Related papers: L'Hopital-Type Rules for Monotonicity with Applica…
We present a perturbative method to estimate the spectral gap for adiabatic quantum optimization, based on the structure of the energy levels in the problem Hamiltonian. We show that for problems that have exponentially large number of…
A general formulation of noncommutative or quantum derivatives for operators in a Banach space is given on the basis of the Leibniz rule, irrespective of their explicit representations such as the G\^ateaux derivative or commutators. This…
Using Schwinger Variational Principle we solve the problem of quantum harmonic oscillator with time dependent frequency. Here, we do not take the usual approach which implicitly assumes an adiabatic behavior for the frequency. Instead, we…
Sum rules have played an important role in the development of many branches of physics since the earliest days of quantum mechanics. We present examples of one-dimensional quantum mechanical sum rules and apply them in two familiar systems,…
Uncertainties $(\Delta x)^2$ and $(\Delta p)^2$ are analytically derived in an $N$-coupled harmonic oscillator system when spring and coupling constants are arbitrarily time-dependent and each oscillator is in an arbitrary excited state.…
In this paper, we present a comprehensive study of the monotonicity and log-concavity of the generalized Marcum and Nuttall Q-functions. More precisely, a simple probabilistic method is firstly given to prove the monotonicity of these two…
It is shown, by considering the case of the harmonic oscillator, that quantum fluctuations may be the most significant contribution to the random walk of a single molecule. From this point, the controversy on the existence of a standard…
We introduce the theory of operator monotone functions and employ it to derive a new inequality relating the quantum relative entropy and the quantum conditional entropy. We present applications of this new inequality and in particular we…
One of the simplest example of non-commutative (NC) spaces is the NC plane. In this article we investigate the consequences of the non-commutativity to the quantum mechanics on a plane. We derive corrections to the standard (commutative)…
Differential equations on spaces of operators are very little developed in Mathematics, being in general very challenging. Here, we study a novel system of such (non-linear) differential equations. We show it has a unique solution for all…
Since there are quantization ambiguities in constructing the Hamiltonian constraint operator in isotropic loop quantum cosmology, it is crucial to check whether the key features of loop quantum cosmology are robust against the ambiguities.…
Ratios of integrals can be bounded in terms of ratios of integrands under certain monotonicity conditions. This result, related with L'H\^{o}pital's monotone rule, can be used to obtain sharp bounds for cumulative distribution functions. We…
We present here an application of a new quantization scheme. We quantize the Taub cosmology by quantizing only the anisotropy parameter $\beta$ and imposing the super-Hamiltonian constraint as an expectation-value equation to recover the…
We study the qualitative behavior of nonlinear Dirac equations arising in quantum field theory on complete Riemannian manifolds. In particular, we derive monotonicity formulas and Liouville theorems for solutions of these equations.…
A new framework of loop quantization that assimilates conformal and scale invariance is constructed and is found to be applicable to a large class of physically important theories of gravity and gravity-matter systems. They include general…
We explore to what extent path-integral quantum Monte Carlo methods can efficiently simulate the tunneling behavior of quantum adiabatic optimization algorithms. Specifically we look at symmetric cost functions defined over n bits with a…
In this work we introduce a criterion for testing general covariance in effective quantum gravity theories. It adapts the analysis of invariance under general spacetime diffeomorphisms of the Einstein-Hilbert action to the case of effective…
A numerical bootstrap method is proposed to provide rigorous and nontrivial bounds in general quantum many-body systems with locality. In particular, lower bounds on ground state energies of local lattice systems are obtained by imposing…
We present a new theorem describing stable solutions for a driven quantum system. The theorem, coined `inertial theorem', is applicable for fast driving, provided the acceleration rate is small. The theorem states that in the inertial limit…
In these notes we address the canonical quantization of the cosmological models which appear as solutions of the low energy effective action of closed bosonic string theory (dilaton models). The analysis is restricted to the quantization of…