey
Received 18 May 2013; Accepted 12 July 2013
Academic Editor: Stanislav Hencl
Copyright © 2013 Ebru Ergun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider on an appropriate Sobolev space a non-Hermitian Hamiltonian depending on the two complex parameters α and β and having real spectrum. We derive a closed formula for a family of the metric operators, which render the Hamiltonian Hermitian. In some particular cases, we calculate the Hermitian counterpart of .
In the quasi-Hermitian quantum mechanics introduced in  and developed further in [2, 3] and other works, an important problem is to construct a new inner product (scalar product) with respect to which the Hamiltonian (non-Hermitian with respect to the original inner product of the Hilbert space ) having real spectrum turns out to be Hermitian (self-adjoint). In other words, one should find an invertible positive bounded operator with bounded inverse, called metric, which fulfills so that if is a new inner product, then becomes Hermitian with respect to , where denotes the adjoint operator of in the Hilbert space with the original inner product . If such an operator exists, then the operator is called quasi-Hermitian.
As is known [3, 4], to construct a metric operator , satisfying (1) the formula can be used provided that the spectrum of is real and pure discrete (i.e., each point of the spectrum of is an eigenvalue of with finite multiplicity and is an isolated point of the spectrum) and the eigenvectors of form a Riesz basis of space . Here the set is constructed as follows. Let us label the eigenvalues of with , and let denote a Riesz basis of consisting of the eigenvectors of . Then one can construct another Riesz basis of that satisfies where is the Kronecker delta and denotes the identity operator in . The operator defined by (2) is manifestly positive, and it is boundedly invertible, with the inverse given by We can use (2) to introduce the positive-definite inner product and identify the physical Hilbert space with the underlying vector space of endowed with this inner product. Let be the unique positive-definite square root of . Then the operator as a mapping of onto is a unitary operator and the operator is Hermitian in the space and is called a Hermitian counterpart of . Thus, under a similarity transformation implemented by , the non-Hermitian Hamiltonian is equivalent to a Hermitian Hamiltonian , according to (6). The “ physical observables” are the Hermitian operators acting in . We can use the unitary-equivalence of and realized by to construct the physical observables using those of the conventional quantum mechanics, that is, Hermitian operators acting in the reference Hilbert space . This is done according to . Since is a unitary operator as a mapping of onto is a Hermitian operator acting in if and only if is a Hermitian operator acting in .
Since the metric operator is of central importance, many attempts have been made to construct it when given only a non-Hermitian Hamiltonian. In general, the construction of a metric operator is a difficult problem, and therefore most of the available formulas for are just approximative, expressed as leading terms of perturbation series [3, 5]. So far, one has only succeeded to compute exact expressions for the metric and Hermitian counterparts in very few cases of non-Hermitian Hamiltonians (for some of them, see [6–12]).
In the present paper, we deal with the eigenvalue problem where is a spectral parameter, and are fixed nonzero complex numbers, and is a function from the Sobolev space . The space consists of all complex-valued functions differentiable on each of the intervals and with the derivative absolutely continuous on each closed subinterval of the intervals and (therefore there exists the second derivative almost everywhere because any function which is absolutely continuous on an interval is differentiable almost everywhere on that interval) and such that
It can easily be shown (see ) that for any function , there exist the finite limit values , and . The conditions in (8) are called the transition conditions (matching conditions or impulse conditions), while the conditions in (9) are called the boundary conditions.
This paper is organized as follows. In Section 2, we introduce the Hamiltonian corresponding to the eigenvalue problem (7), (8), (9) and formulate the main results of the paper in the form of three theorems. The first two of them determine the regions in the space of the complex parameters and where is (i) Hermitian, (ii) -symmetric, while the third theorem presents a closed formula for a family of the metric operators depending on two arbitrary real positive parameters when is non-Hermitian having real spectrum. In Section 3, the adjoint operator of the Hamiltonian is described. Section 4 provides the spectral analysis of the Hamiltonian and its adjoint calculating explicitly the eigenvalues and eigenfunctions of these operators. Section 5 contains a proof of Theorem 3 deriving a closed formula for a family of the metric operators. In Section 6, we calculate a Hermitian counterpart of in two particular cases. Finally, in Section 7, we make a short conclusion.
2. Main Results
To introduce the Hamiltonian corresponding to the eigenvalue problem (7), (8), and (9), consider the standard Hilbert space with the usual inner product where the bar over a function (or over a number) denotes the complex conjugate. Note that the Sobolev space forms a subspace of .
In the Hilbert space , we consider the Hamiltonian defined as follows. The domain of definition of the operator consists of all functions satisfying the transition conditions (8) and the boundary conditions (9). On the functions , the Hamiltonian is given by
Let and be the parity and the time reversal operators defined by The Hamiltonian is called -symmetric if for every , we have that and
Main results of the present paper can be formulated as the following theorems.
Theorem 1. The operator is Hermitian if and only if .
Theorem 2. The operator is -symmetric if and only if and .
Theorem 3. Assume that the complex numbers and are nonzero. If , then any complex number is an eigenvalue of the operator , but if , then the spectrum of is real and consists of the simple eigenvalues Furthermore, let and be a linear operator defined on by where and are arbitrary positive real numbers. Then is an invertible positive bounded operator with bounded inverse and satisfies so that the Hamiltonian is quasi-Hermitian in this case and is a metric operator for it.
3. The Adjoint of the Hamiltonian
Theorem 4. For any nonzero complex numbers and such that , the relation holds. Therefore, the domain of definition of consists of all functions satisfying the transition conditions and the boundary conditions and on such functions ,
Proof. Recall that by definition of the adjoint operator, the domain of definition of consists of those functions in such that
for all in and some in depending on . Since is dense in is uniquely determined by and, by definition, .
The inclusion is easy to see. Indeed, take any . Hence, and satisfies conditions (20) and (21). Then for every we have, integrating by parts twice and taking into account that satisfies (8) and (9), The sum of terms outside the integrals became zero because of the fact that satisfies (20) and (21). This shows that belongs to (therefore (24) holds), and on such elements , we have equality (22). Note that in this part, we did not use the condition .
To complete the proof of the theorem, where this is the major part of the proof, we should show that the opposite of (24) also holds and that on any we have equality (22). This can be done similarly to that given in  in the proof of Proposition 4.
Applying Theorem 4 to the operator , we get that . Therefore, for any nonzero complex numbers and with , the operator is closed because, as is known (and it can easily be seen), the adjoint of any densely defined linear operator is a closed operator.
4. Spectral Analysis of the Hamiltonian and Its Adjoint
In this section, we study the spectrum of the operator .
Theorem 5. Let and be nonzero complex numbers. Then, one has the following. If , then the spectrum of the operator is real and consists of the (simple) eigenvalues The corresponding eigenfunctions are found by for , where are nonzero complex constants which may depend on and . If , then any complex number is an eigenvalue of the operator with the corresponding eigenfunction where is an arbitrary nonzero complex constant which may depend on .
From Theorems 4 and 5, it follows that if the spectrum of the adjoint operator coincides with the set of eigenvalues given in (26), and the corresponding eigenfunctions are given by for , where are nonzero complex constants which may depend on and .
From it follows, by a standard way, that and are orthogonal to each other provided . We can normalize the eigenfunctions appropriately to get where is the Kronecker delta. Namely, taking into account it is easy to see that (31) follows by choosing the coefficients and in (27) and (29) according to the equations These equations can obviously be satisfied. We choose the coefficient by the requirement where and are arbitrary nonzero complex numbers, determining by the equations in (33): Then we have for , where Note that forms a set of orthonormal in eigenfunctions of the Hermitian Sturm-Liouville eigenvalue problem and forms a set of orthonormal in eigenfunctions of the Hermitian Sturm-Liouville eigenvalue problem
As is well known, the sets and form complete orthonormal families (orthonormal bases) in the Hilbert space . In particular, we have the expansions and the Parseval equalities for every .
Let us calculate the resolvent of the operator , where and are nonzero complex numbers with . To this end, we introduce the solutions and of (7) satisfying the transition conditions (8) and the initial conditions respectively. Note that satisfies the boundary condition in (9) at and at . Putting , we can find these solutions explicitly as follows: The Wronskian of these solutions is Therefore, by the standard Green function approach, the resolvent of the operator has the form, for , where the function is defined for by the formula It follows from the explicit form of that Therefore, the resolvent of is a compact operator (moreover, a Hilbert-Schmidt operator). This implies that the spectrum of is purely discrete and consists of the eigenvalues .
5. Proof of Theorem 3
To use formula (2) for constructing a metric operator for our Hamiltonian , we have to show that the set of eigenfunctions of forms a Riesz basis of the Hilbert space . For this purpose, we will use the following result (see [13, Chapter 6, Theorem 2.1]). Suppose a sequence is complete in the Hilbert space , there corresponds to it a complete biorthogonal sequence , and for any , one has Then the sequence forms a Riesz basis of the space .
To prove the completeness in of the functions defined by (27), assume that an element is orthogonal to all the functions so that we have for all . We have to show that then . From (52), we get for all . Since the sets and form complete orthogonal families (orthogonal bases) in the Hilbert space as the sets of eigenfunctions of the Hermitian Sturm-Liouville eigenvalue problems (41) and (42), respectively, we conclude from the last equations that Hence by the condition , we get . Therefore, , and the desired completeness is shown.
Now, we are in a position to use formula (2) for calculating the metric operator for the Hamiltonian . Using formulas (38) and (39) for , we find that for any , Hence, if , then and if , then Next, using the expansion formulas given in (43) and taking into account that , we have the following.
6. The Hermitian Counterpart of a Non-Hermitian Hamiltonian
If for complex numbers and we have , and , then by Theorem 3, for the Hamiltonian , we have a family of metric operators defined by (17) and depending on two arbitrary positive real parameters and . Each concrete choice of the metric operator determines a Hermitian counterpart of . In this section, we will calculate in two particular cases.
Assume that where is a positive real number.
Obviously, if is a metric operator for , then any positive scalar multiple of is also a metric operator for the same Hamiltonian . Therefore, dividing both sides of (17) by and setting , we get, due to condition (60), that the operator defined by is a metric operator for the Hamiltonian . The inner product determined by this metric operator has the form
It follows that the operator is given by and its inverse by We can now calculate the operator The domain of definition of the operator consists of all functions such that . Therefore consists of all functions satisfying the transition conditions and the boundary conditions Moreover, (65) implies by (63) and (64) that if , then We see that Since is Hermitian by Theorem 1.
As a second example, consider the case where and are arbitrary positive real numbers. By Theorem 1, the Hamiltonian is non-Hermitian if . Taking in (17), we find that the operator defined by is a metric operator for the Hamiltonian . The inner product determined by this metric operator has the form Further reasoning as in the case of the previous example, we find that the operator is a Hermitian counterpart of the Hamiltonian .
An axiom of the classical quantum mechanics dictates that physical observables and in particular the Hamiltonian must be Hermitian (self-adjoint) because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). However, during the past two decades, it was understood that non-Hermitian Hamiltonians having real spectrum can also be used in the quantum mechanics. For this purpose, one should construct a new inner product (scalar product) with respect to which the Hamiltonian, non-Hermitian with respect to the original inner product of the Hilbert space, turns out to be Hermitian. In this paper, for a class of concrete non-Hermitian Hamiltonians with real spectrum, we have constructed explicitly such inner products.
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