# Finding of the Metric Operator for a Quasi-Hermitian Model

Authors: Ebru Ergun
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

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


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## Body

## 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
(1)H*η+=η+H(i.e.,H*=η+Hη+-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
(2)η+=∑n=1∞‍〈·,ϕn〉ϕn
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 {ϕn}n=1∞ is constructed as follows. Let us label the eigenvalues of H with En, and let {ψn}n=1∞ denote a Riesz basis of ℋ consisting of the eigenvectors ψn of H,Hψn=Enψn. Then one can construct another Riesz basis {ϕn}n=1∞ of ℋ that satisfies
(3)H*ϕn=Enϕn,〈ψn,ϕm〉=δnm,∑n=1∞‍〈·,ϕn〉ψn=I,
where δnm 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
(4)η+-1=∑n=1∞‍〈·,ψn〉ψn.
We can use (2) to introduce the positive-definite inner product
(5)〈·,·〉+=〈η+·,·〉
and identify the physical Hilbert space ℋphys 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 ℋphys onto ℋ is a unitary operator and the operator
(6)h=ρHρ-1
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 ℋphys. We can use the unitary-equivalence of ℋphys 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 ℋphys onto ℋ,O is a Hermitian operator acting in ℋphys if and only if o is a Hermitian operator acting in ℋ.Since the metric operatorη+ 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(7)-ψ′′(x)=λψ(x),x∈[-1,0)∪(0,1],(8)ψ(0-)=αψ(0+),ψ′(0-)=βψ′(0+),(9)ψ′(-1)=ψ′(1)=0,
where λ is a spectral parameter, α and β are fixed nonzero complex numbers, and ψ is a function from the Sobolev space W2,2([-1,0)∪(0,1]). The space W2,2([-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
(10)∫-10|ψ′′(x)|2dx<∞,∫01‍|ψ′′(x)|2dx<∞.It can easily be shown (see [11]) that for any function ψ∈W2,2([-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
(11)ψ(-1)=ψ(1)=0
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 Section2, we introduce the Hamiltonian Hα,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 Hα,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 Hα,β is non-Hermitian having real spectrum. In Section 3, the adjoint operator of the Hamiltonian Hα,β is described. Section 4 provides the spectral analysis of the Hamiltonian Hα,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 Hα,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
(12)〈ψ,ϕ〉=∫-11‍ψ(x)ϕ¯(x)dx,
where the bar over a function (or over a number) denotes the complex conjugate. Note that the Sobolev space W2,2([-1,0)∪(0,1]) forms a subspace of L2(-1,1).In the Hilbert spaceℋ=L2(-1,1), we consider the Hamiltonian Hα,β defined as follows. The domain of definition D(Hα,β) of the operator Hα,β consists of all functions ψ∈W2,2([-1,0)∪(0,1]) satisfying the transition conditions (8) and the boundary conditions (9). On the functions ψ∈D(Hα,β), the Hamiltonian Hα,β is given by
(13)(Hα,βψ)(x)=-ψ′′(x),x∈[-1,0)∪(0,1].LetP and T be the parity and the time reversal operators defined by
(14)(Pψ)(x)=ψ(-x),(Tψ)(x)=ψ¯(x).
The Hamiltonian Hα,β is called PT-symmetric if for every ψ∈D(Hα,β), we have that PTψ∈D(Hα,β) and
(15)Hα,βPTψ=PTHα,βψ.Main results of the present paper can be formulated as the following theorems.Theorem 1.
The operatorHα,β is Hermitian if and only if αβ¯=1.Theorem 2.
The operatorHα,β is PT-symmetric if and only if |α|=1 and |β|=1.Theorem 3.
Assume that the complex numbersα and β are nonzero. If α+β=0, then any complex number is an eigenvalue of the operator Hα,β, but if α+β≠0, then the spectrum of Hα,β is real and consists of the simple eigenvalues
(16)λn=(n2π)2,wheren=0,1,2,….
Furthermore, let α+β≠0 and η+ be a linear operator defined on ℋ by
(17)(η+ψ)(x)=12{(A+B)ψ(x)+(Aβ-Bα)ψ(-x)ifx∈[-1,0),(A|β|2+B|α|2)ψ(x)+(Aβ¯-Bα¯)ψ(-x)ifx∈(0,1],
where A and B are arbitrary positive real numbers. Then η+ is an invertible positive bounded operator with bounded inverse and satisfies
(18)∀ψ∈D(Hα,β)⟹η+ψ∈D(Hα,β*),Hα,β*η+ψ=η+Hα,βψ
so that the Hamiltonian Hα,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 numbersα and β such that α+β≠0, the relation
(19)Hα,β*=H1/β¯,1/α¯
holds. Therefore, the domain of definition D(Hα,β*) of Hα,β* consists of all functions ϕ∈W2,2([-1,0)∪(0,1]) satisfying the transition conditions
(20)ϕ(0-)=1β¯ϕ(0+),ϕ′(0-)=1α¯ϕ′(0+)
and the boundary conditions
(21)ϕ′(-1)=ϕ′(1)=0,
and on such functions ϕ,
(22)(Hα,β*ϕ)(x)=-ϕ′′(x),x∈[-1,0)∪(0,1].Proof.
Recall that by definition of the adjoint operator, the domain of definitionD(Hα,β*) of Hα,β* consists of those functions ϕ in L2(-1,1) such that
(23)〈Hα,βψ,ϕ〉=〈ψ,f〉
for all ψ in D(Hα,β) and some f in L2(-1,1) depending on ϕ. Since D(Hα,β) is dense in L2(-1,1),f is uniquely determined by ϕ and, by definition, Hα,β*ϕ=f.
The inclusion(24)D(H1/β¯,1/α¯)⊂D(Hα,β*)
is easy to see. Indeed, take any ϕ∈D(H1/β¯,1/α¯). Hence, ϕ∈W2,2([-1,0)∪(0,1]) and satisfies conditions (20) and (21). Then for every ψ∈D(Hα,β) we have, integrating by parts twice and taking into account that ψ satisfies (8) and (9),
(25)〈Hα,βψ,ϕ〉=-∫-11‍ψ′′(x)ϕ¯(x)dx=-∫-10‍ψ′′(x)ϕ¯(x)dx-∫01‍ψ′′(x)ϕ¯(x)dx=-ψ′(x)ϕ¯(x)|-10-+ψ(x)ϕ¯′(x)|-10--∫-10‍ψ(x)ϕ¯′′(x)dx-ψ′(x)ϕ¯(x)|0+1+ψ(x)ϕ¯′(x)|0+1-∫01‍ψ(x)ϕ¯′′(x)dx=[-βϕ¯(0-)+ϕ¯(0+)]ψ′(0+)+[αϕ¯′(0-)-ϕ¯′(0+)]ψ(0+)+ϕ¯′(-1)ψ(-1)-ϕ¯′(1)ψ(1)-∫-11‍ψ(x)ϕ¯′′(x)dx=-∫-11‍ψ(x)ϕ¯′′(x)dx.
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(Hα,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(Hα,β*) we have equality (22). This can be done similarly to that given in [11] in the proof of Proposition 4.From Theorem4, we get Theorem 1 as a corollary under the condition α+β≠0. On the other hand, if α+β=0, the operator Hα,β cannot be Hermitian because in this case Hα,β has nonreal eigenvalues (see Theorem 5(ii)).Theorem2 follows exactly in the same way as in [11, Section 4].Applying Theorem4 to the operator H1/β¯,1/α¯, we get that H1/β¯,1/α¯*=Hα,β. Therefore, for any nonzero complex numbers α and β with α+β≠0, the operator Hα,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 operatorHα,β.Theorem 5.
Letα and β be nonzero complex numbers. Then, one has the following. (i)
Ifα+β≠0, then the spectrum of the operator Hα,β is real and consists of the (simple) eigenvalues
(26)λn=(n2π)2,wheren=0,1,2,….
The corresponding eigenfunctions ψn are found by
(27)ψ2j(x)=c2j{αcosjπxifx∈[-1,0),cosjπxifx∈(0,1],ψ2j+1(x)=c2j+1{βsin(j+12)πxifx∈[-1,0),sin(j+12)πxifx∈(0,1],
for j=0,1,2,…, where cn are nonzero complex constants which may depend on α and β.(ii)
Ifα+β=0, then any complex number λ=k2(k∈ℂ) is an eigenvalue of the operator Hα,β with the corresponding eigenfunction
(28)ψ(x)=c{αcosk(x+1)ifx∈[-1,0),cosk(x-1)ifx∈(0,1],
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 Theorems4 and 5, it follows that if α+β≠0(α≠0,β≠0) the spectrum of the adjoint operator Hα,β* coincides with the set of eigenvalues λn given in (26), and the corresponding eigenfunctions ϕn are given by
(29)ϕ2j(x)=d2j{1β¯cosjπxifx∈[-1,0),cosjπxifx∈(0,1],ϕ2j+1(x)=d2j+1{1α¯sin(j+12)πxifx∈[-1,0),sin(j+12)πxifx∈(0,1],
for j=0,1,2,…, where dn are nonzero complex constants which may depend on α and β.From(30)Hα,βψn=λnψn,Hα,β*ϕm=λmϕm
it follows, by a standard way, that ψn and ϕm are orthogonal to each other provided n≠m. We can normalize the eigenfunctions appropriately to get
(31)〈ψn,ϕm〉=δnm(n,m=0,1,2,…),
where δnm is the Kronecker delta. Namely, taking into account
(32)∫01‍cos2jπxdx=∫01‍sin2(j+12)πxdx=12,
it is easy to see that (31) follows by choosing the coefficients cn and dn in (27) and (29) according to the equations
(33)1=α+β2βc2jd¯2j,1=α+β2αc2j+1d¯2j+1(j=0,1,2,…).
These equations can obviously be satisfied. We choose the coefficient dn by the requirement
(34)d2j=aβ¯,d2j+1=bα¯(j=0,1,2,…),
where a and b are arbitrary nonzero complex numbers, determining cn by the equations in (33):
(35)c2j=2a¯α+β,c2j+1=2b¯α+β(j=0,1,2,…).
Then we have
(36)ψ2j(x)=a¯2α+β{αuj(x)ifx∈[-1,0),uj(x)ifx∈(0,1],(37)ψ2j+1(x)=b¯2α+β{βvj(x)ifx∈[-1,0),vj(x)ifx∈(0,1],(38)ϕ2j(x)=a2{uj(x)ifx∈[-1,0),β¯uj(x)ifx∈(0,1],(39)ϕ2j+1(x)=b2{vj(x)ifx∈[-1,0),α¯vj(x)ifx∈(0,1],
for j=0,1,2,…, where
(40)uj(x)=2cosjπx,vj(x)=2sin(j+12)πx(j=0,1,2,…).
Note that {uj}j=0∞ forms a set of orthonormal in L2(0,1) eigenfunctions of the Hermitian Sturm-Liouville eigenvalue problem
(41)-u′′(x)=λu(x),x∈[0,1],u′(0)=u′(1)=0,
and {vj}j=0∞ forms a set of orthonormal in L2(0,1) eigenfunctions of the Hermitian Sturm-Liouville eigenvalue problem
(42)-v′′(x)=λv(x),x∈[0,1],v(0)=v′(1)=0.As is well known, the sets{uj}j=0∞ and {vj}j=0∞ form complete orthonormal families (orthonormal bases) in the Hilbert space L2(0,1). In particular, we have the expansions
(43)f(x)=∑j=0∞‍uj(x)∫01‍f(t)uj(t)dt,f(x)=∑j=0∞‍vj(x)∫01‍f(t)vj(t)dt
and the Parseval equalities
(44)∫01‍|f(x)|2dx=∑j=0∞‍|∫01‍f(t)uj(t)dt|2=∑j=0∞‍|∫01‍f(t)vj(t)dt|2
for every f∈L2(0,1).Let us calculate the resolvent of the operatorHα,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
(45)φ(-1,λ)=1,φ′(-1,λ)=0,θ(1,λ)=1,θ′(1,λ)=0,
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:
(46)φ(x,λ)={cosk(x+1),x∈[-1,0),1αcoskcoskx-1βsinksinkx,x∈(0,1],θ(x,λ)={αcoskcoskx+βsinksinkx,x∈[-1,0),cosk(x-1),x∈(0,1].
The Wronskian of these solutions is
(47)Wx(φ,θ)=φ(x,λ)θ′(x,λ)-φ′(x,λ)θ(x,λ)={12(α+β)ksin2k,x∈[-1,0),12α+βαβksin2k,x∈(0,1].
Therefore, by the standard Green function approach, the resolvent Rλ=(Hα,β-λI)-1 of the operator Hα,β has the form, for λ≠λn(n=0,1,2,…),
(48)Rλf(x)=∫-11‍G(x,y,λ)f(y)dy,x∈[-1,0)∪(0,1],
where the function G(x,y,λ) is defined for x,y∈[-1,0)∪(0,1] by the formula
(49)G(x,y,λ)=-1Wy(φ,θ)×{φ(x,λ)θ(y,λ),-1≤x≤y≤1,φ(y,λ)θ(x,λ),-1≤y≤x≤1.
It follows from the explicit form of G(x,y,λ) that
(50)∫-11‍∫-11‍|G(x,y,λ)|2dxdy<∞.
Therefore, the resolvent of Hα,β is a compact operator (moreover, a Hilbert-Schmidt operator). This implies that the spectrum of Hα,β is purely discrete and consists of the eigenvalues λn=(πn/2)2,n=0,1,2,….


## 5. Proof of Theorem3

To use formula (2) for constructing a metric operator for our Hamiltonian Hα,β, we have to show that the set {ψn}n=0∞ of eigenfunctions ψn of Hα,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 {ψn}n=0∞ is complete in the Hilbert space ℋ, there corresponds to it a complete biorthogonal sequence {ϕn}n=0∞, and for any f∈ℋ, one has
(51)∑n=0∞‍|〈f,ψn〉|2<∞,∑n=0∞‍|〈f,ϕn〉|2<∞.
Then the sequence {ψn}n=0∞ forms a Riesz basis of the space ℋ.To prove the completeness inL2(-1,1) of the functions ψn defined by (27), assume that an element ψ∈L2(-1,1) is orthogonal to all the functions ψn so that we have
(52)α¯∫-10‍ψ(x)cosjπxdx+∫01‍ψ(x)cosjπxdx=0,β¯∫-10‍ψ(x)sin(j+12)πxdx+∫01‍ψ(x)sin(j+12)πxdx=0,
for all j=0,1,2,…. We have to show that then ψ=0. From (52), we get
(53)∫01‍[α¯ψ(-x)+ψ(x)]cosjπxdxdx=0,∫01‍[-β¯ψ(-x)+ψ(x)]sin(j+12)πxdx=0,
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
(54)α¯ψ(-x)+ψ(x)=0,-β¯ψ(-x)+ψ(x)=0,x∈[0,1].
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{ϕn}n=0∞ also is complete in L2(-1,1). Next, {ψn}n=0∞ and {ϕn}n=0∞ are biorthogonal by (31), and the conditions in (51) are fulfilled for the functions ψn and ϕn 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 Hα,β. Using formulas (38) and (39) for ϕn, we find that for any ψ∈L2(-1,1),
(55)(η+ψ)(x)=∑n=0∞‍〈ψ,ϕn〉ϕn=∑j=0∞‍〈ψ,ϕ2j〉ϕ2j+∑j=0∞‍〈ψ,ϕ2j+1〉ϕ2j+1=a¯2∑j=0∞‍{∫-10‍ψ(t)uj(t)dt+β∫01‍ψ(t)uj(t)dt}ϕ2j(x)+b¯2∑j=0∞‍{∫-10‍ψ(t)vj(t)dt+b¯2∑j=0∞‍22+α∫01‍ψ(t)vj(t)dt}ϕ2j+1(x).
Hence, if x∈[-1,0), then
(56)(η+ψ)(x)=|a|22∑j=0∞‍{∫-10‍ψ(t)uj(t)dt+β∫01‍ψ(t)uj(t)dt}uj(x)+|b|22∑j=0∞‍{∫-10‍ψ(t)vj(t)dt+|b|22∑j=0∞‍+α∫01‍ψ(t)vj(t)dt}vj(x),
and if x∈(0,1], then
(57)(η+ψ)(x)=|a|22∑j=0∞‍{∫-10‍ψ(t)uj(t)dt=|a|22∑j=0∞‍+β∫01‍ψ(t)uj(t)dt}β¯uj(x)+|b|22∑j=0∞‍{∫-10‍ψ(t)vj(t)dt+|b|22∑j=0∞‍+α∫01‍ψ(t)vj(t)dt}α¯vj(x).
Next, using the expansion formulas given in (43) and taking into account that uj(-x)=uj(x),vj(-x)=-vj(x), we have the following.Ifx∈[-1,0), then
(58)∑j=0∞‍uj(x)∫-10‍ψ(t)uj(t)dt=∑j=0∞‍uj(-x)∫01‍ψ(-t)uj(t)dt=ψ(x),∑j=0∞‍uj(x)∫01‍ψ(t)uj(t)dt=∑j=0∞‍uj(-x)∫01‍ψ(t)uj(t)dt=ψ(-x),∑j=0∞‍vj(x)∫-10‍ψ(t)vj(t)dt=∑j=0∞‍vj(-x)∫01‍ψ(-t)vj(t)dt=ψ(x),∑j=0∞‍vj(x)∫01‍ψ(t)vj(t)dt=-∑j=0∞‍vj(-x)∫01‍ψ(t)vj(t)dt=-ψ(-x).
If x∈(0,1], then
(59)∑j=0∞‍uj(x)∫-10‍ψ(t)uj(t)dt=∑j=0∞‍uj(x)∫01‍ψ(-t)uj(t)dt=ψ(-x),∑j=0∞‍uj(x)∫01‍ψ(t)uj(t)dt=ψ(x),∑j=0∞‍vj(x)∫01‍ψ(t)vj(t)dt=ψ(x),∑j=0∞‍vj(x)∫-10‍ψ(t)vj(t)dt=-∑j=0∞‍vj(x)∫01‍ψ(-t)vj(t)dt=-ψ(-x).
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 numbersα and β we have α≠0,β≠0, and α+β≠0, then by Theorem 3, for the Hamiltonian Hα,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 Hα,β. In this section, we will calculate h in two particular cases.(
1
) Assume that
(60)β=cα,
where c is a positive real number.Obviously, ifη+ is a metric operator for Hα,β, then any positive scalar multiple of η+ is also a metric operator for the same Hamiltonian Hα,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
(61)(η+ψ)(x)={ψ(x)ifx∈[-1,0),c|α|2ψ(x)ifx∈(0,1]
is a metric operator for the Hamiltonian Hα,cα. The inner product determined by this metric operator has the form
(62)〈ψ,ϕ〉+=∫-10‍ψ(x)ϕ¯(x)dx+c|α|2∫01‍ψ(x)ϕ¯(x)dx.It follows that the operatorρ=η+ is given by
(63)(ρψ)(x)={ψ(x)ifx∈[-1,0),|α|cψ(x)ifx∈(0,1]
and its inverse ρ-1 by
(64)(ρ-1ψ)(x)={ψ(x)ifx∈[-1,0),1|α|cψ(x)ifx∈(0,1].
We can now calculate the operator
(65)h=ρHα,cαρ-1.
The domain of definition D(h) of the operator h consists of all functions ψ∈L2(-1,1) such that ρ-1ψ∈D(Hα,cα). Therefore D(h) consists of all functions ψ∈W2,2([-1,0)∪(0,1]) satisfying the transition conditions
(66)ψ(0-)=α|α|cψ(0+),ψ′(0-)=αc|α|ψ′(0+)
and the boundary conditions
(67)ψ′(-1)=ψ′(1)=0.
Moreover, (65) implies by (63) and (64) that if ψ∈D(h), then
(68)(hψ)(x)=-ψ′′(x),x∈[-1,0)∪(0,1].
We see that
(69)h=Hz,wwithz=α|α|c,w=αc|α|.
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 Hα,β is non-Hermitian if αβ≠1. Taking B=Aβ/α in (17), we find that the operator η+ defined by
(70)(η+ψ)(x)={ψ(x)ifx∈[-1,0),αβψ(x)ifx∈(0,1]
is a metric operator for the Hamiltonian Hα,β. The inner product determined by this metric operator has the form
(71)〈ψ,ϕ〉+=∫-10‍ψ(x)ϕ¯(x)dx+αβ∫01‍ψ(x)ϕ¯(x)dx.
Further reasoning as in the case of the previous example, we find that the operator
(72)h=Hz,wwithz=αβ,w=βα
is a Hermitian counterpart of the Hamiltonian Hα,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.



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*Source: 716109-2013-08-27.xml*