Finding of the Metric Operator for a Quasi-Hermitian Model

Abstract

We consider on an appropriate Sobolev space a non-Hermitian Hamiltonian 𝐻_𝛼,𝛽 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 𝐻_𝛼,𝛽 Hermitian. In some particular cases, we calculate the Hermitian counterpart of 𝐻_𝛼,𝛽.

1. Introduction

In the quasi-Hermitian quantum mechanics introduced in [1] 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 𝐻 (non-Hermitian with respect to the original inner product of the Hilbert space ℋ) having real spectrum turns out to be Hermitian (self-adjoint). In other words, one should find an invertible positive bounded operator 𝜂₊ with bounded inverse, called metric, which fulfills 𝐻^∗𝜂₊=𝜂₊𝐻i.e.,𝐻^∗=𝜂₊𝐻𝜂₊⁻¹ (1) so that if ⟨⋅,⋅⟩₊=⟨𝜂₊⋅,⋅⟩  is a new inner product, then 𝐻 becomes Hermitian with respect to ⟨⋅,⋅⟩₊, where 𝐻^∗ denotes the adjoint operator of 𝐻 in the Hilbert space ℋ with the original inner product ⟨⋅,⋅⟩. If such an operator 𝜂₊ exists, then the operator 𝐻 is called quasi-Hermitian.

As is known [3, 4], to construct a metric operator 𝜂₊, satisfying (1) the formula 𝜂₊=^∞_𝑛=1⋅,𝜙_𝑛𝜙_𝑛 (2) can be used provided that the spectrum of 𝐻 is real and pure discrete (i.e., each point 𝐸 of the spectrum of 𝐻 is an eigenvalue of 𝐻 with finite multiplicity and 𝐸 is an isolated point of the spectrum) and the eigenvectors of 𝐻 form a Riesz basis of space ℋ. Here the set {𝜙_𝑛}^∞_𝑛=1 is constructed as follows. Let us label the eigenvalues of 𝐻 with 𝐸_𝑛, and let {𝜓_𝑛}^∞_𝑛=1 denote a Riesz basis of ℋ consisting of the eigenvectors 𝜓_𝑛 of 𝐻,𝐻𝜓_𝑛=𝐸_𝑛𝜓_𝑛. Then one can construct another Riesz basis {𝜙_𝑛}^∞_𝑛=1 of ℋ that satisfies 𝐻^∗𝜙_𝑛=𝐸_𝑛𝜙_𝑛,𝜓_𝑛,𝜙_𝑚=𝛿_𝑛𝑚,^∞_𝑛=1⋅,𝜙_𝑛𝜓_𝑛=𝐼, (3) where 𝛿_𝑛𝑚 is the Kronecker delta and 𝐼 denotes the identity operator in ℋ. The operator 𝜂₊ defined by (2) is manifestly positive, and it is boundedly invertible, with the inverse given by 𝜂₊⁻¹=^∞_𝑛=1⋅,𝜓_𝑛𝜓_𝑛. (4) We can use (2) to introduce the positive-definite inner product ⟨⋅,⋅⟩₊=𝜂₊⋅,⋅ (5) and identify the physical Hilbert space ℋ_phys with the underlying vector space of ℋ endowed with this inner product. Let 𝜌=√𝜂₊ be the unique positive-definite square root of 𝜂₊. Then the operator 𝜌 as a mapping of ℋ_phys onto ℋ is a unitary operator and the operator ℎ=𝜌𝐻𝜌⁻¹ (6) is Hermitian in the space ℋ and is called a Hermitian counterpart of 𝐻. Thus, under a similarity transformation implemented by 𝜌=√𝜂₊, the non-Hermitian Hamiltonian 𝐻 is equivalent to a Hermitian Hamiltonian ℎ, according to (6). The “ physical observables” are the Hermitian operators 𝑂 acting in ℋ_phys. We can use the unitary-equivalence of ℋ_phys and ℋ realized by 𝜌 to construct the physical observables 𝑂 using those of the conventional quantum mechanics, that is, Hermitian operators 𝑜 acting in the reference Hilbert space ℋ. This is done according to 𝑂=𝜌⁻¹𝑜𝜌 [3]. Since 𝜌 is a unitary operator as a mapping of ℋ_phys onto ℋ,𝑂 is a Hermitian operator acting in ℋ_phys if and only if 𝑜 is a Hermitian operator acting in ℋ.

Since the metric operator 𝜂₊ 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 𝜂₊ 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 −𝜓^′′(𝑥)(𝑥)[)∪(],𝜓(0=𝜆𝜓,𝑥∈−1,00,1₋)=𝛼𝜓0⁺,𝜓^′(0⁻)=𝛽𝜓^′0⁺,𝜓^′(−1)=𝜓^′(1)=0, (7)(8)(9) where 𝜆 is a spectral parameter, 𝛼 and 𝛽 are fixed nonzero complex numbers, and 𝜓 is a function from the Sobolev space 𝑊^2,2([−1,0)∪(0,1]). The space 𝑊^2,2([−1,0)∪(0,1]) consists of all complex-valued functions 𝜓∈𝐿²(−1,1) differentiable on each of the intervals [−1,0) and (0,1] with the derivative 𝜓^′ absolutely continuous on each closed subinterval of the intervals [−1,0) and (0,1] (therefore there exists the second derivative 𝜓^′′ almost everywhere because any function which is absolutely continuous on an interval is differentiable almost everywhere on that interval) and such that ⁰₋₁𝜓^′′(𝑥)²𝑑𝑥<∞,¹₀𝜓^′′(𝑥)²𝑑𝑥<∞. (10)

It can easily be shown (see [11]) that for any function 𝜓∈𝑊^2,2([−1,0)∪(0,1]), there exist the finite limit values 𝜓(0⁻),𝜓^′(0⁻),𝜓(0⁺), and 𝜓^′(0⁺). The conditions in (8) are called the transition conditions (matching conditions or impulse conditions), while the conditions in (9) are called the boundary conditions.

Problem (7), (8), and (9) with the Dirichlet boundary conditions 𝜓(−1)=𝜓(1)=0 (11) instead of the Neumann boundary conditions (9) was before investigated by the author in [11, 12].

The model (8) in the 𝑃𝑇-symmetric case |𝛼|=|𝛽|=1 had been investigated for the first time by Siegl in [9] where explicit formulas for the metric and its square root are presented.

This paper is organized as follows. In Section 2, we introduce the Hamiltonian 𝐻_𝛼,𝛽 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 𝛼 and 𝛽 where 𝐻_𝛼,𝛽 is (i) Hermitian, (ii) 𝑃𝑇-symmetric, while the third theorem presents a closed formula for a family of the metric operators depending on two arbitrary real positive parameters when 𝐻_𝛼,𝛽 is non-Hermitian having real spectrum. In Section 3, the adjoint operator of the Hamiltonian 𝐻_𝛼,𝛽 is described. Section 4 provides the spectral analysis of the Hamiltonian 𝐻_𝛼,𝛽 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 𝐻_𝛼,𝛽 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 ℋ=𝐿²(−1,1) with the usual inner product 𝜓,𝜙=¹₋₁𝜓(𝑥)𝜙(𝑥)𝑑𝑥, (12) where the bar over a function (or over a number) denotes the complex conjugate. Note that the Sobolev space 𝑊^2,2([−1,0)∪(0,1]) forms a subspace of 𝐿²(−1,1).

In the Hilbert space ℋ=𝐿²(−1,1), we consider the Hamiltonian 𝐻_𝛼,𝛽 defined as follows. The domain of definition 𝐷(𝐻_𝛼,𝛽) of the operator 𝐻_𝛼,𝛽 consists of all functions 𝜓∈𝑊^2,2([−1,0)∪(0,1]) satisfying the transition conditions (8) and the boundary conditions (9). On the functions 𝜓∈𝐷(𝐻_𝛼,𝛽), the Hamiltonian 𝐻_𝛼,𝛽 is given by 𝐻_𝛼,𝛽𝜓(𝑥)=−𝜓^′′(𝑥),𝑥∈[−1,0)∪(0,1]. (13)

Let 𝑃 and 𝑇 be the parity and the time reversal operators defined by 𝑃𝜓(𝑥)=𝜓(−𝑥),𝑇𝜓(𝑥)=𝜓(𝑥). (14) The Hamiltonian 𝐻_𝛼,𝛽 is called 𝑃𝑇-symmetric if for every 𝜓∈𝐷(𝐻_𝛼,𝛽), we have that 𝑃𝑇𝜓∈𝐷(𝐻_𝛼,𝛽) and 𝐻_𝛼,𝛽𝑃𝑇𝜓=𝑃𝑇𝐻_𝛼,𝛽𝜓. (15)

Main results of the present paper can be formulated as the following theorems.

Theorem 1. The operator 𝐻_𝛼,𝛽 is Hermitian if and only if 𝛼𝛽=1.

Theorem 2. The operator 𝐻_𝛼,𝛽 is 𝑃𝑇-symmetric if and only if |𝛼|=1 and |𝛽|=1.

Theorem 3. Assume that the complex numbers 𝛼 and 𝛽 are nonzero. If 𝛼+𝛽=0, then any complex number is an eigenvalue of the operator 𝐻_𝛼,𝛽, but if 𝛼+𝛽≠0, then the spectrum of 𝐻_𝛼,𝛽 is real and consists of the simple eigenvalues 𝜆_𝑛=𝑛2𝜋²,𝑤ℎ𝑒𝑟𝑒𝑛=0,1,2,…. (16) Furthermore, let 𝛼+𝛽≠0 and 𝜂₊ be a linear operator defined on ℋ by 𝜂₊𝜓(𝑥)=12()𝜓(𝑥)+𝜓()[),𝐴𝛽𝐴+𝐵𝐴𝛽−𝐵𝛼−𝑥𝑖𝑓𝑥∈−1,0₂+𝐵|𝛼|²𝜓(𝑥)+𝐴𝛽−𝐵𝛼𝜓()(],−𝑥𝑖𝑓𝑥∈0,1 (17) where 𝐴 and 𝐵 are arbitrary positive real numbers. Then 𝜂₊ is an invertible positive bounded operator with bounded inverse and satisfies ∀𝜓∈𝐷𝐻_𝛼,𝛽⟹𝜂₊𝜓∈𝐷𝐻^∗_𝛼,𝛽,𝐻^∗_𝛼,𝛽𝜂₊𝜓=𝜂₊𝐻_𝛼,𝛽𝜓 (18) so that the Hamiltonian 𝐻_𝛼,𝛽 is quasi-Hermitian in this case and 𝜂₊ is a metric operator for it.

3. The Adjoint of the Hamiltonian

Theorem 4. For any nonzero complex numbers 𝛼 and 𝛽 such that 𝛼+𝛽≠0, the relation 𝐻^∗_𝛼,𝛽=𝐻_{1/𝛽,1/𝛼} (19) holds. Therefore, the domain of definition 𝐷(𝐻^∗_𝛼,𝛽) of 𝐻^∗_𝛼,𝛽 consists of all functions 𝜙∈𝑊^2,2([−1,0)∪(0,1]) satisfying the transition conditions 𝜙(0⁻)=1𝛽𝜙0⁺,𝜙^′(0⁻)=1𝛼𝜙^′0⁺ (20) and the boundary conditions 𝜙^′(−1)=𝜙^′(1)=0, (21) and on such functions 𝜙, 𝐻^∗_𝛼,𝛽𝜙(𝑥)=−𝜙^′′(𝑥),𝑥∈[−1,0)∪(0,1]. (22)

Proof. Recall that by definition of the adjoint operator, the domain of definition 𝐷(𝐻^∗_𝛼,𝛽) of 𝐻^∗_𝛼,𝛽 consists of those functions 𝜙 in 𝐿²(−1,1) such that 𝐻_𝛼,𝛽𝜓,𝜙=𝜓,𝑓 (23) for all 𝜓 in 𝐷(𝐻_𝛼,𝛽) and some 𝑓 in 𝐿²(−1,1) depending on 𝜙. Since 𝐷(𝐻_𝛼,𝛽) is dense in 𝐿²(−1,1),𝑓 is uniquely determined by 𝜙 and, by definition, 𝐻^∗_𝛼,𝛽𝜙=𝑓.
The inclusion 𝐷𝐻_{1/𝛽,1/𝛼}⊂𝐷𝐻^∗_𝛼,𝛽 (24) is easy to see. Indeed, take any 𝜙∈𝐷(𝐻_{1/𝛽,1/𝛼}). Hence, 𝜙∈𝑊^2,2([−1,0)∪(0,1]) and satisfies conditions (20) and (21). Then for every 𝜓∈𝐷(𝐻_𝛼,𝛽) we have, integrating by parts twice and taking into account that 𝜓 satisfies (8) and (9), 𝐻_𝛼,𝛽𝜓,𝜙=−¹₋₁𝜓^′′(𝑥)𝜙(𝑥)𝑑𝑥=−⁰₋₁𝜓^′′(𝑥)𝜙(𝑥)𝑑𝑥−¹₀𝜓^′′(𝑥)𝜙(𝑥)𝑑𝑥=−𝜓^′(𝑥)𝜙(𝑥)|⁰⁻₋₁+𝜓(𝑥)𝜙^′(𝑥)|⁰⁻₋₁−⁰₋₁𝜓(𝑥)𝜙^′′(𝑥)𝑑𝑥−𝜓^′(𝑥)𝜙(𝑥)|¹₀⁺+𝜓(𝑥)𝜙^′(𝑥)|¹₀⁺−¹₀𝜓(𝑥)𝜙^′′(𝑥)=𝑑𝑥−𝛽𝜙(0⁻)+𝜙0⁺𝜓^′0⁺+𝛼𝜙^′(0⁻)−𝜙^′0⁺𝜓0⁺+𝜙^′(−1)𝜓(−1)−𝜙^′(1)𝜓(1)−¹₋₁𝜓(𝑥)𝜙^′′(𝑥)𝑑𝑥=−¹₋₁𝜓(𝑥)𝜙^′′(𝑥)𝑑𝑥. (25) The sum of terms outside the integrals became zero because of the fact that 𝜙 satisfies (20) and (21). This shows that 𝜙 belongs to 𝐷(𝐻^∗_𝛼,𝛽) (therefore (24) holds), and on such elements 𝜙, we have equality (22). Note that in this part, we did not use the condition 𝛼+𝛽≠0.
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 𝜙∈𝐷(𝐻^∗_𝛼,𝛽) we have equality (22). This can be done similarly to that given in [11] in the proof of Proposition 4.

From Theorem 4, we get Theorem 1 as a corollary under the condition 𝛼+𝛽≠0. On the other hand, if 𝛼+𝛽=0, the operator 𝐻_𝛼,𝛽 cannot be Hermitian because in this case 𝐻_𝛼,𝛽 has nonreal eigenvalues (see Theorem 5(ii)).

Theorem 2 follows exactly in the same way as in [11, Section 4].

Applying Theorem 4 to the operator 𝐻_{1/𝛽,1/𝛼}, we get that 𝐻^∗_{1/𝛽,1/𝛼}=𝐻_𝛼,𝛽. Therefore, for any nonzero complex numbers 𝛼 and 𝛽 with 𝛼+𝛽≠0, the operator 𝐻_𝛼,𝛽 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 𝐻_𝛼,𝛽.

Theorem 5. Let 𝛼 and 𝛽 be nonzero complex numbers. Then, one has the following. (i) If 𝛼+𝛽≠0, then the spectrum of the operator 𝐻_𝛼,𝛽 is real and consists of the (simple) eigenvalues 𝜆_𝑛=𝑛2𝜋²,𝑤ℎ𝑒𝑟𝑒𝑛=0,1,2,…. (26) The corresponding eigenfunctions 𝜓_𝑛 are found by 𝜓_2𝑗(𝑥)=𝑐_2𝑗[),(],𝜓𝛼cos𝑗𝜋𝑥𝑖𝑓𝑥∈−1,0cos𝑗𝜋𝑥𝑖𝑓𝑥∈0,1_2𝑗+1(𝑥)=𝑐_2𝑗+11𝛽sin𝑗+2[),1𝜋𝑥𝑖𝑓𝑥∈−1,0sin𝑗+2𝜋𝑥𝑖𝑓𝑥∈(0,1], (27) for 𝑗=0,1,2,…, where 𝑐_𝑛 are nonzero complex constants which may depend on 𝛼 and 𝛽.(ii) If 𝛼+𝛽=0, then any complex number 𝜆=𝑘²(𝑘∈ℂ) is an eigenvalue of the operator 𝐻_𝛼,𝛽 with the corresponding eigenfunction 𝜓(𝑥)()[),()(],=𝑐𝛼cos𝑘𝑥+1𝑖𝑓𝑥∈−1,0cos𝑘𝑥−1𝑖𝑓𝑥∈0,1 (28) where 𝑐 is an arbitrary nonzero complex constant which may depend on 𝜆.

This theorem is elementary and can be proved as in [11]. Note that it can also be verified directly by substituting (27) and (28) in (7), (8), and (9).

From Theorems 4 and 5, it follows that if 𝛼+𝛽≠0(𝛼≠0,𝛽≠0) the spectrum of the adjoint operator 𝐻^∗_𝛼,𝛽 coincides with the set of eigenvalues 𝜆_𝑛 given in (26), and the corresponding eigenfunctions 𝜙_𝑛 are given by 𝜙_2𝑗(𝑥)=𝑑_2𝑗1𝛽[),(],𝜙cos𝑗𝜋𝑥if𝑥∈−1,0cos𝑗𝜋𝑥if𝑥∈0,1_2𝑗+1(𝑥)=𝑑_2𝑗+11𝛼1sin𝑗+2[),1𝜋𝑥if𝑥∈−1,0sin𝑗+2𝜋𝑥if𝑥∈(0,1], (29) for 𝑗=0,1,2,…, where 𝑑_𝑛 are nonzero complex constants which may depend on 𝛼 and 𝛽.

From 𝐻_𝛼,𝛽𝜓_𝑛=𝜆_𝑛𝜓_𝑛,𝐻^∗_𝛼,𝛽𝜙_𝑚=𝜆_𝑚𝜙_𝑚 (30) it follows, by a standard way, that 𝜓_𝑛 and 𝜙_𝑚 are orthogonal to each other provided 𝑛≠𝑚. We can normalize the eigenfunctions appropriately to get 𝜓_𝑛,𝜙_𝑚=𝛿_𝑛𝑚(𝑛,𝑚=0,1,2,…), (31) where 𝛿_𝑛𝑚 is the Kronecker delta. Namely, taking into account ¹₀cos²𝑗𝜋𝑥𝑑𝑥=¹₀sin²1𝑗+21𝜋𝑥𝑑𝑥=2, (32) it is easy to see that (31) follows by choosing the coefficients 𝑐_𝑛 and 𝑑_𝑛 in (27) and (29) according to the equations 1=𝛼+𝛽𝑐2𝛽^2𝑗𝑑_2𝑗,1=𝛼+𝛽𝑐2𝛼^2𝑗+1𝑑_2𝑗+1𝑗=0,1,2,…. (33) These equations can obviously be satisfied. We choose the coefficient 𝑑_𝑛 by the requirement 𝑑_2𝑗=𝑎𝛽,𝑑_2𝑗+1=𝑏𝛼𝑗=0,1,2,…, (34) where 𝑎 and 𝑏 are arbitrary nonzero complex numbers, determining 𝑐_𝑛 by the equations in (33): 𝑐_2𝑗=2𝑎𝛼+𝛽,𝑐_2𝑗+1=2𝑏.𝛼+𝛽𝑗=0,1,2,… (35) Then we have 𝜓_2𝑗(𝑥)=𝑎√2𝛼+𝛽𝛼𝑢_𝑗(𝑥)[),𝑢if𝑥∈−1,0_𝑗(𝑥)(],𝜓if𝑥∈0,1_2𝑗+1(𝑥)=𝑏√2𝛼+𝛽𝛽𝑣_𝑗(𝑥)[),𝑣if𝑥∈−1,0_𝑗(𝑥)(],𝜙if𝑥∈0,1_2𝑗(𝑥)=𝑎√2𝑢_𝑗(𝑥)if𝑥∈[−1,0),𝛽𝑢_𝑗(𝑥)(],𝜙if𝑥∈0,1_2𝑗+1(𝑥)=𝑏√2𝑣_𝑗(𝑥)if𝑥∈[−1,0),𝛼𝑣_𝑗(𝑥)if𝑥∈(0,1], (36)(37)(38)(39) for 𝑗=0,1,2,…, where 𝑢_𝑗(𝑥)=√𝑣2cos𝑗𝜋𝑥,_𝑗(𝑥)=√12sin𝑗+2𝜋𝑥𝑗=0,1,2,…. (40) Note that {𝑢_𝑗}^∞_𝑗=0 forms a set of orthonormal in 𝐿²(0,1) eigenfunctions of the Hermitian Sturm-Liouville eigenvalue problem −𝑢^′′(𝑥)(𝑥)[],𝑢=𝜆𝑢,𝑥∈0,1_′(0)=𝑢^′(1)=0, (41) and {𝑣_𝑗}^∞_𝑗=0 forms a set of orthonormal in 𝐿²(0,1) eigenfunctions of the Hermitian Sturm-Liouville eigenvalue problem −𝑣^′′(𝑥)(𝑥)[],𝑣(0)=𝜆𝑣,𝑥∈0,1=𝑣^′(1)=0. (42)

As is well known, the sets {𝑢_𝑗}^∞_𝑗=0 and {𝑣_𝑗}^∞_𝑗=0 form complete orthonormal families (orthonormal bases) in the Hilbert space 𝐿²(0,1). In particular, we have the expansions 𝑓(𝑥)=^∞_𝑗=0𝑢_𝑗(𝑥)¹₀𝑓(𝑡)𝑢_𝑗(𝑡)𝑓(𝑥)=𝑑𝑡,_∞𝑗=0𝑣_𝑗(𝑥)¹₀𝑓(𝑡)𝑣_𝑗(𝑡)𝑑𝑡 (43) and the Parseval equalities ¹₀𝑓(𝑥)²𝑑𝑥=^∞_𝑗=0¹₀𝑓(𝑡)𝑢_𝑗(𝑡)𝑑𝑡²=^∞_𝑗=0¹₀𝑓(𝑡)𝑣_𝑗(𝑡)𝑑𝑡² (44) for every 𝑓∈𝐿²(0,1).

Let us calculate the resolvent of the operator 𝐻_𝛼,𝛽, where 𝛼 and 𝛽 are nonzero complex numbers with 𝛼+𝛽≠0. To this end, we introduce the solutions 𝜑(𝑥,𝜆) and 𝜃(𝑥,𝜆) of (7) satisfying the transition conditions (8) and the initial conditions 𝜑(−1,𝜆)=1,𝜑^′()𝜃()−1,𝜆=0,1,𝜆=1,𝜃^′(1,𝜆)=0, (45) respectively. Note that 𝜑(𝑥,𝜆) satisfies the boundary condition in (9) at 𝑥=−1 and 𝜃(𝑥,𝜆) at 𝑥=1. Putting 𝜆=𝑘², we can find these solutions explicitly as follows: 𝜑()=()[),𝑥,𝜆1cos𝑘𝑥+1,𝑥∈−1,0𝛼1cos𝑘cos𝑘𝑥−𝛽(],𝜃()=[),()(].sin𝑘sin𝑘𝑥,𝑥∈0,1𝑥,𝜆𝛼cos𝑘cos𝑘𝑥+𝛽sin𝑘sin𝑘𝑥,𝑥∈−1,0cos𝑘𝑥−1,𝑥∈0,1 (46) The Wronskian of these solutions is 𝑊_𝑥𝜑,𝜃=𝜑(𝑥,𝜆)𝜃^′(𝑥,𝜆)−𝜑^′()𝜃()=1𝑥,𝜆𝑥,𝜆2[),1𝛼+𝛽𝑘sin2𝑘,𝑥∈−1,02𝛼+𝛽(].𝛼𝛽𝑘sin2𝑘,𝑥∈0,1 (47) Therefore, by the standard Green function approach, the resolvent 𝑅_𝜆=(𝐻_𝛼,𝛽−𝜆𝐼)⁻¹ of the operator 𝐻_𝛼,𝛽 has the form, for 𝜆≠𝜆_𝑛(𝑛=0,1,2,…), 𝑅_𝜆𝑓(𝑥)=¹₋₁𝐺𝑥,𝑦,𝜆𝑓𝑦𝑑𝑦,𝑥∈[−1,0)∪(0,1], (48) where the function 𝐺(𝑥,𝑦,𝜆) is defined for 𝑥,𝑦∈[−1,0)∪(0,1] by the formula 𝐺1𝑥,𝑦,𝜆=−𝑊_𝑦×𝜑()𝜃𝜑𝜃()𝜑,𝜃𝑥,𝜆𝑦,𝜆,−1≤𝑥≤𝑦≤1,𝑦,𝜆𝑥,𝜆,−1≤𝑦≤𝑥≤1. (49) It follows from the explicit form of 𝐺(𝑥,𝑦,𝜆) that ¹₋₁¹₋₁𝐺𝑥,𝑦,𝜆²𝑑𝑥𝑑𝑦<∞. (50) Therefore, the resolvent of 𝐻_𝛼,𝛽 is a compact operator (moreover, a Hilbert-Schmidt operator). This implies that the spectrum of 𝐻_𝛼,𝛽 is purely discrete and consists of the eigenvalues 𝜆_𝑛=(𝜋𝑛/2)²,𝑛=0,1,2,….

5. Proof of Theorem 3

To use formula (2) for constructing a metric operator for our Hamiltonian 𝐻_𝛼,𝛽, we have to show that the set {𝜓_𝑛}^∞_𝑛=0 of eigenfunctions 𝜓_𝑛 of 𝐻_𝛼,𝛽 forms a Riesz basis of the Hilbert space ℋ=𝐿²(−1,1). For this purpose, we will use the following result (see [13, Chapter 6, Theorem 2.1]). Suppose a sequence {𝜓_𝑛}^∞_𝑛=0 is complete in the Hilbert space ℋ, there corresponds to it a complete biorthogonal sequence {𝜙_𝑛}^∞_𝑛=0, and for any 𝑓∈ℋ, one has ^∞_𝑛=0𝑓,𝜓_𝑛²<∞,^∞_𝑛=0𝑓,𝜙_𝑛²<∞. (51) Then the sequence {𝜓_𝑛}^∞_𝑛=0 forms a Riesz basis of the space ℋ.

To prove the completeness in 𝐿²(−1,1) of the functions 𝜓_𝑛 defined by (27), assume that an element 𝜓∈𝐿²(−1,1) is orthogonal to all the functions 𝜓_𝑛 so that we have 𝛼⁰₋₁𝜓(𝑥)cos𝑗𝜋𝑥𝑑𝑥+¹₀𝜓(𝑥)cos𝑗𝜋𝑥𝑑𝑥=0,𝛽⁰₋₁𝜓(𝑥)1sin𝑗+2+𝜋𝑥𝑑𝑥₁₀𝜓(𝑥)1sin𝑗+2𝜋𝑥𝑑𝑥=0, (52) for all 𝑗=0,1,2,…. We have to show that then 𝜓=0. From (52), we get 𝛼𝜓(¹₀−𝑥)+𝜓(𝑥)cos𝑗𝜋𝑥𝑑𝑥𝑑𝑥=0,₁₀−()(𝑥1𝛽𝜓−𝑥+𝜓)sin𝑗+2𝜋𝑥𝑑𝑥=0, (53) for all 𝑗=0,1,2,…. Since the sets {cos𝑗𝜋𝑥}^∞_𝑗=0 and {sin(𝑗+1/2)𝜋𝑥}^∞_𝑗=0 form complete orthogonal families (orthogonal bases) in the Hilbert space 𝐿²(0,1) as the sets of eigenfunctions of the Hermitian Sturm-Liouville eigenvalue problems (41) and (42), respectively, we conclude from the last equations that ()(𝑥)−𝛼𝜓−𝑥+𝜓=0,𝛽𝜓(−𝑥)+𝜓(𝑥)=0,𝑥∈[0,1]. (54) Hence by the condition 𝛼+𝛽≠0, we get 𝜓(−𝑥)=𝜓(𝑥)=0,𝑥∈[0,1]. Therefore, 𝜓(𝑥)=0,𝑥∈[−1,1], and the desired completeness is shown.

We can show similarly that the system {𝜙_𝑛}^∞_𝑛=0 also is complete in 𝐿²(−1,1). Next, {𝜓_𝑛}^∞_𝑛=0 and {𝜙_𝑛}^∞_𝑛=0 are biorthogonal by (31), and the conditions in (51) are fulfilled for the functions 𝜓_𝑛 and 𝜙_𝑛 defined by (36)–(39) in virtue of (44).

Now, we are in a position to use formula (2) for calculating the metric operator 𝜂₊ for the Hamiltonian 𝐻_𝛼,𝛽. Using formulas (38) and (39) for 𝜙_𝑛, we find that for any 𝜓∈𝐿²(−1,1), 𝜂₊𝜓(𝑥)=^∞_𝑛=0𝜓,𝜙_𝑛𝜙_𝑛=^∞_𝑗=0𝜓,𝜙_2𝑗𝜙_2𝑗+^∞_𝑗=0𝜓,𝜙_2𝑗+1𝜙_2𝑗+1=𝑎√2^∞_𝑗=0⁰⁻¹𝜓(𝑡)𝑢_𝑗(𝑡)𝑑𝑡+𝛽¹₀𝜓(𝑡)𝑢_𝑗(𝑡)𝑑𝑡𝜙_2𝑗(𝑥)+𝑏√2^∞_𝑗=0⁰⁻¹𝜓(𝑡)𝑣_𝑗(𝑡)𝑑𝑡+𝛼¹₀𝜓(𝑡)𝑣_𝑗(𝑡)𝑑𝑡𝜙_2𝑗+1(𝑥). (55) Hence, if 𝑥∈[−1,0), then 𝜂₊𝜓(𝑥)=|𝑎|²2^∞_𝑗=0⁰₋₁𝜓(𝑡)𝑢_𝑗(𝑡)𝑑𝑡+𝛽¹₀𝜓(𝑡)𝑢_𝑗(𝑡)𝑑𝑡𝑢_𝑗(𝑥)+|𝑏|²2^∞_𝑗=0⁰₋₁𝜓(𝑡)𝑣_𝑗(𝑡)𝑑𝑡+𝛼¹₀𝜓(𝑡)𝑣_𝑗(𝑡)𝑑𝑡𝑣_𝑗(𝑥), (56) and if 𝑥∈(0,1], then 𝜂₊𝜓(𝑥)=|𝑎|²2^∞_𝑗=0⁰₋₁𝜓(𝑡)𝑢_𝑗(𝑡)𝑑𝑡+𝛽¹₀𝜓(𝑡)𝑢_𝑗(𝑡)𝑑𝑡𝛽𝑢_𝑗(𝑥)+|𝑏|²2^∞_𝑗=0⁰₋₁𝜓(𝑡)𝑣_𝑗(𝑡)𝑑𝑡+𝛼¹₀𝜓(𝑡)𝑣_𝑗(𝑡)𝑑𝑡𝛼𝑣_𝑗(𝑥). (57) Next, using the expansion formulas given in (43) and taking into account that 𝑢_𝑗(−𝑥)=𝑢_𝑗(𝑥),𝑣_𝑗(−𝑥)=−𝑣_𝑗(𝑥), we have the following.

If 𝑥∈[−1,0), then ^∞_𝑗=0𝑢_𝑗(𝑥)⁰₋₁𝜓(𝑡)𝑢_𝑗(𝑡)=𝑑𝑡_∞𝑗=0𝑢_𝑗(−𝑥)¹₀𝜓(−𝑡)𝑢_𝑗(𝑡)𝑑𝑡=𝜓_∞𝑗=0(𝑥),𝑢_𝑗(𝑥)¹₀𝜓(𝑡)𝑢_𝑗(𝑡)=𝑑𝑡_∞𝑗=0𝑢_𝑗(−𝑥)¹₀𝜓(𝑡)𝑢_𝑗(𝑡)𝑑𝑡=𝜓(−𝑥_∞𝑗=0),𝑣_𝑗(𝑥)⁰₋₁𝜓(𝑡)𝑣_𝑗(𝑡)=𝑑𝑡_∞𝑗=0𝑣_𝑗(−𝑥)¹₀𝜓(−𝑡)𝑣_𝑗(𝑡)𝑑𝑡=𝜓_∞𝑗=0(𝑥),𝑣_𝑗(𝑥)¹₀𝜓(𝑡)𝑣_𝑗(𝑡)𝑑𝑡=−^∞_𝑗=0𝑣_𝑗(−𝑥)¹₀𝜓(𝑡)𝑣_𝑗(𝑡)𝑑𝑡=−𝜓(−𝑥). (58) If 𝑥∈(0,1], then ^∞_𝑗=0𝑢_𝑗(𝑥)⁰₋₁𝜓(𝑡)𝑢_𝑗(𝑡)=𝑑𝑡_∞𝑗=0𝑢_𝑗(𝑥)¹₀𝜓(−𝑡)𝑢_𝑗(𝑡)𝑑𝑡=𝜓(−𝑥_∞𝑗=0),𝑢_𝑗(𝑥)¹₀𝜓(𝑡)𝑢_𝑗(𝑡)𝑑𝑡=𝜓_∞𝑗=0(𝑥),𝑣_𝑗(𝑥)¹₀𝜓(𝑡)𝑣_𝑗(𝑡)𝑑𝑡=𝜓_∞𝑗=0(𝑥),𝑣_𝑗(𝑥)⁰₋₁𝜓(𝑡)𝑣_𝑗(𝑡)𝑑𝑡=−^∞_𝑗=0𝑣_𝑗(𝑥)¹₀𝜓(−𝑡)𝑣_𝑗(𝑡)𝑑𝑡=−𝜓(−𝑥). (59) Substituting these in (56) and (57), we arrive at formula (17) with 𝐴=|𝑎|² and 𝐵=|𝑏|². Since 𝑎 and 𝑏 were arbitrary nonzero complex numbers, 𝐴 and 𝐵 are arbitrary positive real numbers.

6. The Hermitian Counterpart of a Non-Hermitian Hamiltonian

If for complex numbers 𝛼 and 𝛽 we have 𝛼≠0,𝛽≠0, and 𝛼+𝛽≠0, then by Theorem 3, for the Hamiltonian 𝐻_𝛼,𝛽, we have a family of metric operators defined by (17) and depending on two arbitrary positive real parameters 𝐴 and 𝐵. Each concrete choice of the metric operator 𝜂₊ determines a Hermitian counterpart ℎ of 𝐻_𝛼,𝛽. In this section, we will calculate ℎ in two particular cases.

(1) Assume that 𝛽=𝑐𝛼, (60) where 𝑐 is a positive real number.

Obviously, if 𝜂₊ is a metric operator for 𝐻_𝛼,𝛽, then any positive scalar multiple of 𝜂₊ is also a metric operator for the same Hamiltonian 𝐻_𝛼,𝛽. Therefore, dividing both sides of (17) by 𝐴 and setting 𝑐=𝐵/𝐴, we get, due to condition (60), that the operator 𝜂₊ defined by 𝜂₊𝜓(𝑥)=𝜓(𝑥)[),𝑐|𝛼|if𝑥∈−1,0₂𝜓(𝑥)if𝑥∈(0,1] (61) is a metric operator for the Hamiltonian 𝐻_{𝛼,𝑐𝛼}. The inner product determined by this metric operator has the form 𝜓,𝜙₊=⁰₋₁𝜓(𝑥)𝜙(𝑥)𝑑𝑥+𝑐|𝛼|²¹₀𝜓(𝑥)𝜙(𝑥)𝑑𝑥. (62)

It follows that the operator 𝜌=√𝜂₊ is given by 𝑥)=𝜓(𝑥)[),|𝛼|√𝜌𝜓(if𝑥∈−1,0𝑐𝜓(𝑥)if𝑥∈(0,1] (63) and its inverse 𝜌⁻¹ by 𝜌⁻¹𝜓(𝑥)=𝜓(𝑥)[),1if𝑥∈−1,0|𝛼|√𝑐𝜓(𝑥)if𝑥∈(0,1]. (64) We can now calculate the operator ℎ=𝜌𝐻_{𝛼,𝑐𝛼}𝜌⁻¹. (65) The domain of definition 𝐷(ℎ) of the operator ℎ consists of all functions 𝜓∈𝐿²(−1,1) such that 𝜌⁻¹𝜓∈𝐷(𝐻_{𝛼,𝑐𝛼}). Therefore 𝐷(ℎ) consists of all functions 𝜓∈𝑊^2,2([−1,0)∪(0,1]) satisfying the transition conditions 𝜓(0⁻)=𝛼|𝛼|√𝑐𝜓0⁺,𝜓^′(0⁻)=𝛼√𝑐|𝛼|𝜓^′0⁺ (66) and the boundary conditions 𝜓^′(−1)=𝜓^′(1)=0. (67) Moreover, (65) implies by (63) and (64) that if 𝜓∈𝐷(ℎ), then ℎ𝜓(𝑥)=−𝜓^′′(𝑥),𝑥∈[−1,0)∪(0,1]. (68) We see that ℎ=𝐻_𝑧,𝑤𝛼with𝑧=|𝛼|√𝑐𝛼√,𝑤=𝑐|𝛼|. (69) Since 𝑧𝑤=1,ℎ is Hermitian by Theorem 1.

(2) As a second example, consider the case where 𝛼 and 𝛽 are arbitrary positive real numbers. By Theorem 1, the Hamiltonian 𝐻_𝛼,𝛽 is non-Hermitian if 𝛼𝛽≠1. Taking 𝐵=𝐴𝛽/𝛼 in (17), we find that the operator 𝜂₊ defined by 𝜂₊𝜓𝑥)=𝜓(𝑥)[),(𝑥)(](if𝑥∈−1,0𝛼𝛽𝜓if𝑥∈0,1 (70) is a metric operator for the Hamiltonian 𝐻_𝛼,𝛽. The inner product determined by this metric operator has the form 𝜓,𝜙₊=⁰₋₁𝜓(𝑥)𝜙(𝑥)𝑑𝑥+𝛼𝛽¹₀𝜓(𝑥)𝜙(𝑥)𝑑𝑥. (71) Further reasoning as in the case of the previous example, we find that the operator ℎ=𝐻_𝑧,𝑤with𝑧=𝛼𝛽,𝑤=𝛽𝛼 (72) is a Hermitian counterpart of the Hamiltonian 𝐻_𝛼,𝛽.

7. Conclusion

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.