Imaginary eigenvalues of Hermitian Hamiltonian with an inverted potential well and transition to the real spectrum at exceptional point by a non-Hermitian interaction

In quantum mechanics the Hamiltonian has to be Hermitian to guarantee real energy spectrum for the closed systems. The non-Hermitian Hamiltonian describes usually open systems, which exchange energy continuously with external source and thus possess complex energy spectrum. Bender and his collaborators proposed for the first time a creative finding that the Hermitian Hamiltonian is sufficient but not necessary to have real eigenvalues CM98 ; CM99 ; CM02 . According to them the non-Hermitian Hamiltonian with real spectrum has the parity-time ($PT$) reflection symmetry CM98 ; CM99 ; CM02 ; GuResult ; GuAnn ; LiuResult . Mostafazadeh named the non-Hermitian Hamiltonian with real spectrum as pseudo-Hermitian, which satisfies a necessary condition Mosta1 ; Mosta2 ; Mosta3 . While the $PT$ symmetry is neither necessary nor sufficient condition for a non-Hermitian Hamiltonian to have real spectrum. It is found that the $PT$ -asymmetric Hamiltonian can also possess real spectraLiuPysica ; Amaa .

For the $PT$ -symmetric non-Hermitian Hamiltonians there exists an exceptional point, which separates the unbroken $PT$ symmetry region with pure real-spectrum and a broken $PT$ symmetry region, where eigenvalues are complexCM98 ; CM04 ; CM07 . This transition at the exceptional point has been observed in numerous laboratory experimentsCM18 ; DN18 for various non-Hermitian Hamiltonians. The non-Hermitian Hamiltonian has become a rapidly developing subject in different branches of physicsCH13 ; ZZ16 ; SB12 ; CM13 ; CE10 ; YW19 . The spectrum property becomes complicated at the exceptional pointsLC14 ; PG15 ; HX16 ; WL17 due to the symmetry breaking, which can be modulated by dissipative mediaSK19 ; JZ18 ; YZ17 ; FQ18 . The exceptional point and complex domain of eigenvalues have been well studied particularly in relation with optical systemsCM18 ; DN18 .

As a matter of fact the content of quantum mechanics is more rich than what we are known. Not only the non-Hermitian Hamiltonians of open systems but also the Hermitian Hamiltonian of a conservative system can have complex spectrum if the potential is unbounded from below. The complex spectrum of an inverted double-well potential was studied long ago in terms of the instanton method under semiclassical approximation LM92 ; LM94 . The imaginary part of energy eigenvalues characterizes the decay rate or life time of metastable states LM92 ; LM94 . As of yet the rigorous complex eigenvalues and associated eigenstates have not been given in the framework of quantum mechanics for the Hermitian Hamiltonians.

The gain and loss of energy are not the only mechanism for complex eigenvalues. We propose in the present paper a solvable Hamiltonian consisting of $SU(1,1)$ generators, which can be realized by single-mode boson operators. In the absence of external field the Hamiltonian, which is Hermitian however with imaginary eigenvalues, describes a particle in an inverted potential well. Classically the particle moves acceleratingly away from the center of the inverted potential well. The quantum wave functions are, of course, spatially non-localized. Inverted potential well unbounded from below is found in the minisuperpace model of expanding universe, which is established from Einstein equation by including the cosmology constant Liang .

We demonstrate in the present paper that the non-Hermitian interaction of an external field can decrease continuously the slope of inverted potential well by the increase of coupling constant. At the end the potential vanishes at a critical coupling-value, namely the exceptional point, where all eigenstates are degenerate with a zero eigenvalue. Beyond the exceptional point the eigenvalues become real as that of a normal oscillator. We discover a peculiar transition from imaginary to real spectra at the exceptional point induced by the non-Hermitian interaction.

The Hamiltonian with a non-Hermitian interaction term can be converted by a similarity transformation to a Hermitian one. The transformation operator is not unitary but Hermitian GuAnn different from the unitary transformation in the ordinary quantum mechanics. The classical counterpart of the non-Hermitian Hamiltonian is seen to be a complex function of canonical variables. It becomes under the canonical transformation of variables a real function indicating strict one to one correspondence of the quantum and classical Hamiltonians GuAnn .

The article is organized as follows. In Sec.II, the imaginary spectrum and eigenstates are obtained by means of the algebraic method for the Hermitian Hamiltonian with an inverted potential well. We demonstrate in Sec.III the transition from imaginary to real spectra at exceptional point by the non-Hermitian interaction. The quantum-classical correspondence for the Hermitian and non-Hermitian Hamiltonians is revealed in Sec.IV. The conclusion and discussion are presented in Sec.V.

The hermiticity of Hamiltonian is not sufficient condition for real eigenvalues. We assume that the Hermitian Hamiltonian possesses discrete complex eigenvalues. There must exist dual-set of eigenstates such that

	$\displaystyle\widehat{H}|u_{n}\rangle_{r}$	$\displaystyle=$	$\displaystyle E_{n}|u_{n}\rangle_{r},\quad_{r}\langle u_{n}|\widehat{H}=_{r}\langle u_{n}|E_{n}^{\ast}$
	$\displaystyle\widehat{H}|u_{n}\rangle_{l}$	$\displaystyle=$	$\displaystyle E_{n}^{\ast}|u_{n}\rangle_{l},\quad_{l}\langle u_{n}|\widehat{H}=_{l}\langle u_{n}|E_{n},$		(1)

in which the subscripts “$r$”, “$l$” denote respectively the “ket” and “bra” states. The two sets of eigenstates are mutually orthonormal

$${}_{l}\langle u_{n}|u_{m}\rangle_{r}=_{r}\langle u_{n}|u_{m}\rangle_{l}=\delta_{n,m}.$$

The density operators defined by $\widehat{\rho}=|\psi\rangle_{rl}\langle\psi|$, $\widehat{\rho}^{{\dagger}}=|\psi\rangle_{lr}\langle\psi|$ are non-Hermitian invariants

$$\frac{d\widehat{\rho}}{dt}=\frac{d\widehat{\rho}^{{\dagger}}}{dt}=0.$$

They obey the quantum Liouvill equation (in the unit convention $\hbar=1$ throughout the paper),

$$i\frac{\partial\widehat{\rho}}{\partial t}=\left[\widehat{H},\quad\widehat{\rho}\right],\quad i\frac{\partial\widehat{\rho}^{{\dagger}}}{\partial t}=\left[\widehat{H},\quad\widehat{\rho}^{{\dagger}}\right],$$

which are in consistence with the Schrödinger equations

$$i\frac{\partial}{\partial t}|\psi\rangle_{\gamma}=\widehat{H}|\psi\rangle_{\gamma},$$

for both the “bra” and “ket” states ($\gamma=l,r$). The state densities $\langle x|\widehat{\rho}|x\rangle$, $\langle x|\widehat{\rho}^{{\dagger}}|x\rangle$ are conserved quantities satisfying the basic requirement of quantum mechanics.

We consider the Hermitian Hamiltonian for a particle of unit mass in the inverted potential well

$$\widehat{H}=\Omega\left(\frac{\widehat{p}^{2}}{2}-\frac{\widehat{x}^{2}}{2}\right),$$

(2)

in which $\widehat{x}$, $\widehat{p}$ are dimensionless operator with the usual commutation relation $\left[\widehat{x},\widehat{p}\right]=i$. The classical orbit of the particle in the inverted potential well is simply

$$x\left(t\right)=\frac{v_{0}}{\Omega}\sinh(\Omega t),$$

with initial position $x_{0}=0$ and velocity $v_{0}$. A minisuperspace model Liang of the universe is found also as an inverted potential well, which results in the accelerating expansion. The Hamiltonian Eq.(2) becomes with the imaginary frequency boson-operators

$$\widehat{H}=i\Omega\left(\widehat{b}_{+}\widehat{b}_{-}+\frac{1}{2}\right),$$

where

$$\widehat{b}_{-}=\sqrt{\frac{i}{2}}\left(\widehat{x}+\widehat{p}\right),\widehat{b}_{+}=\sqrt{\frac{i}{2}}\left(\widehat{x}-\widehat{p}\right).$$

(3)

The imaginary-frequency boson operators satisfy the same commutation relation as the usual boson operators

$$\left[\widehat{b}_{-},\widehat{b}_{+}\right]=1.$$

(4)

The commutation relation of $SU(1,1)$ generators GuAnn ; LLM is invariant under the realization of imaginary-frequency boson-operators

$$\widehat{S}_{z}=\frac{1}{2}\left(\widehat{b}_{+}\widehat{b}_{-}+\frac{1}{2}\right),\widehat{S}_{+}=\frac{1}{2}\left(\widehat{b}_{+}\right)^{2},\widehat{S}_{-}=\frac{1}{2}\left(\widehat{b}_{-}\right)^{2}$$

(5)

such that

$$\left[\widehat{S}_{z},\widehat{S}_{\pm}\right]=\pm\widehat{S}_{\pm},\left[\widehat{S}_{+},\widehat{S}_{-}\right]=-2\widehat{S}_{z}.$$

(6)

Thus the Hamiltonian Eq.(2), is represented as

$$\widehat{H}=2i\Omega\widehat{S}_{z}.$$

(7)

Since

$$\widehat{S}_{z}^{{\dagger}}=\frac{1}{2}\left(-\widehat{b}_{-}\widehat{b}_{+}+\frac{1}{2}\right)=-\widehat{S}_{z},$$

the Hamiltonian Eq.(7) is Hermitian as it should be. While the $SU(1,1)$ generator $\widehat{S}_{z}$ is no longer Hermitian different from the representation of normal boson operators GuAnn ; LLM . The imaginary-frequency boson-number operators

$$\widehat{n}_{I}=\widehat{b}_{+}\widehat{b}_{-};\quad\widehat{n}_{I}^{{\dagger}}=-\widehat{b}_{-}\widehat{b}_{+}$$

(8)

are non-Hermitian with eigenstates given by

	$\displaystyle\widehat{n}_{I}|n\rangle_{r}$	$\displaystyle=$	$\displaystyle n|n\rangle_{r},_{r}\langle n|n=_{r}\langle n|\widehat{n}_{I}^{{\dagger}},$		(9)
	$\displaystyle\widehat{n}_{I}^{{\dagger}}|n\rangle_{l}$	$\displaystyle=$	$\displaystyle n|n\rangle_{l},_{l}\langle n|n=_{l}\langle n|\widehat{n}_{I}^{{\dagger}}$

The orthogonal condition is

$${}_{l}\langle n|m\rangle_{r}=\delta_{nm}.$$

(10)

The Hermitian Hamiltonian possesses discrete imaginary eigenvalues

$$\widehat{H}|n\rangle_{\gamma}=\pm i\Omega\left(n+\frac{1}{2}\right)|n\rangle_{\gamma},$$

(11)

with $\gamma=r,l$ respectively for the “ket” and “bra” states. Indeed the Hermitian Hamiltonian with imaginary eigenvalues Eq.(11) has a dual-set of eigenstates in agreement with the general theory Eq.(1).

Based on the commutation relation of imaginary-frequency boson-operators Eq.(4) the arbitrary order eigenstates can be generated from the ground states such that (see Appendix for the derivation)

	$\displaystyle|n\rangle_{r}$	$\displaystyle=$	$\displaystyle\frac{\left(\widehat{b}_{+}\right)^{n}}{\sqrt{n!}}|0\rangle_{r},$
	$\displaystyle|n\rangle_{l}$	$\displaystyle=$	$\displaystyle\frac{\left(\widehat{b}_{-}\right)^{n}}{i^{n}\sqrt{n!}}|0\rangle_{l}.$

From the ground-state equations

$$\widehat{b}_{-}|0\rangle_{r}=0;\widehat{b}_{+}|0\rangle_{l}=0$$

the arbitrary order eigenfunctions are seen to be imaginary-frequency polynomials

	$\displaystyle\psi_{r,n}\left(x,\Omega\right)$	$\displaystyle=$	$\displaystyle\frac{1}{\pi^{\frac{1}{4}}\sqrt{2^{n}n!}}\sqrt{i^{\left(n+\frac{1}{2}\right)}}\left(x-i\frac{d}{dx}\right)^{n}e^{-i\frac{x^{2}}{2}},$
	$\displaystyle\psi_{l,n}\left(x,\Omega\right)$	$\displaystyle=$	$\displaystyle\frac{1}{\pi^{\frac{1}{4}}\sqrt{2^{n}n!i^{\left(n-\frac{1}{2}\right)}}}\left(x+i\frac{d}{dx}\right)^{n}e^{i\frac{x^{2}}{2}},$		(12)

which derived analytically in the Appendix are spatially non-localized different from the Hermit polynomials of the normal oscillator. It may be better to remark that the wave functions are represented in the dimensionless coordinate $x$, which is scaled by the square root of frequency $\sqrt{\Omega}$. In the ordinary space coordinate, $x$ should be replaced by $\sqrt{\Omega}x$ (see Appendix). Thus the eigenfunctions Eq.(12) are actually frequency dependent in the usual space coordinate.

We apply a non-Hermitian interaction to the Hamiltonian for a particle in the inverted potential well

$$\widehat{H}=\widehat{H}_{0}+G\left(\widehat{S}_{+}-\widehat{S}_{-}\right),$$

(13)

where

$$\widehat{H}_{0}=2i\Omega\widehat{S}_{z}$$

and $G$ is the coupling constant. $\widehat{H}_{0}$ is Hermitian however with imaginary eigenvalue as shown in the previous section, while total Hamiltonian is non-Hermitian,

$$\widehat{H}^{{\dagger}}=\widehat{H}_{0}-G\left(\widehat{S}_{+}-\widehat{S}_{-}\right)$$

(14)

where $\widehat{S}_{+}$, $\widehat{S}_{-}$ given in Eq.(5) are anti-Hermitian $\widehat{S}_{+}^{{\dagger}}=-\widehat{S}_{+}$, $\widehat{S}_{-}^{{\dagger}}=-\widehat{S}_{-}$ seen from the definition of boson operators Eq.(3).

The non-Hermitian Hamiltonian with complex eigenvalues can be also transformed to a Hermitian one in general. To see this we assume that

$$\widehat{H}|u_{n}\rangle_{r}=\varepsilon_{n}|u_{n}\rangle_{r},\quad\widehat{H}^{{\dagger}}|u_{n}\rangle_{l}=\varepsilon_{n}^{\ast}|u_{n}\rangle_{l}.$$

Under a similarity transformation with a Hermitian operator $\widehat{R}^{{\dagger}}=\widehat{R}$, we have

$$\widehat{R}\widehat{H}\widehat{R}^{-1}|\widetilde{u}_{n}\rangle_{r}=\varepsilon_{n}|\widetilde{u}_{n}\rangle_{r},\quad\widehat{R}^{-1}\widehat{H}^{{\dagger}}\widehat{R}|\widetilde{u}_{n}\rangle_{l}=\varepsilon_{n}^{\ast}|\widetilde{u}_{n}\rangle_{l},$$

with

$$|\widetilde{u}_{n}\rangle_{l}=\widehat{R}^{-1}|u_{n}\rangle_{l},\quad|\widetilde{u}_{n}\rangle_{r}=\widehat{R}|u_{n}\rangle_{r}.$$

(15)

If the equality

$$\widehat{R}\widehat{H}\widehat{R}^{-1}=\widehat{R}^{-1}\widehat{H}^{{\dagger}}\widehat{R}\equiv\widetilde{H},$$

(16)

is satisfied, the transformed Hamiltonian $\widetilde{H}=\widetilde{H}^{{\dagger}}$ is Hermitian with complex eigenvalues.

To this end the Hamiltonian Eq.(13) can be diagonalized in terms of a similarity transformation with the Hermitian operator

$$\widehat{R}=e^{-i\frac{\eta}{2}\left(\widehat{S}_{+}+\widehat{S}_{-}\right)},$$

(17)

where the real-value parameter $\eta$ is to be determined. The $SU(1,1)$ generators are realized by the imaginay-frequency boson operators in Eq.(5). It is easy to check that the transformation operator Eq.(17) is Hermitian $\widehat{R}=\widehat{R}^{{\dagger}}$. Using the transformation relations GuAnn ; LLM

	$\displaystyle\widehat{R}\widehat{S}_{+}\widehat{R}^{-1}$	$\displaystyle=$	$\displaystyle\widehat{S}_{+}\cosh^{2}\frac{\eta}{2}-\widehat{S}_{-}\sinh^{2}\frac{\eta}{2}-i\widehat{S}_{z}\sinh\eta,$
	$\displaystyle\widehat{R}\widehat{S}_{-}\widehat{R}^{-1}$	$\displaystyle=$	$\displaystyle\widehat{S}_{-}\cosh^{2}\frac{\eta}{2}-\widehat{S}_{+}\sinh^{2}\frac{\eta}{2}+i\widehat{S}_{z}\sinh\eta,$
	$\displaystyle\widehat{R}\widehat{S}_{z}\widehat{R}^{-1}$	$\displaystyle=$	$\displaystyle\widehat{S}_{z}\cosh\eta+\frac{i}{2}(\widehat{S}_{+}-\widehat{S}_{-})\sinh\eta,$

the diagonalized Hamiltonian is indeed Hermitian

$$\widetilde{H}=\widehat{R}\widehat{H}\widehat{R}^{-1}=\widehat{R}^{-1}\widehat{H}^{{\dagger}}\widehat{R}=2i\Gamma_{I}\widehat{S}_{z},$$

(18)

with an effective imaginary frequency

$$\Gamma_{I}=\sqrt{\Omega^{2}-G^{2}}.$$

(19)

The non-Hermitian coupling constant $G$ decreases the slope of potential well and the effective frequency. There exists an exceptional point

$$G_{c}=\Omega,$$

where all eigenstates are degenerate with vanishing eigenvalue. The pending parameter $\eta$ is determined from the equation

$$\sinh\eta=\pm\frac{G}{\sqrt{\Omega^{2}-G^{2}}},$$

(20)

which for a real value of $\eta$ is valid below the exceptional point $G<G_{c}$. The Hamiltonian Eq.(18) describes the particle in a deformed inverted potential well with the reduced slope, which tends to zero at the exceptional point. The variation of potential-well shape is illustrated in Fig.1.

The eigenstates of transformed Hamiltonian are obviously

$$\widetilde{H}|\widetilde{u}_{n}\rangle_{\gamma}=\pm i\Gamma_{I}\left(n+\frac{1}{2}\right)|\widetilde{u}_{n}\rangle_{\gamma},$$

respectively for the “ket” and “bra” states, $\gamma=r,l$. In our case it is simply the eigenstate of imaginary-frequency boson-number $\widehat{n}_{I}$

$$|\widetilde{u}_{n}\rangle_{\gamma}=|n\rangle_{\gamma},$$

defined in equation Eq.(9). The eigenstates of the original non-Hermitian Hamiltonians in equations Eq.(13),Eq.(14) are obtained respectively by the inverse transforms

$$|u_{n}\rangle_{r}=\widehat{R}^{-1}|n\rangle_{r},\quad|u_{n}\rangle_{l}=\widehat{R}|n\rangle_{l},$$

with corresponding eigenvalues

$$\varepsilon_{n}=i\Gamma_{I}\left(n+\frac{1}{2}\right),$$

and

$$\varepsilon_{n}^{\ast}=-i\Gamma_{I}\left(n+\frac{1}{2}\right).$$

The lowest-layer energy spectrum with respect to interaction constant $G$ is displayed in Fig.2 for $n=0$ (black)$,1$ (blue)$,2$ (red). All eigenvalues vanish at the exceptional point $G_{c}$, which separates the real and imaginary domains of energy spectrum. We emphasize that the open orbits below the exceptional point $G_{c}$ are also quantized, however with two branches of imaginary discrete-eigenvalues.

With the coordinate representation of $SU(1,1)$ generators $\widehat{S}_{+}$ and $\widehat{S}_{-}$ in equations Eq.(3),Eq.(5)the eigenfunctions of ”bra” and ”ket” states are obtained as

	$\displaystyle u_{r,n}\left(x\right)$	$\displaystyle=$	$\displaystyle e^{-\frac{\eta}{2}\left(x^{2}-\frac{d^{2}}{dx^{2}}\right)}\psi_{r,n}\left(x,\Gamma_{I}\right),$
	$\displaystyle u_{l,n}\left(x\right)$	$\displaystyle=$	$\displaystyle e^{\frac{\eta}{2}\left(x^{2}-\frac{d^{2}}{dx^{2}}\right)}\psi_{l,n}\left(x,\Gamma_{I}\right),$

in which the value of parameter $\eta$ is solved from Eq.(20). The eigenfunctions $\psi_{\gamma,n}\left(x,\Gamma_{I}\right)=\langle x|\widetilde{u}_{n}\rangle_{\gamma}$ for $\gamma=r,l$  are the same as in the equation Eq.(12), however with the reduced frequency $\Gamma_{I}$ instead of $\Omega$. The state density probability is

	$\displaystyle\rho_{r,l}$	$\displaystyle=$	$\displaystyle\langle x|u_{r,n}\rangle\langle u_{l,n}|x\rangle=\psi_{r,n}\left(x,\Gamma_{I}\right)\psi_{l,n}^{\ast}\left(x,\Gamma_{I}\right),$
	$\displaystyle\rho_{l,r}$	$\displaystyle=$	$\displaystyle\rho_{r,l}^{\ast}.$

Beyond $G_{c}$ the potential becomes a normal oscillator-well as shown in Fig.(1). By analytical continuation extension the transformed Hamiltonian describes a normal oscillator

$$\widetilde{H}_{ext}=2\Gamma\widehat{S}_{z},$$

(21)

with

$$\Gamma=\sqrt{G^{2}-\Omega^{2}}.$$

The $SU(1,1)$ generators in this region are realized by the normal boson operators $\widehat{a}$, $\widehat{a}^{{\dagger}}$

$$\widehat{S}_{z}=\frac{1}{2}\left(\widehat{n}+\frac{1}{2}\right),\quad\widehat{n}=\widehat{a}^{{\dagger}}\widehat{a}$$

where

$$\widehat{a}=\frac{1}{\sqrt{2}}\left(\widehat{x}+i\widehat{p}\right),\widehat{a}^{{\dagger}}=\frac{1}{\sqrt{2}}\left(\widehat{x}-i\widehat{p}\right).$$

The commutation relation Eq.(6) of $SU(1,1)$ generators is invariant under the realization of normal boson-operators.

The eigenstates of transformed Hamiltonian Eq.(21) are that of normal oscillator

$$\widetilde{H}_{ext}|n\rangle=\Gamma\left(n+\frac{1}{2}\right)|n\rangle,$$

where $\widehat{n}|n\rangle=n|n\rangle$.

The classical counterpart of the quantum Hamiltonian Eq.(13) in the realization of imaginary-frequency boson operators Eq.(5),Eq.(3) is seen to be

$$H\left(x,p\right)=\Omega\left(\frac{p^{2}}{2}-\frac{x^{2}}{2}\right)-iGxp,$$

(22)

which is a complex function of canonical variables $x$, $p$ corresponding to the non-Hermitian interaction. We are going to find what is the classical counterpart of the transformed Hermitian Hamiltonian Eq.(18). To this end we adopt a canonical transformation of variables

	$\displaystyle x$	$\displaystyle=$	$\displaystyle X\cosh\frac{\eta}{2}-iP\sinh\frac{\eta}{2},$
	$\displaystyle p$	$\displaystyle=$	$\displaystyle iX\sinh\frac{\eta}{2}+P\cosh\frac{\eta}{2},$		(23)

in which parameter $\eta$ is to be determined. Substituting the canonical variable transformation Eq.(23) into the phase space Lagrangian

$$\mathcal{L=}\overset{.}{x}p-H\left(x,p\right)$$

(24)

with the classical Hamiltonian given by Eq.(22) yields

$$\mathcal{L}\left(X,P\right)=\mathcal{L}^{\prime}\left(X,P\right)+\frac{dF(X,P)}{dt},$$

(25)

where $F(X,P)$ is called the gauge or generating function. Two Lagrangians $\mathcal{L}\left(X,P\right)$ and

$$\mathcal{L}^{\prime}\left(X,P\right)=\overset{.}{X}P-H^{\prime}\left(X,P\right)$$

differing by a total time derivative of canonical-variable function $F(X,P)$ are gauge equivalent LW2023 , since they give rise to the same equation of motion. The gauge transformation of the Hamiltonian Eq.(22) is found as

$$H^{\prime}=\Gamma_{I}\left(\frac{P^{2}}{2}-\frac{X^{2}}{2}\right),$$

(26)

under the condition that the pending parameter $\eta$ is determined by the equation Eq.(20). The Hamiltonian $H^{\prime}$, which indeed is a real function corresponding to the Hermitian Hamiltonian, describes a particle in an inverted potential well. It is exactly the classical counterpart of quantum version $\widetilde{H}$ given in Eq.(18). The gauge function

$$F=\frac{1}{2}\left[\frac{i}{2}\left(P^{2}-X^{2}\right)-XP\right]\frac{G}{\Gamma_{I}}+\frac{1}{2}XP$$

is however a complex function of the canonical variables. The quantum-classical correspondence holds strictly XIN14 ; GuAnn for the Hamiltonian of imaginary spectrum with non-Hermitian interaction.

The hermiticity of Hamiltonian is not a necessary condition to possess real spectrum. It is insufficient either. The Hermitian Hamiltonian of a conservative system can also have complex eigenvalues, if the potential is unbounded from below. The complex eigenvalues of metastable states were studied long ago with the semiclassical method LM92 ; LM94 . Rigorous eigenvalues and states are still lacking. The Hermitian Hamiltonian with inverted potential well is solved in the algebraic method by means of imaginary-frequency boson operators. The imaginary spectrum is derived analytically along with a dual-set of eigenstates corresponding respectively to the complex conjugate eigenvalues. The density-operators are non-Hermitian invariants, which give rise to the density probability conservation in agreement with the basic requirement of quantum mechanics. The $SU(1,1)$ generator $\widehat{S}_{z}$, which possesses always real spectrum determined by the commutation relation, however is non-Hermitian in the realization of imaginary-frequency boson operators. An interesting observation is that the Hermitian Hamiltonian possesses imaginary eigenvalues while the spectrum of non-Hermitian operator $\widehat{S}_{z}$ is real. It is well known that the spectrum of the pseudo-Hermitian Hamiltonians is not necessarily real in the whole region of parameter value. It approaches a complex domain at the exceptional point when the coupling constant of non-Hermitian interaction increases. Different from the common belief we find the opposite direction of transition from imaginary to real domains of eigenvalues at the exceptional point. In the considered $SU(1,1)$ Hamiltonian the non-Hermitian interaction decreases the slope of inverted potential well. The potential-well slope vanishes at the exceptional point, where all eigenstates are degenerate with zero eigenvalues. Beyond the exceptional point the potential becomes normal well with a real spectrum. Although the $SU(1,1)$ generator $\widehat{S}_{z}$ possesses real spectrum always, the boson-operator realization of it can be either Hermitian with the normal boson or non-Hermitian with the imaginary-frequency boson operators. The quantum-classical one to one correspondence exists strictly for the Hermitian and non-Hermitian Hamiltonians.

This work was supported by the National Natural Sci ence Foundation of China (Grant No. 12374312), ShanxiScholarship Council of China (Grant No. 2022-014 and 2023-033).

The authors declare no conflict of interest.

The data that support the findings of this study are available from the cor responding author upon reasonable request.

Hamiltonian of inverted potential:

The Hamiltonian for a particle of unit mass in an inverted potential well reads

$$\widehat{H}=\frac{\widehat{p}^{2}}{2}-\frac{1}{2}\Omega^{2}\widehat{x}^{2}$$

(27)

with the conventional unit $\hbar=1$. In the dimensionless variables with $\widehat{p}$ replaced by $\widehat{p}/\sqrt{\Omega}$and $\widehat{x}$ by $\sqrt{\Omega}\widehat{x}$ we have the Hamiltonian Eq.(2). The Hamiltonian Eq.(27) can be regarded as a harmonic oscillator of imaginary frequency,

$$\widehat{H}=\frac{\widehat{p}^{2}}{2}+\frac{1}{2}\left(i\Omega\right)^{2}\widehat{x}^{2}.$$

Imaginary frequency polynomials:

Using the commutation relation of boson operators Eq.(4) it is easy to verify

$$\widehat{n}_{I}\widehat{b}_{-}|n\rangle_{r}=\left(n-1\right)\widehat{b}_{-}|n\rangle_{r}.$$

Thus $\widehat{b}_{-}$ is a lowering operator for the “ket” state. We assume that

$$\widehat{b}_{-}|n\rangle_{r}=c_{n-}^{r}|n-1\rangle_{r},$$

(28)

in which the coefficient $c_{n-}^{r}$ is to be determined. $\widehat{b}_{+}$ acts as a rising operator, since

$$\widehat{n}_{I}\widehat{b}_{+}|n-1\rangle_{r}=n\widehat{b}_{+}|n-1\rangle_{r}.$$

We may define

$$\widehat{b}_{+}|n-1\rangle_{r}=c_{\left(n-1\right)+}^{r}|n\rangle_{r}.$$

Acting the rising operator on the Eq.(28) yields

$$\widehat{n}_{I}|n\rangle_{r}=c_{n-}^{r}c_{\left(n-1\right)+}^{r}|n\rangle_{r},$$

and then we have from the orthogonal condition Eq.(10) the recurrence relation

$$c_{n-}^{r}c_{\left(n-1\right)+}^{r}=n.$$

Repeating the same procedure on the states $|n-1\rangle_{r}$, $|n-2\rangle_{r}$ ; $|n-2\rangle_{r}$, $|n-3\rangle_{r}$ ;$\cdot\cdot\cdot;$we obtain

$$c_{\left(n-1\right)-}^{r}c_{\left(n-2\right)+}^{r}=n-1;$$

$$c_{\left(n-2\right)-}^{r}c_{\left(n-3\right)+}^{r}=n-2;$$

$$\cdot\cdot\cdot\cdot\cdot\cdot$$

$$c_{1-}^{r}c_{0+}^{r}=1.$$

For the boson number ground-state we require

$$\widehat{b}_{-}|0\rangle_{r}=0,$$

(29)

and

$$c_{0-}^{r}=0.$$

The coefficients are determined as

	$\displaystyle c_{n-}^{r}$	$\displaystyle=$	$\displaystyle c_{\left(n-1\right)+}^{r}=\sqrt{n};c_{\left(n-1\right)-}^{r}=c_{\left(n-2\right)+}^{r}=\sqrt{n-1};\cdot\cdot\cdot;$
	$\displaystyle c_{1-}^{r}$	$\displaystyle=$	$\displaystyle c_{0+}^{r}=1.$

Thus the $n$th “ket” state can be generated from ground state with the rising operator

$$|n\rangle_{r}=\frac{(\widehat{b}_{+})^{n}}{c_{\left(n-1\right)+}^{r}...c_{0+}^{r}}|0\rangle_{r}=\frac{(\widehat{b}_{+})^{n}}{\sqrt{n!}}|0\rangle_{r}.$$

For the “bra” states situation is opposite that $\widehat{b}_{-}$ acts as a rising operator

$$\widehat{n}_{I}^{{\dagger}}\widehat{b}_{-}|n\rangle_{l}=\left(n+1\right)\widehat{b}_{-}|n\rangle_{l},$$

and thus

$$\widehat{b}_{-}|n\rangle_{l}=c_{n-}^{l}|n+1\rangle_{l}.$$

While $\widehat{b}_{+}$ is lowering operator

$$\widehat{n}_{I}^{{\dagger}}\widehat{b}_{+}|n\rangle_{l}=\left(n-1\right)\widehat{b}_{+}|n\rangle_{l},$$

and

$$\widehat{b}_{+}|n\rangle_{l}=c_{n+}^{l}|n-1\rangle_{l}.$$

(30)

The recurrence relation for the “bra” states is obtained from the equations Eq.(8),Eq.(30) as

$$n|n\rangle_{l}=\widehat{n}^{{\dagger}}|n\rangle_{l}=-c_{n+}^{l}c_{\left(n-1\right)-}^{l}|n\rangle_{l},$$

and from the orthogonal condition Eq.(10) we have

$$c_{n+}^{l}c_{\left(n-1\right)-}^{l}=-n.$$

Repeating the same procedure on the states $|n-1\rangle_{l}$, $|n-2\rangle_{l}$ ; $|n-2\rangle_{l}$, $|n-3\rangle_{l}$ ;$\cdot\cdot\cdot;$the result is

$$c_{(n-1)+}^{l}c_{\left(n-2\right)-}^{l}=-\left(n-1\right);$$

$$\cdot\cdot\cdot\cdot\cdot\cdot$$

$$c_{1+}^{l}c_{0-}^{l}=-1.$$

For the ground state it is

$$\widehat{b}_{+}|0\rangle_{l}=0,$$

(31)

and

$$c_{0+}^{l}=0.$$

The solutions of coefficients are seen to be

	$\displaystyle c_{n+}^{l}$	$\displaystyle=$	$\displaystyle c_{\left(n-1\right)-}^{l}=i\sqrt{n},c_{(n-1)+}^{l}=c_{\left(n-2\right)-}^{l}=i\sqrt{n-1};\cdot\cdot\cdot;$
	$\displaystyle c_{1+}^{l}$	$\displaystyle=$	$\displaystyle c_{0-}^{l}=i,$

from which the $n$th “bra” state can be generated from the ground state such as

$$|n\rangle_{l}=\frac{\left(\widehat{b}_{-}\right)^{n}}{i^{n}\sqrt{n!}}|0\rangle_{l}.$$

In coordinate representation the ground-state equation Eq.(29) is

$$\left(x-i\frac{d}{dx}\right)\psi_{r,0}=0,$$

from which the ground “ket” state is found as

$$\psi_{r,0}=\frac{1}{\sqrt{N}}e^{-i\frac{x^{2}}{2}},$$

where $N$ is the normalization constant to be determined from the orthonormal condition Eq.(10) between “bra” and “ket” states. The arbitrary order eigenfunctions can be generated from the ground state such as

$$\psi_{r,n}=\frac{1}{\sqrt{N}\sqrt{n!}}\left(\sqrt{\frac{i}{2}}\right)^{n}\left(x-i\frac{d}{dx}\right)^{n}e^{-i\frac{x^{2}}{2}}.$$

With the same procedure we have the “bra” ground-state equation Eq.(31)

$$\left(x+i\frac{d}{dx}\right)\psi_{l,0}=0,$$

in the coordinate representation and the wave function

$$\psi_{l,0}=\frac{1}{\sqrt{N}}e^{i\frac{x^{2}}{2}}.$$

The arbitrary-order wave function of “bra” state is

$$\psi_{l,n}=\frac{1}{i^{n}\sqrt{n!}\sqrt{N}}\left(\sqrt{\frac{i}{2}}\right)^{n}\left(x+i\frac{d}{dx}\right)^{n}e^{i\frac{x^{2}}{2}}.$$

The normalization constant can be determined from the orthonormal condition Eq.(10) between “bra” and “ket” states

$$\int\psi_{l,0}^{\ast}\psi_{r,0}dx=\frac{1}{N}\int e^{-ix^{2}}dx=1.$$

The normalization constant

$$N=\sqrt{\frac{\pi}{i}},$$

is then obtained by means of the imaginary integral measure as in the path integral of quantum mechanics (see for example LW2023 ). The normalized wave functions are given by equation Eq.(12) in Sec.II.