Related papers: Eigenvalue inequalities for Klein-Gordon Operators

We investigate a second order elliptic differential operator $A_{\beta, \mu}$ on a bounded, open set $\Omega\subset\mathbb{R}^{d}$ with Lipschitz boundary subject to a nonlocal boundary condition of Robin type. More precisely we have $0\leq…

The generalized h-dependent operator algebra is defined ($0\leq h \leq h_o$). For h= h_o it becomes equivalent to the quantum mechanical algebra of observables and for h=0 it is equivalent to the classical one. We show this by proposing how…

We construct a generalised expression for the normal ordering of (a+a^{\dagger})^{m} for integral values of m and use the result to study the quantum anharmonic oscillator problem in the Heisenberg approach. In particular, we derive…

This undergraduate thesis is concerned with developing the tools of differential geometry and semiclassical analysis needed to understand the the quantum ergodicity theorem of Schnirelman (1974), Zelditch (1987), and Colin de Verdi\`ere…

Let $\mathcal{P}_{\Omega,tA}$ denoted the Pauli operator on a bounded open region $\Omega\subset\mathbb{R}^2$ with Dirichlet boundary conditions and magnetic potential $A$ scaled by some $t>0$. Assume that the corresponding magnetic field…

We consider a class of homogeneous partial differential operators on a finite-dimensional vector space and study their associated heat kernels. The heat kernels for this general class of operators are seen to arise naturally as the limiting…

We consider a non-self adjoint operator of the form $-h^2 \Delta + i(V(x) + \alpha(x)y)$ on the upper half plane $y > 0$ with Dirichlet boundary conditions on $\{y = 0\}$ with $V \geq 0$, $V$ admitting a non-degenerate minimum at $x = 0$…

We formulate the quantum mechanics of the solutions of a Klein-Gordon-type field equation: (\partial_t^2+D)\psi(t)=0, where D is a positive-definite operator acting in a Hilbert space \tilde H. We determine all the positive-definite inner…

Let $\Omega\subset\R^N$ be an arbitrary open set and denote by $(e^{-t(-\Delta)_{\RR^N}^s})_{t\ge 0}$ (where $0<s<1$) the semigroup on $L^2(\RR^N)$ generated by the fractional Laplace operator. In the first part of the paper we show that if…

In this paper we present a new method of proof of Hardy type inequalities for two-dimensional quantum Hamiltonians with a magnetic field of finite flux. Our approach gives a quantitative lower bound on the best constant in these…

In this paper, we prove the uniform estimates for the resolvent $(\Delta - \alpha)^{-1}$ as a map from $L^q$ to $L^{q'}$ on real hyperbolic space $\mathbb{H}^n$ where $\alpha \in \mathbb{C}\setminus [(n - 1)^2/4, \infty)$ and $2n/(n + 2)…

We review various attempts to localize the discrete spectra of semirelativistic Hamiltonians of the form H = \beta \sqrt{m^2 + p^2} + V(r) (w.l.o.g. in three spatial dimensions) as entering, for instance, in the spinless Salpeter equation.…

We revisit the problem of computing the determinant of Klein-Gordon operator $\Delta = -\nabla^2 + M^2$ on Euclideanized $AdS_3$ with the Euclideanized time coordinate compactified with period $\beta$, $H_3/Z$, by explicitly computing its…

We study the eigenvalues of the magnetic Schroedinger operator associated with a magnetic potential A and a scalar potential q, on a compact Riemannian manifold M, with Neumann boundary conditions if the boundary is not empty. We obtain…

We study the eigenvalues for infinitesimal generators of semigroups of composition operators acting on Hardy spaces, Bergman spaces, and the Dirichlet space. Such semigroups are induced by semigroups of holomorphic functions. Depending on…

We study the Dirichlet eigenvalue problem of homogeneous H\"{o}rmander operators $\triangle_{X}=\sum_{j=1}^{m}X_{j}^{2}$ on a bounded open domain containing the origin, where $X_1, X_2, \ldots, X_m$ are linearly independent smooth vector…

The Hamiltonian operator plays a central role in quantum theory being a generator of unitary quantum dynamics. Its expectation value describes the energy of a quantum system. Typically being a non-unitary operator, the action of the…

We analyze a general class of difference operators $H_\varepsilon = T_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon \mathbb{Z}^d)$, where $V_\varepsilon$ is a one-well potential and $\varepsilon$ is a small parameter. We construct…

In this paper, we establish a fundamental inequality for fourth order partial differential operator $\cal P=\alpha\partial_s+\beta\partial_{ss}+\Delta^2$ ($\alpha, \beta\in\mathbb{R}$) with an abstract exponential-type weight function. Such…

Let $\cl{M}$ be a Hilbert module of holomorphic functions over a natural function algebra $\mathcal{A}(\Omega)$, where $\Omega \subseteq \bb{C}^m$ is a bounded domain. Let $\cl{M}_0\subseteq \cl{M}$ be the submodule of functions vanishing…