Two families of superintegrable and isospectral potentials in two dimensions

Supersymmetric isospectral formalism for the calculation of near-zero energy states: application to the very weakly bound ${^4}$He trimer excited state

We propose a novel mathematical approach for the calculation of near-zero energy states by solving potentials which are isospectral with the original one. For any potential, families of strictly isospectral potentials (with very different shape) having desirable and adjustable features are generated by supersymmetric isospectral formalism. The near-zero energy Efimov state in the original potential is effectively trapped in the deep well of the isospectral family and facilitates more accurate calculation of the Efimov state. Application to the first excited state in 4He trimer is presented.

Self-isospectral tri-supersymmetry in PT-symmetric quantum systems with pure imaginary periodicity

We study a reflectionless PT-symmetric quantum system described by the pair of complexified Scarf II potentials mutually displaced in the half of their pure imaginary period. Analyzing the rich set of intertwining discrete symmetries of the pair, we find an exotic supersymmetric structure based on three matrix differential operators that encode all the properties of the system, including its reflectionless (finite-gap) nature. The structure we revealed particularly sheds new light on the splitting of the discrete states into two families, related to the bound and resonance states in Hermitian …

利用胚芽图（germ graph）概念构造等散射非等谱量子图

Construction of isoscattering non-isospectral quantum graphs using the concept of a germ graph

本文利用胚芽图概念，将含Γ_A2、Γ_A3的等散射非等谱图作为附件连接到胚芽图，构造出Γ_3、Γ_4，验证其非等谱并研究其等散射特性。

Rational deformations of conformal mechanics

We study deformations of the quantum conformal mechanics of De Alfaro-Fubini-Furlan with rational additional potential term generated by applying the generalized Darboux-Crum-Krein-Adler transformations to the quantum harmonic oscillator and by using the method of dual schemes and mirror diagrams. In this way we obtain infinite families of isospectral and non-isospectral deformations of the conformal mechanics model with special values of the coupling constant $g=m(m+1)$, $m\in {\mathbb N}$, in the inverse square potential term, and for each completely isospectral or gapped deformation given b…

Hearing the shape of a quantum boundary condition

We study the isospectrality problem for a free quantum particle confined in a ring with a junction, analyzing all the self-adjoint realizations of the corresponding Hamiltonian in terms of a boundary condition at the junction. In particular, by characterizing the energy spectrum in terms of a spectral function, we classify the self-adjoint realizations in two classes, identifying all the families of isospectral Hamiltonians. These two classes turn out to be discerned by the action of parity (i.e. space reflection), which plays a central role in our discussion.

Graph isomorphism and Gaussian boson sampling

We introduce a connection between a near-term quantum computing device, specifically a Gaussian boson sampler, and the graph isomorphism problem. We propose a scheme where graphs are encoded into quantum states of light, whose properties are then probed with photon-number-resolving detectors. We prove that the probabilities of different photon-detection events in this setup can be combined to give a complete set of graph invariants. Two graphs are isomorphic if and only if their detection probabilities are equivalent. We present additional ways that the measurement probabilities can be combine…

Supersymmetry-generated complex optical potentials with real spectra

We show that the formalism of supersymmetry (SUSY), when applied to parity-time (PT) symmetric optical potentials, can give rise to novel refractive index landscapes with altogether non-trivial properties. In particular, we find that the presence of gain and loss allows for arbitrarily removing bound states from the spectrum of a structure. This is in stark contrast to the Hermitian case, where the SUSY formalism can only address the fundamental mode of a potential. Subsequently we investigate isospectral families of complex potentials that exhibit entirely real spectra, despite the fact that …

Time-dependent rational extensions of the parametric oscillator: quantum invariants and the factorization method

New families of time-dependent potentials related to the parametric oscillator are introduced. This is achieved by introducing some general time-dependent operators that factorize the appropriate constant of motion (quantum invariant) of the parametric oscillator, leading to new families of quantum invariants that are almost-isospectral to the initial one. Then, the respective time-dependent Hamiltonians are constructed, and the solutions of the Schrödinger equation are determined from the intertwining relationships and by finding the appropriate time-dependent complex-phases of the Lewis-Ries…

连接两类非厄米性：利用Lanczos变换从非互易耦合到虚势

Bridging two families of non-Hermiticity: from non-reciprocal couplings to an imaginary potential using the Lanczos transformation

本研究利用Lanczos变换在非互易耦合与虚势两类非厄米性之间建立等谱映射，将广义Hatano-Nelson模型映射为具有虚势的紧束缚模型，提供了理解奇异点的新途径，并实现了无需PT对称的复势实谱系统构建。