Related papers: On one-dimensional G-dynamics and non-Hermitian Ha…
The three simultaneous algebraic equations, $C^2=1$, $[C,PT]=0$, $[C,H]=0$, which determine the $C$ operator for a non-Hermitian $PT$-symmetric Hamiltonian $H$, are shown to have a nonunique solution. Specifically, the $C$ operator for the…
A Wheeler-Dewitt quantum constraint operator for four-dimensional, non-perturbative Lorentzian vacuum quantum gravity is defined in the continuum. The regulated Wheeler-DeWitt constraint operator is densely defined, does not require any…
It is shown that if the C operator for a PT-symmetric Hamiltonian with simple eigenvalues is not unique, then it is unbounded. Apart from the special cases of finite-matrix Hamiltonians and Hamiltonians generated by differential expressions…
We present a general counting result for the unstable eigenvalues of linear operators of the form $JL$ in which $J$ and $L$ are skew- and self-adjoint operators, respectively. Assuming that there exists a self-adjoint operator $K$ such that…
An N-dimensional position-dependent mass Hamiltonian (depending on a parameter \lambda) formed by a curved kinetic term and an intrinsic oscillator potential is considered. It is shown that such a Hamiltonian is exactly solvable for any…
Two-term Weyl-type asymptotic law for the eigenvalues of one-dimensional quasi-relativistic Hamiltonian (-h^2 c^2 d^2/dx^2 + m^2 c^4)^(1/2) + V_well(x) (the Klein-Gordon square-root operator with electrostatic potential) with the infinite…
We consider the problem of finding $\lambda\in \mathbb{R}$ and a function $u:\mathbb{R}^n\rightarrow\mathbb{R}$ that satisfy the PDE $$ \max\left\{\lambda + F(D^2u) -f(x),H(Du)\right\}=0, \quad x\in \mathbb{R}^n. $$ Here $F$ is elliptic,…
The family of Tremblay-Turbiner-Winternitz (TTW) Hamiltonians $H_k$ on a plane, corresponding to any positive real value of $k$, is shown to admit another ${\cal N} = 2$ supersymmetric extension than that previously introduced by the…
We consider a classical Hamiltonian $H$ on $\mathbb{R}^{2d}$, invariant by a finite group of symmetry $G$, whose Weyl quantization $\hat{H}$ is a selfadjoint operator on $L^2(\mathbb{R}^d)$. If $\chi$ is an irreducible character of $G$, we…
We give an explicit characterization of the most general quasi-Hermitian operator H, the associated metric operators \eta_+, and \eta_+-pseudo-Hermitian operators acting in two-dimensional complex Euclidean space C^2. These operators…
In this work we generalise various recent results on the evolution and monotonicity of the eigenvalues of certain geometric operators under specified geometric flows. Given a closed, compact Riemannian manifold $\big(M^n,g(t)\big)$ and a…
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of…
Consider the Cauchy problem for a strictly hyperbolic, $N\times N$ quasilinear system in one space dimension $$ u_t+A(u) u_x=0,\qquad u(0,x)=\bar u(x), \eqno (1) $$ where $u \mapsto A(u)$ is a smooth matrix-valued map, and the initial data…
Consider the family of Schr\"odinger operators (and also its Dirac version) on $\ell^2(\mathbb{Z})$ or $\ell^2(\mathbb{N})$ \[ H^W_{\omega,S}=\Delta + \lambda F(S^n\omega) + W, \quad \omega\in\Omega, \] where $S$ is a transformation on…
We consider the nonlinear eigenvalue problem $Lx + \varepsilon N(x) = \lambda Cx$, $\|x\|=1$, where $\varepsilon,\lambda$ are real parameters, $L, C\colon G \to H$ are bounded linear operators between separable real Hilbert spaces, and…
We address the issue of when generalized quantum dynamics, which is a classical symplectic dynamics for noncommuting operator phase space variables based on a graded total trace Hamiltonian ${\bf H}$, reduces to Heisenberg picture complex…
The connection zeta function of a finite abstract simplicial complex G is defined as zeta_L(s)=sum_x 1/lambda_x^s, where lambda_x are the eigenvalues of the connection Laplacian L defined by L(x,y)=1 if x and y intersect and 0 else. (I) As…
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review…
It is necessary to calculate the C operator for the non-Hermitian PT-symmetric Hamiltonian H=\half p^2+\half\mu^2x^2-\lambda x^4 in order to demonstrate that H defines a consistent unitary theory of quantum mechanics. However, the C…
Given a second-order, holomorphic, linear differential equation $Lf=0$ on a punctured Riemann surface, we say that its monodromy group $G\subset\operatorname{GL}(2,\mathbb{C})$ is `unitary' if it preserves a non-degenerate Hermitian form…