Related papers: A reformulation of Hilbert's tenth problem through…
States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a K\"ahler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics…
In this paper, we introduce a system of split variational inequality problems in real Hilbert spaces. Using projection method, we propose an iterative algorithm for the system of split variational inequality problems. Further, we prove that…
For a ring $R$, Hilbert's Tenth Problem $HTP(R)$ is the set of polynomial equations over $R$, in several variables, with solutions in $R$. We view $HTP$ as an operator, mapping each set $W$ of prime numbers to $HTP(\mathbb Z[W^{-1}])$,…
We review the conceptual problems in quantum mechanics on a fundamental level. It is shown that the proposed model of extended electrons and a clear understanding of rotations in three dimensional space solve a large part of these problems,…
The Helmholtz equation is a prototypical model for time-harmonic wave propagation. Numerical solutions become increasingly challenging as the wave number $k$ grows, due to the equation's elliptic yet noncoercive character and the highly…
We start with a discussion of the use of mathematics to model the real world then justify the role of Hilbert space formalism for such modelling in the general context of quantum logic. Following this, the incompleteness of the…
A number of recent studies have proposed that linear representations are appropriate for solving nonlinear dynamical systems with quantum computers, which fundamentally act linearly on a wave function in a Hilbert space. Linear…
In Quantum Physics, a measurement is represented by a projection on some closed subspace of a Hilbert space. We study algebras of operators that abstract from the algebra of projections on closed subspaces of a Hilbert space. The properties…
These highly informal lecture notes aim at introducing and explaining several closely related problems on zeros of analytic functions defined by ordinary differential equations and systems of such equations. The main incentive for this…
The random matrix ensembles are applied to the quantum statistical two-dimensional systems of electrons. The quantum systems are studied using the finite dimensional real, complex and quaternion Hilbert spaces of the eigenfunctions. The…
Quantum computing has been increasingly applied in nuclear physics. In this work, we combine quantum computing with the complex scaling method to address the resonance problem. Due to the non-Hermiticity introduced by complex scaling,…
We search for a possible mathematical formulation of some of the key ideas of the relational interpretation of quantum mechanics and study their consequences. We also briefly overview some proposals of relational quantum mechanics for an…
We present a brief survey of results where quantum information processing is useful to solve distributed computation tasks. We describe problems that are impossible to solve using classical resources but that become feasible with the help…
A major challenge to the control of infinite dimensional quantum systems is the irreversibility which is often present in the system dynamics. Here we consider systems with discrete-spectrum Hamiltonians operating over a Schwartz space…
Interpretational problems with quantum mechanics can be phrased precisely by only talking about empirically accessible information. This prompts a mathematical reformulation of quantum mechanics in terms of classical mechanics. We survey…
Useful relations describing arbitrary parameters of given quantum systems can be derived from simple physical constraints imposed on the vectors in the corresponding Hilbert space. This is well known and it usually proceeds by partitioning…
We present a Hilbert space perspective to homogenization of standard linear evolutionary boundary value problems in mathematical physics and provide a unified treatment for (non-)periodic homogenization problems in thermodynamics,…
We explain why and how the Hilbert space comes about in quantum theory. The axiomatic structures of vector space, of scalar product, of orthogonality, and of the linear functional are derivable from the statistical description of quantum…
In $\mathbb R^d$, it is well-known that cumulants provide an alternative to moments that can achieve the same goals with numerous benefits such as lower variance estimators. In this paper we extend cumulants to reproducing kernel Hilbert…
Quantum computing offers significant potential for solving NP-hard combinatorial (optimization) problems that are beyond the reach of classical computers. One way to tap into this potential is by reformulating combinatorial problems as a…