Related papers: Short note on the perturbation of operators with d…
The variation of spectral subspaces for linear self-adjoint operators under an additive bounded perturbation is considered. The objective is to estimate the norm of the difference of two spectral projections associated with isolated parts…
Let $T_n^d(A)$ denote a partial upper triangular operator matrix whose diagonal entries are given and the others unknown. In this article we have aim to find characterizations of (left,right) invertibility of $T_n^d(A)$ in terms of diagonal…
It is well known how to define the operator $Q$ for the total charge (i.e., positron number minus electron number) on the standard Hilbert space of the second-quantized Dirac equation. Here we ask about operators $Q_A$ representing the…
Finiteness of the point spectrum of linear operators acting in a Banach space is investigated from point of view of perturbation theory. In the first part of the paper we present an abstract result based on analytical continuation of the…
Let us denote ${\cal V}$, the finite dimensional vector spaces of functions of the form $\psi(x) = p_n(x) + f(x) p_m(x)$ where $p_n(x)$ and $p_m(x)$ are arbitrary polynomials of degree at most $n$ and $m$ in the variable $x$ while $f(x)$…
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…
In this paper, we establish a useful set of formulae for the $\sin\Theta$ distance between the original and the perturbed singular subspaces. These formulae explicitly show that how the perturbation of the original matrix propagates into…
In this paper we present how spectral properties of certain linear operators vary when operators are considered in different Hilbert spaces having common dense domain as the space of polynomials in one real variable with complex…
We use the boundary triplet approach to extend the classical concept of perturbation determinants to a more general setup. In particular, we examine the concept of perturbation determinants to pairs of proper extensions of closed symmetric…
This article offers a study of the Calder\'on type inverse problem of determining up to second order coefficients of the higher order elliptic operator. Here we show that it is possible to determine an anisotropic second order perturbation…
We develop deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors, in a spirit similar to classical results for matrices such as those due to Weyl, Davis, Kahan and Wedin. Our bounds…
In this work we propose a generalization of the Hadamard product between two matrices to a tensor-valued, multi-linear product between k matrices for any $k \ge 1$. A multi-linear dual operator to the generalized Hadamard product is…
This work discusses a variational approach to determining the time evolution operator. We directly see a glimpse of how a generalization of the quantum geometric tensor for unitary operators plays a central role in parameter evolution. We…
The spectral problem of the Dirac equation in an external quadratic vector potential is considered using the methods of the perturbation theory. The problem is singular and the perturbation series is asymptotic, so that the methods for…
We study the Harmonic and Dirac Oscillator problem extended to a three-dimensional noncom- mutative space where the noncommutativity is induced by a shift of the dynamical variables with generators of SL(2;R) in a unitary irreducible…
The algebraic approach to operator perturbation method has been applied to two quantum--mechanical systems ``The Stark Effect in the Harmonic Oscillator'' and ``The Generalized Zeeman Effect''. To that end, two realizations of the…
We analyse a modified Dirac equation based on a noncommutative structure in phase space. The noncommutative structure induces generalised momenta and contributions to the energy levels of the standard Dirac equation. Using techniques of…
Suppose that A is a set of n real numbers, each at least 1 apart. Define the ``perturbed sum and product sets'' S and P to be the sums a + b + f(a,b) and products (a+g(a,b))(b+h(a,b)), where f, g, and h satisfy certain upper bounds in terms…
We present a nonrelativistic one-particle quantum mechanics whose perturbative S-matrix exhibits a renormalon divergence that we explicitely compute. The potential of our model is the sum of the 2d Dirac $\delta$-potential -- known to…
Perturbation theory with respect to the kinetic energy of the heavy component of a two-component quantum system is introduced. An effective Hamiltonian that is accurate to second order in the inverse heavy mass is derived. It contains a new…