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We analyze the behavior of a non-Hermitian opened one-dimensional quantum system with $\mathcal{PT}$ symmetry. This system is built by a dimer, with balanced gains and losses described by a parameter $\gamma$. By varying $\gamma$ the system…
We determine numerically the single-particle and the two-particle spectrum of the three-state quantum Potts model on a lattice by using the density matrix renormalization group method, and extract information on the asymptotic (small…
Recent work by Hintz--Vasy provides a partial asymptotic analysis of the low-energy limit of scattering for Schr\"odinger operators with a short-range potential. Using a slight refinement of Hintz's algorithm, we complete the asymptotic…
We study a very general class of first-order linear hyperbolic systems that both become weakly hyperbolic and contain lower-order coefficients that blow up at a single time $t = 0$. In "critical" weakly hyperbolic settings, it is well-known…
This paper is concerned with the efficient numerical treatment of 1D stationary Schr\"odinger equations in the semi-classical limit when including a turning point of first order. For the considered scattering problems we show that the wave…
We consider semiclassical Schr\"odinger operators on the real line of the form $$H(\hbar)=-\hbar^2 \frac{d^2}{dx^2}+V(\cdot;\hbar)$$ with $\hbar>0$ small. The potential $V$ is assumed to be smooth, positive and exponentially decaying…
Let $H(\hbar)=-\hbar^2d^2/dx^2+V(x)$ be a Schr\"odinger operator on the real line, $W(x)$ be a bounded observable depending only on the coordinate and $k$ be a fixed integer. Suppose that an energy level $E$ intersects the potential $V(x)$…
Exceptional points~(EPs) appear as degeneracies in the spectrum of non-Hermitian matrices at which the eigenvectors coalesce. In general, an EP of order $n$ may find room to emerge if $2(n-1)$ real constraints are imposed. Our results show…
We study transport in quantum systems consisting of a finite array of N identical single-channel scatterers. A general expression of the S matrix in terms of the individual-element data obtained recently for potential scattering is…
We propose an analysis technique for the exceptional points (EPs) occurring in the discrete spectrum of open quantum systems (OQS), using a semi-infinite chain coupled to an endpoint impurity as a prototype. We outline our method to locate…
We show that there is a language in $\mathsf{S}_2\mathsf{E}/_1$ (symmetric exponential time with one bit of advice) with circuit complexity at least $2^n/n$. In particular, the above also implies the same near-maximum circuit lower bounds…
This paper studies the scattering matrix $\Sigma(E;\hbar)$ of the problem \[ -\hbar^2 \psi''(x) + V(x) \psi(x) = E\psi(x) \] for positive potentials $V\in C^\infty(\R)$ with inverse square behavior as $x\to\pm\infty$. It is shown that each…
We consider a two-dimensional analogue of Helmholtz resonator with walls of finite thickness in the critical case when there exists an eigenfrequency equalling to the limit of poles generated by both the bounded component of the resonator…
One of the important features of non-Hermitian Hamiltonians is the existence of a unique type of singularities, the so-called exceptional points (EPs). When the corresponding systems operate around such singularities, they exhibit…
A class of above-barrier quantum-scattering problems is shown to provide an experimentally-accessible platform for studying $\mathcal{PT}$-symmetric Schr\"odinger equations that exhibit spontaneous $\mathcal{PT}$ symmetry breaking despite…
The transfer matrix of a possibly complex and energy-dependent scattering potential can be identified with the $S$-matrix of a two-level time-dependent non-Hermitian Hamiltonian H(t). We show that the application of the adiabatic…
We consider nonadiabatic transitions in explicitly time-dependent systems with Hamiltonians of the form $\hat{H}(t) = \hat{A} +\hat{B} t + \hat{C}/t$, where $t$ is time and $\hat{A}$, $\hat{B}$, $\hat{C}$ are Hermitian $N\times N$ matrices.…
The object of study in this paper is the on-shell scattering matrix $S(E)$ of the Schr\"odinger operator with the potential satisfying assumptions typical in the theory of shape resonances. We study the spectrum of $S(E)$ in the…
Singular exponential nonlinearities of the form $e^{h(x)\epsilon^{-1}}$ with $\epsilon>0$ small occur in many different applications. These terms have essential singularities for $\epsilon=0$ leading to very different behaviour depending on…
The transfer matrix ${\mathbf{M}}$ of a short-range potential may be expressed in terms of the time-evolution operator for an effective two-level quantum system with a time-dependent non-Hermitian Hamiltonian. This leads to a dynamical…