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In this study, our goal is to integrate classical mathematical models with deep neural networks by introducing two novel deep neural network models for image segmentation known as Double-well Nets. Drawing inspirations from the Potts model, our models leverage neural networks to represent a region force functional. We extend the well-known MBO (Merriman--Bence--Osher) scheme to solve the Potts model. The widely recognized Potts model is approximated using a doublewell potential and then solved by an operator-splitting method, which turns out to be an extension of the well-known MBO scheme. Subsequently, we replace the region force functional in the Potts model with a UNet-type network, which is data-driven and is designed to capture multiscale features of images, and also introduce control variables to enhance effectiveness. The resulting algorithm is a neural network activated by a function that minimizes the double-well potential. What sets our proposed Double-well Nets apart from many existing deep learning methods for image segmentation is their strong mathematical foundation. They are derived from the network approximation theory and employ the MBO scheme to approximately solve the Potts model. By incorporating mathematical principles, Double-well Nets bridge the MBO scheme and neural networks, and offer an alternative perspective for designing networks with mathematical backgrounds. Through comprehensive experiments, we demonstrate the performance of Double-well Nets, showcasing their superior accuracy and robustness compared to state-of-the-art neural networks. Overall, our work represents a valuable contribution to the field of image segmentation by combining the strengths of classical variational models and deep neural networks. The Double-well Nets introduce an innovative approach that leverages mathematical foundations to enhance segmentation performance. [ABSTRACT FROM AUTHOR]

Recent progress in studies of Rydberg double-well electronic energy states of MeNg (Me = 12-group atom, Ng = noble gas atom) van der Waals (vdW) molecules is presented and analysed. The presentation covers approaches in experimental studies as well as ab initio-calculations of potential energy curves (PECs). The analysis is shown in a broader context of Rydberg states of hetero- and homo-diatomic molecules with PECs possessing complex 'exotic' structure. Laser induced fluorescence (LIF) excitation spectra and dispersed emission spectra employed in the spectroscopical characterization of Rydberg states are presented on the background of the diverse spectroscopic methods for their investigations such as laser vaporization–optical resonance (LV-OR), pump-and-probe methods, and polarization labelling spectroscopy. Important and current state-of-the-art applications of Rydberg states with irregular potentials in photoassociation (PA), vibrational and rotational cooling, molecular clocks, frequency standards, and molecular wave-packet interferometry are highlighted. [ABSTRACT FROM AUTHOR]

This work studies crack propagation in particulate nanocomposites using the phase-field method. The crack propagation has been simulated in a wide range of loadings and the critical load for the crack growth has been obtained. Surface tension, as an inelastic stress, is introduced in the model in a thermodynamically consistent way. The effect of surface tension on the crack tip velocity and the crack evolution has been discussed. The finite element method via COMSOL multiphysics software has been utilized to solve the coupled phase-field and elasticity equations. Modeling and prediction of crack propagation for nanocomposites including different nanoparticles and under different loadings are the main purposes of this work. It is found that the kinetics and morphology of the crack propagation depend on the elastic moduli and the surface energy of nanoparticles as well as their longitudinal and angular distances to each other. [ABSTRACT FROM AUTHOR]

This study investigated the effect of local structural coupling between the buffer and the superconducting layers on the superconductivity of a GdBa 2 Cu 3 O 7-x (GdBCO)/La 0.7 Sr 0.3 MnO 3 (LSMO) bilayer. Epitaxial bilayers consisting of GdBCO with varying thicknesses (100, 300, and 500 nm), with and without LSMO (100 nm), were fabricated on a (001) SrTiO 3 substrate. To investigate the local structural distortion within the GdBCO/LSMO system, the temperature-dependent extended X-ray absorption fine structure (EXAFS) spectra were measured at the Mn- and Cu–K absorption edges. The EXAFS analyses revealed local structural non-uniformity, which manifested as anomalies in the temperature dependence of both the Mn–O and Cu–O bond distributions. A strong correlation exists between the local structural distortion of the Mn–O bond and the degree of lattice instability in the CuO 2 plane, which exhibits a strong thickness dependence. The inter-atomic potential of oxygen oscillations with an anharmonic approximation was adopted to interpret the structural anomalies. As a result, a double-well potential was obtained near T c , owing to the occurrence of longer Cu–O bonds in the CuO 2 plane. The difference in the degree of structural anomalies depending on the GdBCO thickness manifested as a difference in the distance between the two wells, which closely correlates with the appearance of a superconducting state. These experimental results demonstrate the role of local lattice fluctuations in the superconducting mechanism of the GdBCO/LSMO bilayer. [ABSTRACT FROM AUTHOR]

The spontaneous symmetry breaking (SSB) induced by a specific component of a linearly coupled binary Bose–Einstein condensate was analyzed. The model is based on linearly coupled Schrödinger equations with cubic nonlinearity and double-well potential acting on only one of the atomic components. By numerical simulations, symmetric and asymmetric ground states were obtained, and an induced asymmetry in the partner field was observed. In this sense, it is adequately demonstrated that the linear coupling mixing the two-field component (Rabi coupling) promotes the (in)balance between the atomic species, as well as the appearance of the Josephson and SSB phases. [ABSTRACT FROM AUTHOR]

We have determined conditions when a pair of coupled waveguides, a common element for integrated room-temperature photonics, can act as a qubit based on a system with a double-well potential. Moreover, we have used slow-varying amplitude approximation (SVA) for the "classical" wave equation to study the propagation of electromagnetic beams in a couple of dielectric waveguides both analytically and numerically. As a part of an extension of the optical-mechanical analogy, we have considered examples of "quantum operations" on the electromagnetic wave state in a pair of waveguides. Furthermore, we have provided examples of "quantum-mechanical" calculations of nonlinear transfer functions for the implementation of the considered element in optical neural networks. [ABSTRACT FROM AUTHOR]

Quantum quartic single-well anharmonic oscillator Vao(x) = x2 + g2x4 and double-well anharmonic oscillator Vdw(x) = x2(1−gx)2 are essentially one-parametric, they depend on a combination (g2ℏ). Hence, these problems are reduced to study the potentials Vao = u2 + u4 and Vdw = u2(1 − u)2, respectively. It is shown that by taking uniformly-accurate approximation for anharmonic oscillator eigenfunction Ψao(u), obtained recently, see JPA 54 (2021) 295204 [1] and arXiv 2102.04623 [2], and then forming the function Ψdw(u) = Ψao(u)±Ψao(u−1) allows to get the highly accurate approximation for both the eigenfunctions of the double-well potential and its eigenvalues.

In this paper, the effects of slowly varying external excitation and the sign of the strength of signum term of system with a simple signum nonlinearity function are investigated by using the fast-slow analysis method and the transformation phase portrait method. Firstly, a double-well and single-well non-smooth potential are presented. A comparison of the proposed system and the Duffing system with the same fixed points is presented. Secondly, by taking the periodic excitation as a slow-varying parameter, the non-smooth system can possess two timescales and it can be transformed into two linear autonomous systems. The phase space can be divided into two regions by the non-smooth boundaries, in which the trajectory is governed by two different subsystems, respectively. Based on the analysis of the two subsystems, the stabilities and the bifurcations of the equilibrium branches of the fast subsystem are presented when the sign of parameter b of signum term is positive and negative. The results show that the stability of the system and its bifurcations are most sensitive to the change of sign of parameter b .Thirdly, through numerical simulations, the bursting oscillations with different waveforms are observed in the non-smooth system. It is found that for b > 0 , bursting oscillations presents two types of quiescent state (QS) and spiking state (SP), respectively, while if b < 0 , bursting patterns with four type of quiescent state and spiking state which are represented by Q S ± i (i = 1 , 2) and S P ± i (i = 1 , 2) , respectively. Based on the analysis of the bifurcations and portrait phase, the dynamical mechanism of the bursting oscillations is analyzed. Finally, the signum function is replaced by a sharply varying continuous hyperbolic tangent function. The results show that for a small value of constant parameter n , the system exhibits bursting patterns. It is also found that the amplitude and the number of the spikes of bursting oscillation depend on the value of n . The proposed system is implemented in field programmable gate array (FPGA) using hardware/software code-sign to verify the numerical simulation, because it can be applied in embedded engineering based on bursting. [ABSTRACT FROM AUTHOR]

The quantum states of the Kapitza pendulum are investigated within the framework of the effective potential obtained by the method of averaging over fast oscillations. An analytical estimate of the energy spectrum of stabilized states is given using a model potential. For the lowest states of an inverted pendulum, an expression is obtained for the spectrum in terms of energies of a harmonic oscillator corrected using perturbation theory. Tunneling effect corrections to the energies of resonance states in two wells of the effective potential are found. The results of calculations of the structure of vibrational and rotational spectra of the Kapitza pendulum by the semiclassical method and by the Numerov numerical algorithm are compared. [ABSTRACT FROM AUTHOR]

Hydrogen bonds are one of the most enigmatic intermolecular interactions where the strength of interaction depends very intimately on the local environment of the bonded atoms. One can change the entire nature of the interactions by changing its local environment. Many of the enzymatic reactions work by this above-mentioned mechanism. Hence, it is important to understand how changes in the local environment can modify hydrogen bonds. In line with this, complete structural and vibrational investigations on ammonium ion substituted potassium dihydrogen phosphate mixed crystals K0.988(NH4)0.012H2PO4 and (K0.956(NH4)0.044H2PO4) are performed. The behaviour of O–H–O hydrogen bond in these crystals is very sensitive to changes in the local environment. A small change in the concentration of K+ ion has an observable effect on hydrogen bond geometry and hence on the double-well potential energy contour for these bonds. The dynamics of H atom depends on the potential barrier height. It changes gradually with decreasing temperature from predominantly inter-well to completely intra-well dynamics with an intermediate precursor region, where inter-well and intra-well dynamics are important. [ABSTRACT FROM AUTHOR]

Thermally induced particle hopping in the nanoscale double-well potential is fundamental in material design and device operation. After the proposal of the basic hopping theory, several experimental studies, including some using the optical trapping method, have validated the theoretical approach over various friction ranges of the surrounding medium. However, only external parameters, such as viscosity, temperature, and pressures, have been varied in practical circumstances, and other tools capable of adjusting the potential profile itself to modulate the hopping rate are needed. By using metallic nanoantenna with various gap sizes and different optical pump power, we engineered a double-well potential landscape and directly observed the hopping of a single nanoparticle with a diameter of 4 nm. The distance between the two potential wells was 0.6–5 nm, and the maximum well depth and maximum height of the central potential barrier were approximately 69 and 4 kBT, respectively. The hopping rate was governed by the Arrhenius law and showed a vertex when the barrier height was approximately 2 kBT, which was in good agreement with the computational expectations.

We consider a spin-1 ferromagnetic Bose–Einstein condensate with Rashba spin–orbit coupling (SOC) and dipole–dipole interaction (DDI) in a double-well potential. For the nonrotating case, the ground state of the system is obtained by using the imaginary time propagation method, and a ground-state phase diagram is given with respect to the SOC strength and the DDI strength. It is shown that the nonrotating system sustains rich exotic ground-state phases including the ox-horn phase with hidden vortices and antivortices, density droplet phase with vortex–antivortex pairs, modulated stripe phase with vortex–antivortex chains, checkerboard phase with visible vortex–antivortex chains and hidden vortex–antivortex pairs, and the triangular vortex–antivortex lattice phase. For the rotating case, the dynamics of the system is investigated by using a phenomenological dissipation model. The effects of SOC, DDI and rotation on the unique dynamic behaviors of the rotating system and the final steady-state structures are revealed and analyzed. In particular, the rotating system supports fascinating novel spin textures and skyrmion excitations, and experiences a series of topological structure transitions such as the transitions from an interlaced skyrmion–antiskyrmion lattice pair to a skyrmion–meron lattice pair and then to a meron lattice pair.

We consider a non-linear Hartree energy for bosonic particles in a symmetric double-well potential. In the limit where the wells are far apart and the potential barrier is high, we prove that the ground state and first excited state are given to leading order by an even, respectively odd, superposition of ground states in single wells. The corresponding energies are separated by a small tunneling term that we evaluate precisely. [ABSTRACT FROM AUTHOR]

We investigate the dynamics and stationary states of a semiconductor exciton–polariton condensate in a double-well potential. We find that upon the population build-up of the polaritons by above-threshold laser pumping, coherence relaxation due to the phase fluctuations in the polaritons drives the system into a stable fixed point corresponding to a self-organized PT-symmetric phase. [ABSTRACT FROM AUTHOR]

In this paper, we report the existence of bursting oscillations in systems governed by a second-order differential equation with only one signum nonlinearity term and time-delay feedback, driven by the slowly varying external force. Depending on the sign of the strength of signum nonlinearity, two cases are studied: two-well and single-well potentials. The external force acts as the control parameter, the stability of equilibrium points is first discussed and the condition for Hopf bifurcation is established. Secondly, we present the effect of time delay on bursting oscillations when periodic forcing changes slowly. Our results show that the time delay is responsible for the appearance or disappearance of the bursting phenomenon. It is also found that the amplitude, the number of peaks and period of bursting oscillation depend on the value of the time delay. The bursting shapes obtained theoretically are exhibited experimentally using the real microcontroller simulation. [ABSTRACT FROM AUTHOR]

We analyze a finite element/boundary element procedure for a non-convex contact problem for the double-well potential. After relaxing the associated functional, the degenerate minimization problem is reduced to a boundary/domain variational inequality, a discretized saddle point formulation of which may then be solved numerically. The convergence of the Galerkin approximations to certain macroscopic quantities and a corresponding a posteriori estimate for the approximation error are discussed. Numerical results illustrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

We show that spherical 2-designs give stationary points of the classical mechanical system of particles in a Euclidean space interacting by a double-well potential. [ABSTRACT FROM AUTHOR]

In this article, we present a simple FreeFEM++ code to represent high-resolution boundaries of the optimal shape using reaction-diffusion equation–based topology optimization and adaptive mesh refinement. The design algorithm is divided into two parts to achieve a high quality of the boundary. In the first part, loop 1, the ordinary method for a reaction-diffusion equation–based topology optimization, is applied to obtain the optimal shape. In the second part, loop 2, a modified reaction-diffusion equation combined with a double well potential, is employed as an optimizer and the adaptive mesh refinement is used with a gradual decrease of the diffusion coefficient. Because the code is intended for educational purposes, this paper provides background knowledge for understanding of the finite element method and topology optimization. Various numerical examples are discussed to confirm the robustness of the proposed design algorithm. The corresponding code may be downloaded from the website http://ssd.yonsei.ac.kr. [ABSTRACT FROM AUTHOR]

A thermodynamically consistent phase field model for crack propagation is analyzed. The thermodynamic driving force for the crack propagation is derived based on the laws of thermodynamics. The Helmholtz free energy satisfies the thermodynamic equilibrium and instability conditions for the crack propagation. Analytical solutions for the Ginzburg–Landau equation including the surface profile and the estimation of the kinetic coefficient are found. It is shown how kinetic coefficient affects the local stress field. The local critical stress for the crack propagation is calibrated with the theoretical strength which gives the value of the crack surface width. The finite element method is utilized to solve the complete system of crack phase field and mechanics equations. The analytical expressions for the crack surface profile and the crack tip velocity are verified with the numerical solutions. It is shown that under mode I cracking and proper calibration of parameters, phase field models always agree with Griffith's theory. [ABSTRACT FROM AUTHOR]

Numerical and experimental results concerning the control of chaos in the double-well Duffing system with one external force are presented in this paper. Lyapunov exponent is used as the identifier of chaos in the nonlinear system. As the expression of the potential function depends on the damping coefficient, it is found that the depth of the potential well increases with damping. Thus, this paper presents a physical insight into control damping (or coefficient of damping of a system) to quench, and also to prevent chaotic situations, which are considered disturbances and are harmful for operations. It is found that when the depth of the double-well potential increases, it becomes more difficult for the state to switch from one equilibrium to other, and thus, the chaos is suppressed. The new feedback control method based on damping coefficient and Lyapunov exponent is proposed. The experimental investigation using electronic components shows that the results obtained are similar to those observed from numerical investigation. [ABSTRACT FROM AUTHOR]

• Existence and propagation of soliton solutions are investigated in double-well potentials with cubic nonlinearity. • Soliton solutions are obtained including single and multiple peaks. • Significant discussion is devoted to fundamental, tripole, and quadrupole solitons. • Effect of potential well separation on the formation and stability of solitons is studied. The existence, stability and propagation of spatial solitons in double-well potentials with cubic nonlinearity are investigated. The symmetric, tripole and quadrupole solitons show stability at lower power, while the symmetry-broken and antisymmetric solitons can remain stable throughout their existence curves. In comparison, the centered fundamental solitons are less stable than their symmetry-broken bound states and bifurcate from symmetric solitons. In this article, we study the effect of the separation between the two wells of the potential on the formation and stability of solitons. The threshold power of stable solitons tends to decrease with increasing separation for fundamental and symmetric solitons, while it is more complicated for tripole and quadrupole solitons. In general, there is a specific separation between two potential wells at which there are hardly any stable tripole and quadrupole solitons. Linear stability analysis and direct numerical simulations have provided evidence for the stability and propagation of solitons. [ABSTRACT FROM AUTHOR]

Minimal rotating thermodynamic systems are addressed. Particle m placed into the rotating symmetrical double-well potential (bowl), providing binary logical system is considered. The condition providing the transfer of the particle from one frictionless half-well to another, and, in this way, the possibility to record 1 bit of information is derived. The procedure of recording turns out to be irreversible; it is impossible to return the particle to its initial state under rotation about the same axis. The same rotating double-well system exerted to the thermal noise is considered. A minimal rotating thermal engine built of the rotating chamber, movable partition, and the particle confined within the chamber is treated. Rotation of the system displaces the partition, thus enabling erasing of one bit information. Erasing of 1 bit of information is due to the inertia (centrifugal force) acting on the partition. Isothermal expansion of the “minimal gas” expectedly gives rise to the Landauer bound. Compression of the “gas” with the rotation around the same axis is impossible and demands the additional axis of rotation. The interrelation between the possibility of recording/erasing information and the symmetry of the system is considered.

We investigate the dynamics and stationary states of a semiconductor exciton–polariton condensate in a double-well potential. We find that upon the population build-up of the polaritons by above-threshold laser pumping, coherence relaxation due to the phase fluctuations in the polaritons drives the system into a stable fixed point corresponding to a self-organized PT-symmetric phase.

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We consider the one-dimensional free Fokker-Planck equation ∂μt/∂t = ∂/∂x μt·(1/2 V'] -Hμt)], where H denotes the Hilbert transform and V is a particular double-well quartic potential, namely V (x) = 1/4 x4 + c/2 x², with c ≥-2. We prove that the solution (μt)t≥0 of this PDE converges in Wasserstein distance of any order p ≥ 1 to the equilibrium measure μV as t goes to infinity. This provides a first result of convergence for this equation in a non-convex setting. The proof involves free probability and complex analysis techniques. [ABSTRACT FROM AUTHOR]

In this article, we discuss stability of the one-dimensional overdamped Langevin equation in double-well potential. We determine unstable and stable equilibria, and discuss the rate of convergence to stable ones. Also, we derive conditions for stability of general diffusion processes which generalize the classical and well-known results of Khasminskii ( [8] ). [ABSTRACT FROM AUTHOR]

Confocal annular Josephson tunnel junctions (CAJTJs), which are the natural generalization of the circular annular Josephson tunnel junctions, have a rich nonlinear phenomenology due to the intrinsic non-uniformity of their planar tunnel barrier delimited by two closely spaced confocal ellipses. In the presence of a uniform magnetic field in the barrier plane, the periodically changing width of the elliptical annulus generates a asymmetric double-well for a Josephson vortex trapped in a long and narrow CAJTJ. The preparation and readout of the vortex pinned in one of the two potential minima, which are important for the possible realization of a vortex qubit, have been numerically and experimentally investigated for CAJTJs with the moderate aspect ratio 2:1. In this work, we focus on the impact of the annulus eccentricity on the properties of the vortex potential profile and study the depinning mechanism of a fluxon in more eccentric samples with aspect ratio 4:1. We also discuss the effects of the temperature-dependent losses as well as the influence of the current and magnetic noise. [ABSTRACT FROM AUTHOR]

We investigate the properties of a three-dimensional (3D) dipolar Bose-Einstein condensate (BEC) in a double-well potential (DWP). Symmetry breaking and tunneling dynamics phenomena are demonstrated for 164Dy atoms in the 3D DWP using an effective two-mode model. The results show that the symmetry properties of the dynamics are affected markedly by the long-range nature and anisotropy of the dipolar interaction and the isotropic contact interaction. [ABSTRACT FROM AUTHOR]

We investigate the coherent wave packet dynamics of a two-level atom trapped in a symmetric double-well potential in a near-resonance cavity. Prepared on one side of the double-well potential, the atom wave packet oscillates between the left and right wells, while recoil induced by the emitted photon from the atom entangles the atomic internal and external degrees of freedom. The collapse and revival of the tunneling occurs. Adjusting the width of the wave packets, one can modify the tunneling frequency and suppress the tunneling. [ABSTRACT FROM AUTHOR]

The interaction between a two-level atom trapped in symmetric double-well potential and a single mode cavity field is investigated. By solving Schrödinger equation, the analytical expressions of the total wave function and inversion of the atomic energy level for the whole system are given. Time evolution of the atomic population inversion is analyzed for different initial states of the cavity field, such as fock and coherent states. Influence of atomic mass centre motion on population inversion is considered. Results show that by choosing appropriate cavity field initial state, potential well location and related factors, the spontaneous emission rate of an atom can be effectively controlled and the atomic moving centre of mass can suppress Rabi oscillation. [ABSTRACT FROM AUTHOR]

We obtain eigenvalues and eigenfunctions of the Schrödinger equation with a hyperbolic double-well potential. We consider exact polynomial solutions for some particular values of the potential-strength parameter and also numerical energies for arbitrary values of this model parameter. We test the numerical method by means of a suitable exact asymptotic expression for the eigenvalues and also calculate critical values of the strength parameter that are related to the number of bound states supported by the potential. [ABSTRACT FROM AUTHOR]

While for a long time, iron-based superconductors have been investigated looking at the average crystalline structure, there is now growing interest on the divergence of local from the average structure. Here, we use advanced data analysis of EXAFS, a fast and local probe, which has key advantages compared to crystallographic measurements, in unveiling the dynamics of the local structure. The results show that the Fe-As pair oscillates in the double-well potential both in undoped and in cobalt-doped LaFeAsO crystals. The parameters and the characteristics of the double-well potential are determined by curve fits of the polarized EXAFS data. Temperature-dependent Fe-As pair distribution and tunneling frequency are derived from selected model of two potential wells. We observe the softening of the Fe-As oscillations at the tetragonal to orthorhombic structural phase transition at 150 K in LaFeAsO. A weak broad softening is observed while decreasing the temperature from 70 to 40 K in superconducting LaFeCoAsO which could be correlated with the onset of short-range charge density wave order. [ABSTRACT FROM AUTHOR]

In this work, we demonstrate the application of the variational method by computing the ground- and first-excited state energies of a double-well potential. We start with the proper choice of the trial wave functions using optimized parameters, and notice that accurate expectation values in excellent agreement with the numerical results can be aquired by cautious systematic improvement of trial wave functions. [ABSTRACT FROM AUTHOR]

Theoretical analysis of the effect of an external vibration on proton transfer (PT) in a hydrogen bond (HB) is carried out. It is based on the two-dimensional Schrödinger equation with trigonometric double-well potential. Its solution obtained within the framework of the standard adiabatic approximation is available. An analytic formula is derived that provides the calculation of PT rate with the help of elements implemented in Mathematica. We exemplify the general theory by calculating PT rate constant for the intermolecular HB in the Zundel ion H 5 O 2 + (oxonium hydrate). This object enables one to explore a wide range of the HB lengths. Below some critical value of the frequency of the external vibration the calculated PT rate yields extremely rich resonant behavior (multiple manifestations of bell-shaped peaks). It takes place at symmetric coupling of the external vibration to the proton coordinate. This phenomenon is absent for anti-symmetric and squeezed mode couplings. [Display omitted] • Effect of external vibration on proton transfer in a hydrogen bond is considered. • It is based on solution of Schrödinger equation with double-well potential. • Critical value of external vibration frequency exists for symmetric mode coupling. • Below it the rate constant yields multiple manifestations of bell-shaped peaks. • Proton transfer rate constant is calculated for hydrogen bond in Zundel ion. [ABSTRACT FROM AUTHOR]

We study quantum mechanics problem described by the Schrödinger equation with Kapitza pendulum potential, that is the asymmetric double-well potential on the circle. For the oscillatory states spatially localize around the two stable saddle positions of the potential, we obtain the perturbative eigenvalues and corresponding piecewise wavefunctions. The spectrum is computed by extending the angle coordinate to the complex plane so that the quantization condition is formulated as contour integral along a path extending in the imaginary direction. Quantum tunneling between the wells is computed. • Explicit solutions for rotating and oscillatory states of quantum Kapitza pendulum. • Oscillatory states are studied using generalised Floquet theorem for complex period. • The method provides a new perspective on the quantization rule and zero point energy. [ABSTRACT FROM AUTHOR]