Quantum mechanics of inverted potential well -- Hermitian Hamiltonian with imaginary eigenvalues, quantum-classical correspondence

We in this paper study the quantization of a particle in an inverted potential well. The Hamiltonian is Hermitian, while the potential is unbounded below. Classically the particle moves away acceleratingly from the center of potential top. The existing eigenstates must be unstable with imaginary eigenvalues, which characterize the decay rate of states. We solve the Hamiltonian problem of inverted potential well by the algebraic method with imaginary-frequency raising and lowering boson operators similar to the normal oscillator case. The boson number operator is non-Hermitian, while the integer-number eigenvalues are, of course, real. Dual sets of eigenstates, denoted by "bra" and "ket", are requested corresponding respectively to the complex conjugate number-operators. Orthonormal condition exists between the "bra" and "ket" states. We derive a spatially non-localized generating function, from which $n$-th eigenfunctions can be generated by the raising operators in coordinate representation. The "bra" and "ket" generating functions are mutually normalized with the imaginary integration measure. The probability density operators defined between the "bra" and "ket" states are non-Hermitian invariants, which lead to the Schr\H{o}dinger equations respectively for the "bra" and "ket" states. While probabilities of "bra" and "ket" states themselves are not conserved quantities because of the decay. The imaginary-frequency boson coherent states are defined as eigenstates of lowering operators. The minimum uncertainty relation is proved explicitly in the coherent states. Finally the probability average of Heisenberg equation in the coherent states is shown precisely in agreement with the classical equation of motion. The quantum-classical correspondence exists in the imaginary eigenvalue system.