Exact Solution of Schrödinger equation for Complex Mass Quantum System under Complex Morse Potential to study emergent matter types and its phases

TL;DR

The paper solves the Schrödinger equation for a complex-mass system under a complex Morse potential within an extended complex phase space, yielding exact expressions for real and imaginary energy parts and the corresponding eigenfunctions. By imposing a two-dimensional normalization in ECPS and analyzing parameter-space sectors, it identifies regimes of real-spectrum Hermitian-like matter, quasi-stable resonant states, purely complex non-Hermitian matter, non-physical regions, and a quasi-classical determinate regime with static probability density. A ROS (reality of spectrum) analysis clarifies when eigenvalues remain real and how parameter signs restrict admissible domains. The work provides a unified NH-quantum framework for interpreting stability, resonance, and emergent classicality from complex mass and Morse-parameter competition, with potential links to dark-matter–like behavior in a non-Hermitian setting.

Abstract

We present exact solutions of the Schrödinger equation for a quantum system with complex mass subjected to a complex Morse potential in the extended complex phase space. The normalized eigenfunctions and corresponding eigenspectra are derived within a non-Hermitian framework, ensuring consistent probability densities. Conditions for the reality of the spectra are established and used to analyze the dependence of eigenvalue behaviour on potential parameters. The study reveals distinct regimes of spectral characteristics arising from the interplay of complex mass, the Morse parameter, and eigenvalues, leading to the emergence of five intrinsic matter types. By analysing the energy eigenspectra, normalization conditions, and probability density profiles across parameter space, we identify regimes corresponding to real-spectrum Hermitian-like matter, quasi-stable or resonant states, purely complex quantum matter, non-physical, non-normalizable states, and a quasi-classical determinate regime in which the probability density becomes spatially static. One of these system exhibits a non-dissipative, collisionless state with long-range gravitational-like characteristics, suggesting a theoretical analogue for dark matter within a non-Hermitian quantum framework. Further, the five identified classes of matter may be interpreted as distinct phases of a single quantum system governed by complex mass and Morse parameters This classification elucidates the boundary between physical and non-physical regimes in complex quantum systems and provides a unified approach for interpreting stability, resonance, and emergent classicality arising from complex parameters.

Figures (9)

• Figure 1: The ground state (GS) probability density plots of the hydrogen ($H_2$) molecule with respect to $m_i$. (a) shows variation of peak value of the probability density with respect to $m_i$ for the fixed value of $a_i$ taken as $a_i = a_r$. Here, the minima exists at $m_i = m_r$. (b) shows dependency of peak probability density on complex mass parameter for various values of $a_i$. (c) and (d) reveals the variation of spartial confingement of probability density with varying $m_i$ and fixed $a_i$ as $a_i = a_r$. The value of real parameters for the $H_2$ molecule are taken as $V_{or} = 38266$$cm^{-1}$, $a_r = 1.868$$\AA^{-1}$, $m_r = 0.5039$$\mu$.
• Figure 2: The ground state (GS) probability density plots of the hydrogen ($H_2$) molecule with respect to $a_i$. (a) and (b) reveals the variation of spartial confingement of probability density with varying $a_i$ and fixed value of $m_i$ taken as $m_i = m_r$. (c) shows variation of peak value of the probability density with respect to $a_i$ and fixed value of $m_i$, as $m_i = m_r$. (d) shows dependency of peak probability density on complex Morse parameter for various values of $m_i$. The value of real parameters for the $H_2$ molecule are taken as $V_{or} = 38266$$cm^{-1}$, $a_r = 1.868$$\AA^{-1}$, $m_r = 0.5039$$\mu$.
• Figure 3: The ground state (GS) peak probability density (PPD) region plots of the hydrogen ($H_2$) molecule in the parametric space defined by ($a_i$, $m_i$). The blue shaded region shows permissible values of ($a_i$, $m_i$) for positive PPD and region shaded with black and grey corresponds to the values of ($a_i$, $m_i$) for which PPD is non-normalizable and negative, respectively. Here, The value of real parameters for the $H_2$ molecule are taken as $V_{or} = 38266$$cm^{-1}$, $a_r = 1.868$$\AA^{-1}$, $m_r = 0.5039$$\mu$.
• Figure 4: The Variation of the real part of the ground state (GS) energy eigenfunction ($E_r$) of the hydrogen molecule ($H_2$) with respect to imaginary parameters, ($a_i$, $m_i$). (a) and (b) reveals region plot in the parameteric space of ($a_i$, $m_i$). The region shaded with red and gray corresponds to positive and negative value of $E_r$, i.e, defines permissible region of ($a_i$, $m_i$), whereas the black shaded region corresponds to region where PPD is non-normalizable or negative. (c) reveals the nature of $E_r$ with respect to $m_i$ for various values of $a_i$ and (d) shows the nature of $E_r$ with respect to $a_i$ for various values of $m_i$. Here, The value of real parameters for the $H_2$ molecule are taken as $V_{or} = 38266$$cm^{-1}$, $a_r = 1.868$$\AA^{-1}$, $m_r = 0.5039$$\mu$.
• Figure 5: The Variation of the Imaginary part of the ground state (GS) energy eigenfunction ($E_i$) of the hydrogen ($H_2$) molecule with respect to imaginary parameters, ($a_i$, $m_i$). (a) and (b) reveals region plot in the parameteric space of ($a_i$, $m_i$). The region shaded with orange and gray corresponds to positive and negative value of $E_i$, i.e, defines permissible region of ($a_i$, $m_i$), whereas the black shaded region corresponds to region where PPD is non-normalizable or negative. (c) reveals the nature of $E_i$ with respect to $m_i$ for various values of $a_i$ and (d) shows the nature of $E_i$ with respect to $a_i$ for various values of $m_i$. Here, The value of real parameters for the $H_2$ molecule are taken as $V_{or} = 38266$$cm^{-1}$, $a_r = 1.868$$\AA^{-1}$, $m_r = 0.5039$$\mu$.