Related papers: Solvable analogue of the imaginary cubic oscillato…
We prove that if a proper metric space is quasi-isometric to a finitely generated group and to a space with a horoball over a finitely generated group, then that space is quasi-isometric to a rank-one symmetric space or the real line.
We examine various generalizations, e.g. exactly solvable, quasi-exactly solvable and non-Hermitian variants, of a quantum nonlinear oscillator. For all these cases, the same mass function has been used and it has also been shown that the…
The Hermiticity axiom of quantum mechanics guarantees that the energy spectrum is real and the time evolution is unitary (probability-preserving). Nevertheless, non-Hermitian but $\mathcal{PT}$-symmetric Hamiltonians may also have real…
We consider a PT Symmetric Partner to Khare Mandal's recently proposed non-Hermitian potential with complex eigen values. Our potential is Quasi-Exactly solvable and is shown to possess only real eigen values.
We introduce a one-dimensional PT-symmetric system, which includes the cubic self-focusing, a double-well potential in the form of an infinitely deep potential box split in the middle by a delta-functional barrier of an effective height…
We in this paper study the hermiticity of Hamiltonian and energy spectrum for the SU(1; 1) systems. The Hermitian Hamiltonian can possess imaginary eigenvalues in contrast with the common belief that hermiticity is a suffcient condition for…
We present a solution of the quantum mechanics problem of the allowable energy levels of a bound particle in a one-dimensional finite square well. The method is a geometric-analytic technique utilizing the conformal mapping $w \to z = w…
We prove existence of static solutions to the cylindrically symmetric Einstein-Vlasov system, and we show that the matter cylinder has finite extension. The same results are also proved for a quite general class of equations of state for…
A new type of quantum simulator is proposed which can simulate any quantum many-body system in an isomorphic manner. It can actually synthesize a duplicate of the system to be simulated. The isomorphic simulation has the great advantage…
It is known that there exist a limited number of analytic potentials with the unusual property that any bound quantum state therein will be periodic in time. This is known as a perfect quantum state revival. Examples of such potentials are…
We compare two recent approaches of quasi-exactly solvable Schr\" odinger equations, the first one being related to finite-dimensional representations of $sl(2,R)$ while the second one is based on supersymmetric developments. Our results…
We prove that a closed arithmetic hyperbolic 3-manifold with positive first betti number has virtually infinite first betti number.
A new version of PT-symmetric quantum theory is proposed and illustrated by an N-site-lattice Legendre oscillator. The essence of the innovation lies in the replacement of parity P (serving as an indefinite metric in an auxiliary Krein…
By applying the higher order Darboux algorithm to an exactly solvable non Hermitian ${\cal{PT}}$ symmetric potential, we obtain a hierarchy of new exactly solvable non Hermitian ${\cal{PT}}$ symmetric potentials with real spectra. It is…
The Schr\"{o}dinger equation is solved for the case of a particle confined to a small region of a box with infinite walls. If walls of the well are moved, then, due to an effective quantum nonlocal interaction with the boundary, even though…
A pseudo-Hermitian coupled-channel square-well model is proposed, solved and discussed. The domain of parameters is determined where all the bound-state energies (twice degenerate with respect to the second observable which we call "spin")…
Representation theory is shown to be incomplete in terms of enumerating all integrable limits of quantum systems. As a consequence, one can find exactly solvable Hamiltonians which have apparently strongly broken symmetry. The number of…
The coupling of non-Hermitian PT-symmetric Hamiltonians to standard Hermitian Hamiltonians, each of which individually has a real energy spectrum, is explored by means of a number of soluble models. It is found that in all cases the energy…
A purely imaginary potential can provide a phenomenological description of creation and absorption of quantum mechanical particles. PT-invariance of such a potential ensures that the non-unitary phenomena occur in a balanced manner. In…
We consider a Parity-time (PT) invariant non-Hermitian quasi-exactly solvable (QES) potential which exhibits PT phase transition. We numerically study this potential in a complex plane classically to demonstrate different quantum effects.…