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The two-dimensional hydrogen-like atom in a constant magnetic field is considered. It is found that this is actually two separate problems. One for which the magnetic field causes an effective attraction between the nucleus and the electron and one for which it causes an effective repulsion. Each of the two problems has three separate cases depending on the sign of a shifted energy eigenvalue. For two of the six possibilities (shifted energy eigenvalue that is negative) it is shown that the first four solutions can be obtained exactly. For another two of the six possibilities (shifted energy eigenvalue that is positive) it is shown that the first eight solutions can be obtained exactly. For higher order states the energy eigenvalue is the root of a fifth or higher order polynomial, hence, the eigenvalue must be obtained numerically. Once the energy eigenvalue is known the solution to the radial wave equation is also known. Exact solutions for the radial wave equation, for the remaining two possibilities (shifted energy eigenvalue that is zero), are given to any desired order by means of a recursion relation.

The quantum systems with finite-dimensional Hilbert space have several applications and are intensively explored theoretically and experimentally. The mathematical description of these systems follows the analogy with the usual infinite-dimensional case. There exist finite versions for most of the elements used in the continuous case, but (to our knowledge) there does not exist a finite version corresponding to the mixed Gaussian states. Our aim is to fill this gap. The definition we propose for the mixed discrete Gaussian states is based on the explicit formulas we have obtained in the case of pure discrete variable Gaussian states., Comment: 12 pages, 1 figure. Any comments or suggestion are welcome. More details are available at https://unibuc.ro/user/nicolae.cotfas/

The aim of this paper is to analyze from a mathematical perspective some existing schemes to partition a molecular density into several atomic contributions, with a specific focus on Iterative Stockholder Atom (ISA) methods. We provide a unified mathematical framework to describe the latter family of methods and propose a new scheme, named L-ISA (for linear approximation of ISA). We prove several important mathematical properties of the ISA and L-ISA minimization problems and show that the so-called ISA algorithms can be viewed as alternating minimization schemes, which in turn enables us to obtain new convergence results for these numerical methods. Specific mathematical properties of the ISA decomposition for diatomic systems are also presented. We also review the basis-space oriented Distributed Multipole Analysis method, the mathematical formulation of which is also clarified. Different schemes are numerically compared on different molecules and we discuss the advantages and drawbacks of each approach.

We continue to study doubled aspects of algebroid structures equipped with the C-bracket in double field theory (DFT). We find that a family of algebroids, the Vaisman (metric or pre-DFT), the pre- and the ante-Courant algebroids are constructed by the analogue of the Drinfel'd double of Lie algebroid pairs. We examine geometric implementations of these algebroids in the para-Hermitian manifold, which is a realization of the doubled space-time in DFT. We show that the strong constraint in DFT is necessary to realize the doubled and non-trivial Poisson structures but can be relaxed for some algebroids. The doubled structures of twisted brackets and those associated with group manifolds are briefly discussed., Comment: version appeared in JMP

A remarkable discrete counterpart of the Gaussian function of one continuous variable can be defined by using a Jacobi theta function, that is, as the sum of a convergent series. We extend this approach to Gaussian functions of two variables, and investigate the Fourier transform and Wigner function of the functions of discrete variable defined in this way., Comment: 9 pages, 1 figure. For more details see the section "Documente" at https://unibuc.ro/user/nicolae.cotfas/

We continue to discuss the following problem: For which bounded measurable real functions f and g is the commutator i[f(P),g(Q)] a non-negative operator on L^2(R)? In this work we concentrate on the situation where the commutator is a finite rank operator.

In this paper, a new physical notion, intensity, is introduced. The notion of intensity occurs in a special statistics, known as Gentile statistics, which is asymptotically close to ordinary thermodynamics. The introduction of the new notion of intensity in the theory of nuclear matter essentially changes the thermodynamical picture. Moreover, we can say that the thermodynamics of nuclear matter is the antipode of standard thermodynamics. On the basis of the "intensity--compressibility factor" diagram and mean square fluctuations of energy and time, a new table of properties of stable chemical elements is obtained and presented in this paper., Comment: 20 pages, 2 figures and a long table

The Schr\"odinger theory of electrons in an external electromagnetic field can be described from the perspective of the individual electron via the `Quantal Newtonian' laws (or differential virial theorems). These laws are in terms of `classical' fields whose sources are quantal expectations of Hermitian operators taken with respect to the wave function. The laws reveal the following physics: (a) In addition to the external field, each electron experiences an internal field whose components are representative of a specific property of the system such as the correlations due to the Pauli exclusion principle and Coulomb repulsion, the electron density, kinetic effects, and an internal magnetic field component. (The response of the electron is described by the current density field.); (b) The scalar potential energy of an electron is the work done in a conservative field which is the sum of the internal and Lorentz fields. It is thus inherently related to the properties of the system. Its constituent property-related components are hence known. It is a known functional of the wave function; (c) As such the Hamiltonian is a functional of the wave function, thereby revealing the intrinsic self-consistent nature of the Schr\"odinger equation. This then provides a path for the determination of the exact wave function. (d) With the Schr\"odinger equation written in self-consistent form, the Hamiltonian now admits via the Lorentz field a new term that explicitly involves the external magnetic field. The new understandings are explicated for the stationary state case by application to a quantum dot in a magnetostatic field in both a ground and excited state. For the time-dependent case, the same states of the quantum dot in both a magnetostatic and a time-dependent electric field are considered., Comment: 16 pages, 3 figures

The convex set of quantum states of a composite $K \times K$ system with positive partial transpose is analysed. A version of the hit and run algorithm is used to generate a sequence of random points covering this set uniformly and an estimation for the convergence speed of the algorithm is derived. For $K\ge 3$ this algorithm works faster than sampling over the entire set of states and verifying whether the partial transpose is positive. The level density of the PPT states is shown to differ from the Marchenko-Pastur distribution, supported in [0,4] and corresponding asymptotically to the entire set of quantum states. Based on the shifted semi--circle law, describing asymptotic level density of partially transposed states, and on the level density for the Gaussian unitary ensemble with constraints for the spectrum we find an explicit form of the probability distribution supported in [0,3], which describes well the level density obtained numerically for PPT states., Comment: 11 pages, 4 figures

Symplectic and optical joint probability representations of quantum mechanics are considered, in which the functions describing the states are the probability distributions with all random arguments (except the argument of time ). The general formalism of quantizers and dequantizers determining the star product quantization scheme in these representations is given. Taking the Gaussian functions as the distributions of the tomographic parameters the correspondence rules for most interesting physical operators are found and the expressions of the dual symbols of operators in the form of singular and regular generalized functions are derived. Evolution equations and stationary states equations for symplectic and optical joint probability distributions are obtained., Comment: 13 pages, 0 figures

The Hohenberg-Kohn theorem, a cornerstone of electronic density functional theory, concerns uniqueness of external potentials yielding given ground densities of an ${\mathcal N}$-body system. The problem is rigorously explored in a universe of three-dimensional Kato-class potentials, with emphasis on trade-offs between conditions on the density and conditions on the potential sufficient to ensure uniqueness. Sufficient conditions range from none on potentials coupled with everywhere strict positivity of the density, to none on the density coupled with something a little weaker than local $3{\mathcal N}/2$-power integrability of the potential on a connected full-measure set. A second theme is localizability, that is, the possibility of uniqueness over subsets of ${\mathbb R}^3$ under less stringent conditions., Comment: 21 pages, version accepted by Journal of Mathematical Physics

Discrete-time quantum walks can be regarded as quantum dynamical simulators since they can simulate spatially discretized Schr\"{o}dinger, massive Dirac, and Klein-Gordon equations. Here, two different types of Fibonacci discrete-time quantum walks are studied analytically. The first is the Fibonacci coin sequence with a generalized Hadamard coin and demonstrates six-step periodic dynamics. The other model is assumed to have three- or six-step periodic dynamics with the Fibonacci sequence. We analytically show that these models have ballistic transportation properties and continuous limits identical to those of the massless Dirac equation with coin basis change., Comment: 6 pages, 3 figures

For atoms described by a 1-dimensional Friedrichs model we present the following result: pre-existing resonances, when subjected to a small constant electric field (DC Stark effect), are unstable in the weak field limit. In contrast, pre-existing resonances under the influence of a small time-periodic electric field (AC Stark effect) are stable in this limit. This article is a review of our results in "Instability of pre-existing resonances under a small constant electric field", arXiv:1310.4745., Comment: review article, 10 pages, 4 figures

Bell's Theorem from 1964 and the (Strong) Free Will Theorem of Conway and Kochen from 2009 both exclude deterministic hidden variable theories (or, in modern parlance, `ontological models') that are compatible with some small fragment of quantum mechanics, admit `free' settings of the archetypal Alice & Bob experiment, and satisfy a locality condition called Parameter Independence. We clarify the relationship between these theorems by giving reformulations of both that exactly pinpoint their resemblance and their differences. Our reformulation imposes determinism in what we see as the only consistent way, in which the `ontological state' initially determines both the settings and the outcome of the experiment. The usual status of the settings as `free' parameters is subsequently recovered from independence assumptions on the pertinent (random) variables. Our reformulation also clarifies the role of the settings in Bell's later generalization of his theorem to stochastic hidden variable theories., Comment: 10 pages, v2: references [19] and [37] added, footnote 2 expanded. Accepted by Foundations of Physics

Two simple model operators are considered which have pre-existing resonances. A potential corresponding to a small electric field, $f$, is then introduced and the resonances of the resulting operator are considered as $f\to0$. It is shown that these resonances are not continuous in this limit. It is conjectured that a similar behavior will appear in more complicated models of atoms and molecules. Numerical results are presented., Comment: 54 pages, 11 figures. No changes w.r.t. preprint version of 03/29/2014, except publisher information added. The final publication is available at link.springer.com

We formulate, with full generality, the asymptotic estimation theory for Gaussian states in terms of their first and second moments. By expressing the quantum Fisher information (QFI) and the elusive symmetric logarithmic derivative (SLD) in terms of the state's moments (and their derivatives) we are able to obtain the noncommutative extension of the well known expression for the Fisher information of a Gaussian probability distribution. Focusing on models with fixed first moments and identical Williamson 'diagonal' states --which include pure state models--, we obtain their SLD and QFI, and elucidate what features of the Wigner function are fundamentally accessible, and at what rates. In addition, we find the optimal homodyne detection scheme for all such models, and show that for pure state models they attain the fundamental limit., Comment: 7 pages. 2 Appendices

The Hamiltonian of the harmonic oscillator is usually defined as a differential operator, but an integral representation can be obtained by using the coherent state quantization. The finite frame quantization is a finite counterpart of the coherent state quantization and it allows us to define a finite oscillator by starting from the integral representation of the harmonic oscillator. Our purpose is to investigate the oscillator obtained in this way, and to present a possible application to the discrete fractional Fourier transform., Comment: Several new results and figures have been added

In this paper we are discussing the question how a continuous quantum system can be simulated by mean field fluctuations of a finite number of qubits. On the kinematical side this leads to a convergence result which states that appropriately chosen fluctuation operators converge in a certain weak sense (i.e. we are comparing expectation values) to canonical position and momentum Q, P of one-degree of freedom, continuous quantum system. This result is substantially stronger than existing methods which rely either on central limit theorem arguments (and are therefore restricted to the Gaussian world) or are valid only if the states of the ensembles are close to the "fully polarized" state. Dynamically this relationship keeps perfectly intact (at least for small times) as long as the continuous system evolves according to a quadratic Hamiltonian. In other words we can approximate the corresponding (Heisenberg picture) time evolution of the canonical operators Q, P up to arbitrary accuracy by the appropriately chosen time evolution of fluctuation operators of the finite systems., Comment: 29 pages

The geometric measure of entanglement for a symmetric pure state with nonnegative amplitudes has attracted much attention. On the other hand, the spectral theory of nonnegative tensors (hypermatrices) has been developed rapidly. In this paper, we show how the spectral theory of nonnegative tensors can be applied to the study of the geometric measure of entanglement for a pure state with nonnegative amplitudes. Especially, an elimination method for computing the geometric measure of entanglement for symmetric pure multipartite qubit or qutrit states with nonnegative amplitudes is given. For symmetric pure multipartite qudit states with nonnegative amplitudes, a numerical algorithm with randomization is presented and proven to be convergent. We show that for the geometric measure of entanglement for pure states with nonnegative amplitudes, the nonsymmetric ones can be converted to the symmetric ones., Comment: 21 pages

We obtain a general expression for a Wigner transform (Wigner function) on symmetric spaces of non-compact type and study the Weyl calculus of pseudodifferential operators on them.

In this paper, we give a frequency interpretation of negative probability, as well as of extended probability, demonstrating that to a great extent, these new types of probabilities, behave as conventional probabilities. Extended probability comprises both conventional probability and negative probability. The frequency interpretation of negative probabilities gives supportive evidence to the axiomatic system built in (Burgin, 2009; arXiv:0912.4767) for extended probability as it is demonstrated in this paper that frequency probabilities satisfy all axioms of extended probability.

After a brief introduction to Probability Bracket Notation (PBN), indicator operator and conditional density operator (CDO), we investigate probability spaces associated with various quantum systems: system with one observable (discrete or continuous), system with two commutative observables (independent or dependent) and a system of indistinguishable non-interacting many-particles. In each case, we derive unified expressions of conditional expectation (CE), conditional probability (CP), and absolute probability (AP): they have the same format for discrete or continuous spectrum; they are defined in both Hilbert space (using Dirac notation) and probability space (using PBN); and they may be useful to deal with CE of non-commutative observables.

We prove quantitative bounds on the eigenvalues of non-selfadjoint unbounded operators obtained from selfadjoint operators by a perturbation that is relatively-Schatten. These bounds are applied to obtain new results on the distribution of eigenvalues of Schroedinger operators with complex potentials.

The standard (``fine-grained'') interpretation of quantum density functional theory, in which densities are specified with infinitely-fine spatial resolution, is mathematically unruly. Here, a coarse-grained version of DFT, featuring limited spatial resolution, and its relation to the fine-grained theory in the $L^1\cap L^3$ formulation of Lieb, is studied, with the object of showing it to be not only mathematically well-behaved, but consonant with the spirit of DFT, practically (computationally) adequate and sufficiently close to the standard interpretation as to accurately reflect its non-pathological properties. The coarse-grained interpretation is shown to be a good model of formal DFT in the sense that: all densities are (ensemble)-V-representable; the intrinsic energy functional $F$ is a continuous function of the density and the representing external potential is the (directional) functional derivative of the intrinsic energy. Also, the representing potential $v[\rho]$ is quasi-continuous, in that $v[\rho]\rho$ is continuous as a function of $\rho$. The limit of coarse-graining scale going to zero is studied to see if convergence to the non-pathological aspects of the fine-grained theory is adequate to justify regarding coarse-graining as a good approximation. Suitable limiting behaviors or intrinsic energy, densities and representing potentials are found. Intrinsic energy converges monotonically, coarse-grained densities converge uniformly strongly to their low-intrinsic-energy fine-grainings, and $L^{3/2}+L^\infty$ representability of a density is equivalent to the existence of a convergent sequence of coarse-grained potential/ground-state density pairs., Comment: All sections have been revised at least a little, but the significant technical improvements are in section 6

Fermi transport of spinors can be precisely understood in terms of 2-spinor geometry. By using a partly original, previously developed treatment of 2-spinors and classical fields, we describe the family of all transports, along a given 1-dimensional timelike submanifold of spacetime, which yield the standard Fermi transport of vectors. Moreover we show that this family has a distinguished member, whose relation to the Fermi transport of vectors is similar to the relation between the spinor connection and spacetime connection. Various properties of the Fermi transport of spinors are discussed, and applied to the construction of free electron states for a detector-dependent QED formalism introduced in a previous paper., Comment: 18 pages

The theory of Schroedinger bridges for diffusion processes is extended to classical and quantum discrete-time Markovian evolutions. The solution of the path space maximum entropy problems is obtained from the a priori model in both cases via a suitable multiplicative functional transformation. In the quantum case, nonequilibrium time reversal of quantum channels is discussed and space-time harmonic processes are introduced., Comment: 34 pages

Some of the important inequalities associated with quantum entropy are immediate algebraic consequences of the Hansen-Pedersen-Jensen inequality. A general argument is given using matrix perspectives of operator convex functions. A matrix analogue of Mar\'{e}chal's extended perspectives provides additional inequalities, including a $p+q\leq 1$ result of Lieb., Comment: 8 pages

The Ghirardi-Rimini-Weber (GRW) theory is a physical theory that, when combined with a suitable ontology, provides an explanation of quantum mechanics. The so-called collapse of the wave function is problematic in conventional quantum theory but not in the GRW theory, in which it is governed by a stochastic law. A possible ontology is the flash ontology, according to which matter consists of random points in space-time, called flashes. The joint distribution of these points, a point process in space-time, is the topic of this work. The mathematical results concern mainly the existence and uniqueness of this distribution for several variants of the theory. Particular attention is paid to the relativistic version of the GRW theory that I developed in 2004., Comment: 72 pages LaTeX, 3 figures

We prove a theorem about positive-operator-valued measures (POVMs) that is an analog of the Kolmogorov extension theorem, a standard theorem of probability theory. According to our theorem, if a sequence of POVMs G_n on $\mathbb{R}^n$ satisfies the consistency (or projectivity) condition $G_{n+1}(A\times \mathbb{R}) = G_n(A)$ then there is a POVM G on the space $\mathbb{R}^\mathbb{N}$ of infinite sequences that has G_n as its marginal for the first n entries of the sequence. We also describe an application in quantum theory., Comment: 6 pages LaTeX, no figures

The position and momentum information entropies of $D$-dimensional quantum systems with central potentials, such as the isotropic harmonic oscillator and the hydrogen atom, depend on the entropies of the (hyper)spherical harmonics. In turn, these entropies are expressed in terms of the entropies of the Gegenbauer (ultraspherical) polynomials $C_n^{(\lambda)}(x)$, the parameter $\lambda$ being either an integer or a half-integer number. Up to now, however, the exact analytical expression of the entropy of Gegenbauer polynomials of arbitrary degree $n$ has only been obtained for the particular values of the parameter $\lambda=0,1,2$. Here we present a novel approach to the evaluation of the information entropy of Gegenbauer polynomials, which makes use of trigonometric representations for these polynomials and complex integration techniques. Using this method, we are able to find the analytical expression of the entropy for arbitrary values of both $n$ and $\lambda\in\mathbb{N}$., Comment: 19 pages, 1 Postscript figure

Spaces $S_{\omega}, S_{\{\omega\}}, S_{(\omega)}$ of ultradecreasing ultradifferentiable (or for short, ultra-S) functions, depending on a weight $e^{\omega(x)}$, are introduced in the context of quantum statistics. The corresponding coefficient spaces in the Fock basis are identified, and it is shown that the Hermite expansion is a tame isomorphism between these spaces. These results are used to link decrease properties of density matrices to corresponding properties of the Wigner distribution.

The Veneziano amplitude for the tree-level scattering of four tachyonic scalar of open string theory has an arithmetic analogue in terms of the p-adic gamma function. We propose a quantum extension of this amplitude using the q-extended p-adic gamma function given by Koblitz. This provides a one parameter deformation of the arithmetic Veneziano amplitude. We also comment on the dificulty in generalising this to higher point amplitudes., Comment: 8 pages (harvmac b)

The volume of the quantum mechanical state space over $n$-dimensional real, complex and quaternionic Hilbert-spaces with respect to the canonical Euclidean measure is computed, and explicit formulas are presented for the expected value of the determinant in the general setting too. The case when the state space is endowed with a monotone metric or a pull-back metric is considered too, we give formulas to compute the volume of the state space with respect to the given Riemannian metric. We present the volume of the space of qubits with respect to various monotone metrics. It turns out that the volume of the space of qubits can be infinite too. We characterize those monotone metrics which generates infinite volume., Comment: 17 pages

The aim of the paper is to extend the notion of $\alpha$-geometry in the classical and in the noncommutative case by introducing a more general class of pull-back metrics and to give concrete formulas for the scalar curvature of these Riemannian manifolds. We introduce a more general class of pull-back metrics of the noncommutative state spaces, we pull back the Euclidean Riemannian metric of the space of self-adjoint matrices with functions which have an analytic extension to a neighborhood of the interval $]0,1[$ and whose derivative are nowhere zero. We compute the scalar curvature in this setting, and as a corollary we have the scalar curvature of the classical probability space when it is endowed with such a general pull-back metric. In the noncommutative setting we consider real and complex state spaces too. We give a simplification of Gibilisco's and Isola's conjecture for the first nontrivial classical probability space and we present the result of a numerical computation which indicate that the conjecture may be true for the space of real and complex qubits., Comment: 18 pages

Monogenic functions are functions of null vector derivative and are here analysed in the geometric algebra of 5-dimensional spacetime, G(4,1), in order to derive several laws of fundamental physics. The paper introduces the working algebra and the definition of monogenic functions, showing that these generate two 4-dimensional spaces, one with Euclidean signature and the other one with Minkowski signature. The equivalence conditions between the two spaces are studied and relativistic dynamics, not entirely coincident with Einstein's general theory of relativity, is demonstrated. The monogenic condition is then shown to produce Maxwell's equations and electrodynamics both classical and quantized., Comment: 13 pages