Abstract The extent to which quantum computers can simulate physical phenomena and solve the partial differential equations (PDEs) that govern them remains a central open question. In this work, one of the most fundamental PDEs is addressed: the multidimensional transport equation with space- and time-dependent coefficients. We present a quantum numerical scheme based on three steps: quantum state preparation, evolution, and measurement of relevant observables. The evolution step combines a high-order centered finite difference with a time-splitting scheme based on product formula approximations, also known as Trotterization. We introduce a novel vector-norm analysis and prove that the number of time-steps can be reduced by a factor exponential in the number of qubits, compared with previously established operator-norm analysis. This new scaling significantly reduces the projected computational resources, independently of the circuit implementation of the trotterized evolution operator. We also present efficient quantum circuits based on (sparse) Walsh approximations along with numerical simulations that confirm the predicted vector-norm scaling. We report results on real quantum hardware for the one-dimensional convection equation, and solve a non-linear ordinary differential equation via its associated Liouville equation, a particular case of transport equations. This work provides a practical framework for efficiently simulating transport phenomena on quantum computers, with potential applications in plasma physics, molecular gas dynamics and non-linear dynamical systems, including chaotic systems.

The mixed quantum-classical Liouville equation (QCLE) provides an approximate perturbative framework for describing the dynamics of systems with coupled quantum and classical degrees of freedom of disparate thermal wavelengths. The evolution governed by the Liouville operator preserves many properties of full quantum dynamics, including the conservation of total population, energy, and purity, and has shown quantitative agreement with exact quantum results for the expectation values of many observables where direct comparisons are feasible. However, since the QCLE density matrix operator is obtained from the partial Wigner transform of the full quantum density matrix, its matrix elements can have negative values, implying that the diagonal matrix elements behave as pseudo-densities rather than densities of classical phase space. Here, we compare phase-space distributions generated by exact quantum dynamics with those produced by QCLE evolution from pure quantum initial states. We show that resonance effects in the off-diagonal matrix elements differ qualitatively, particularly for low-energy states. Furthermore, numerical and analytical results for low-dimensional models reveal that the QCLE can violate the positivity of marginal phase-space densities, a property that should hold at all times for any physical system. A perturbative analysis of a model system confirms that such violations arise generically. We also show that the violations of positivity of the marginal densities vanish as the initial energy of the system increases relative to the energy gap between subsystem states. These findings suggest that a negativity index, quantifying deviations from positivity, may provide a useful metric for assessing the validity of mixed quantum-classical descriptions.

ABSTRACT A general approach to modeling irreversibility starting from microscopic reversibility is presented. The time t s up to which relevant degrees of freedom of a system are tracked is extremely much shorter than the spectral resolution time t e that would be necessary to resolve the spectrum of all degrees of freedom involved. A relaxator that breaks reversibility condenses in the Liouville operator of the relevant degrees of freedom. The irrelevant degrees of freedom act as an environment for the system. The irreversible relaxator Liouville equation contains memory effects and initial correlations of all degrees of freedom. Stationary states turn out to be generically unique and independent of the initial conditions and exceptions are due to degeneracies. Equilibrium states lie in the relaxator's kernel yielding a stationary Pauli master equation. Kinetic equations for one‐particle densities are constructed as special cases of relaxator Liouville dynamics. Kubo's linear response theory is generalized to relaxator Liouville dynamics and related to irreversibility within the system. In a weak coupling approximation between system and environment the relaxator can be reduced to environmental correlations and bilinear system operators. Markov approximation turns the relaxator Liouville dynamics into a semi‐group dynamics.

ABSTRACT In this study, inverse nodal problems are studied for Sturm–Liouville equations with point δ and δ′ interactions. The study proves the existence of a solution to the inverse nodal problem and provides a practical method for finding the solution.

In this paper, we study the dynamics of a system consisting of two identical qubits interacting non-resonantly with a single-mode field of an ideal resonator in the presence of direct dipole-dipole and Ising interactions between the qubits. In the representation of "dressed" states, we find an exact solution of the quantum time Liouville equation for the full density matrix of the system under consideration for the initial separable and entangled states of the qubits and the thermal initial state of the resonator field. Based on the exact solution, the criterion for qubit entanglement, negativity, is calculated. The results of numerical modeling of negativity show that the inclusion of dipole-dipole and Ising coupling in the case of separable initial states of qubits can lead to a significant increase in the maximum degree of their entanglement, and in the case of entangled initial states of qubits, to stabilization of the initial entanglement. For the model with resonant interaction of qubits and the resonator field, direct interaction of qubits can also lead to the disappearance of the sudden death of entanglement effect.

ABSTRACT Stochastic evolution underpins several approaches to the dynamics of open quantum systems, such as random modulation of Hamiltonian parameters, the stochastic Schrödinger equation (SSE), and the stochastic Liouville equation (SLE). These approaches replace the explicit system–environment coupling with an effective system‐only dynamics, where dissipative behavior emerges from ensemble averaging. Stochastic Hamiltonians, in particular, have long served as phenomenological tools in physical chemistry to include environmental effects without recourse to an explicit microscopic derivation. In this work, we aim at a self‐contained and accessible presentation of these approaches to further elaborate on their common roots in essential concepts of stochastic calculus and to delineate the conditions under which they are equivalent. We also discuss how different formulations naturally lead to different numerical time‐integration schemes, better suited for either classical simulation platforms, based on finite‐difference approximations, or quantum algorithms, that employ random unitary maps. Our analysis aims at providing a unified perspective and actionable recipes for classical and quantum implementations of stochastic evolution in the simulation of open quantum systems.

ABSTRACT In this research, the inverse nodal problem is investigated for second‐order differential operators defined on a finite interval, incorporating discontinuity conditions within the interval. For each point of discontinuity , the existence of a solution to the inverse nodal problem is established, and a constructive method for determining the solution is presented.

We present a new polarization-selective detection scheme that enables simultaneous dual-frequency laser stabilization and implementation of high-contrast coherent population trapping (CPT) resonance. The proposed configuration splits elliptically polarized light into two optical paths after atomic interaction. The component reflected from the polarizing beam splitter, containing Doppler-broadened spectral features, is employed for precise laser frequency locking. The reflected signal exhibits a frequency discrimination slope nine times steeper than that of the transmitted polarizing beam splitter component. Simultaneously, the transmitted component isolates the CPT resonance signals from the positive and negative first-order sidebands for microwave frequency stabilization. Experiment demonstrates excellent agreement with theoretical simulations derived from the Liouville equation. Compared with conventional circular polarization methods, this approach achieves a fourfold enhancement in short-term frequency stability. These advancements make the technique particularly promising for developing compact CPT atomic clocks with improved metrological performance.

A steady-state approach for analysis of high-resolution relaxometry.

In this work, singular impulsive dynamic Sturm{Liouville equations are studied. Weyl’s famous classification was obtained for equations of this type. In this context, firstly, it is defined as the maximal operator that corresponding to the problem. Then, under impulsive conditions, we studied the square integrability of the solutions to the dynamic Sturm{Liouville problem. Weyl’s well-known limit-circle/limit-point situations were examined last.

ABSTRACT We present the construction of a complete system of functions associated with the Sturm–Liouville equation in impedance form on a finite interval , given an impedance function . The system, known as the formal powers , is generated through recursive integration of the impedance function and its reciprocal. We establish the completeness of this system in the space with the weight function . Under additional conditions on , we extend this completeness to Sobolev spaces , along with a generalized Taylor formula. We show that the completeness of the formal powers implies key analytic properties for a transmutation operator associated with the Sturm–Liouville equation in impedance form, including the existence of a continuous inverse. Finally, we introduce a formulation of the Darboux‐transformed equation and establish a relation between the transmutation operators to the original and transformed equations.

Critical Point Approaches for the Existence of Multiple Solutions of a Mixed Boundary Value Problem for a Complete Sturm–Liouville Equation

In this paper, we study the line-shape (LS), which indicates the amount of absorbed energy, and the line-width (LW), which indicates the scattering factor, according to the vibrational direction of the externally applied energy in the electron–phonon potential interaction system of representative semiconductor bonding types, group II–VI (ZnO) and group III–V (GaN) bonded compound semiconductors and pure group IV (Si) bonded semiconductors. One of the two systems receives the externally applied energy of right-handed circular polarization vibration, and the other receives the externally applied energy of left-handed circular polarization vibration. To analyze the quantum transport, we first employ quantum transport theory (QTR) for an electron system confined within a square-well potential, where the projected Liouville equation is addressed using the balanced-average projection method. In analyzing quantum transitions, phonon emission is linked to the transition line-width (LW), whereas phonon absorption is evaluated through the transition line-shape (LS), highlighting its sensitivity to temperature and magnetic field variations. As a result of analyzing the line-width (LW), which is a quantum scattering coefficient, and the line-shape (LS), which represents the absorbed power, the absorbed power and scattering coefficient were higher for the left circularly polarized vibration under the influence of the external magnetic field. In contrast, the right polarization produced smaller values. In addition, the scattering coefficient (LW) and the absorbed power according to the bonding type of the semiconductor were the largest in Si, a group IV bonded semiconductor, followed by group III–V (GaN) and group II–VI (ZnO) bonded semiconductors.

We consider solutions to some semilinear elliptic equations on complete noncompact Riemannian manifolds and study their classification as well as the effect of their presence on the underlying manifold. When the Ricci curvature is nonnegative, we prove both the classification of positive solutions to the critical equation and the rigidity for the ambient manifold. The same results are obtained when we consider solutions to the Liouville equation on Riemannian surfaces. The results are obtained via a suitable P -function whose constancy implies the classification of both the solutions and the underlying manifold. The analysis carried out on the P -function also makes it possible to classify nonnegative solutions for subcritical equations on manifolds enjoying a Sobolev inequality and satisfying an integrability condition on the negative part of the Ricci curvature. Some of our results are new even in the Euclidean case.

ABSTRACT In this paper, we consider a nonlinear eigenvalue problem consisting of nonlinear Sturm–Liouville equation with Dirichlet boundary conditions on a symmetric interval, where is the eigenparameter. We study the inverse problem associated with this problem in ‐framework and provide a procedure for constructing the nonlinear function .

Mixed solutions to the Liouville equation

Abstract We examine four dimensional, near-extremal black hole solutions in the presence of a finite boundary obeying conformal boundary conditions, where the conformal class of the induced metric and the trace of the extrinsic curvature are fixed. Working in Euclidean signature and at fixed charge, we find the near-extremal regime is dominated by a double-scaling limit which reveals new scaling laws for the quasi-local conformal entropy at low temperatures. Upon spherical dimensional reduction, we obtain the effective two-dimensional dilaton-gravity theory that describes the near-extremal regime. In contrast to Dirichlet boundaries, for conformal boundaries a linear dilaton potential is not sufficient to capture the leading correction away from extremality and higher orders are needed. We also examine near-Nariai solutions and the circular reduction of pure three-dimensional gravity in (Anti-) de Sitter space. In the latter, provided the boundary is placed near the conformal boundary of three-dimensional Anti-de Sitter space, the dynamics of the spherically symmetric boundary mode is governed by a Liouville equation that descends from a (minus) Schwarzian effective action.

Abstract In this paper, we study the fractional Caffarelli–Kohn–Nirenberg (CKN) inequality in one dimension when the parameter $\gamma $ converges (from the left) to its critical value $1/2$, obtaining Onofri’s inequality in the unit disk as the limit. A difficulty that we encounter is the lack of an explicit expression for the extremal function at which the CKN inequality is attained, which we address by studying solutions of the weighted Liouville equation for the half-Laplacian in dimension one.

Classical Solution of the First Mixed Problem for the Inhomogeneous Liouville Equation in a Half-Strip

Abstract We prove that Palais–Smale sequences for Liouville-type functionals on closed surfaces are precompact whenever they satisfy a bound on their Morse index. As a byproduct, we obtain a new proof of existence of solutions for Liouville-type mean-field equations in a supercritical regime. Moreover, we also discuss an extension of this result to the case of singular Liouville equations.