1,721,164 research outputs found
Multiplication of distributions in any dimension: applications to -function and its derivatives
No full textIn two previous papers the author introduced a multiplication of distributions in one dimension and
he proved that two one-dimensional Dirac delta functions and their derivatives can be multiplied, at least under
certain conditions. Here, mainly motivated by some engineering applications in the analysis of the structures,
we propose a different definition of multiplication of
distributions which can be easily extended to any spatial
dimension. In particular we prove that with this new definition
delta functions and their derivatives can still be multiplied
Finite-dimensional pseudo-bosons: A non-Hermitian version of the truncated harmonic oscillator
Full text linkWe propose a deformed version of the commutation rule introduced in 1967 by Buchdahl to describe a particular model of the truncated harmonic oscillator. The rule we consider is defined on a N-dimensional Hilbert space H N , and produces two biorthogonal bases of H N which are eigenstates of the Hamiltonians h=[Formula presented](q 2 +p 2 ), and of its adjoint h † . Here q and p are non-Hermitian operators obeying [q,p]=i(1−Nk), where k is a suitable orthogonal projection operator. These eigenstates are connected by ladder operators constructed out of q, p, q † and p † . Some examples are discussed
Weak pseudo-bosons
Full text linkWe show how the notion of pseudo-bosons, originally introduced as operators acting on some Hilbert space, can be extended to a distributional settings. In doing so, we are able to construct a rather general framework to deal with generalized eigenvectors of the multiplication and of the derivation operators. Connections with the quantum damped harmonic oscillator are also briefly considered
FOURIER TRANSFORMS, FRACTIONAL DERIVATIVES, AND A LITTLE BIT OF QUANTUM MECHANICS
Full text linkWe discuss some of the mathematical properties of the fractional derivative defined by means of Fourier transforms. We first consider its action on the set of test functions i(R), and then we extend it to its dual set, i'(R), the set of tempered distributions, provided they satisfy some mild conditions. We discuss some examples, and we show how our definition can be used in a quantum mechanical context
Some results on the rotated infinitely deep potential and its coherent states
Full text linkThe Swanson model is an exactly solvable model in quantum mechanics with a manifestly non self-adjoint Hamiltonian whose eigenvalues are all real. Its eigenvectors can be deduced easily, by means of suitable ladder operators. This is because the Swanson Hamiltonian is deeply connected with that of a standard quantum Harmonic oscillator, after a suitable rotation in configuration space is performed. In this paper we consider a rotated version of a different quantum system, the infinitely deep potential, and we consider some of the consequences of this rotation. In particular, we show that differences arise with respect to the Swanson model, mainly because of the technical need of working, here, with different Hilbert spaces, rather than staying in L2(R). We also construct Gazeau–Klauder coherent states for the system, and analyze their properties
Ladder operators with no vacuum, their coherent states, and an application to graphene
Full text linkIn literature ladder operators of different nature exist. The most famous are those obeying canonical (anti-) commutation relations, but they are not the only ones. In our knowledge, all ladder operators have a common feature: the lowering operators annihilate a non zero vector, the {\em vacuum}. This is connected to the fact that operators of these kind are often used in factorizing some positive operators, or some operators which are { bounded from below}. This is the case, of course, of the harmonic oscillator, but not only. In this paper we discuss what happens when considering lowering operators with no vacua. In particular, after a general analysis of this situation, we propose a possible construction of coherent states, and we apply our construction to graphene
Abstract ladder operators and their applications
Full text linkWe consider a rather general version of ladder operator Z used by some authors in few recent papers, [H 0, Z] = λZ for some, H0=H0†, and we show that several interesting results can be deduced from this formula. Then we extend it in two ways: first we replace the original equality with formula [H 0, Z] = λZ[Z †, Z], and secondly we consider [H, Z] = λZ for some C, H ≠ H †. In both cases many applications are discussed. In particular we consider factorizable Hamiltonians and Hamiltonians written in terms of operators satisfying the generalized Heisenberg algebra or the D pseudo-bosonic commutation relations
Abstract ladder operators for non self-adjoint Hamiltonians, with applications
No full textLadder operators are useful, if not essential, in the analysis of some given physical system since they can be used to find easily eigenvalues and eigenvectors of its Hamiltonian. In this paper we extend our previous results on abstract ladder operators considering in many details what happens if the Hamiltonian of the system is not self-adjoint. Among other results, we give an existence criterion for coherent states constructed as eigenstates of our lowering operators. In the second part of the paper we discuss two different examples of our framework: pseudo-quons and a deformed generalized Heisenberg algebra. Incidentally, and interestingly enough, we show that pseudo-quons can be used to diagonalize an oscillator-like Hamiltonian written in terms of (non self-adjoint) position and momentum operators which obey a deformed commutation rule of the kind often considered in minimal length quantum mechanics
One-directional quantum mechanical dynamics and an application to decision making
Full text linkIn recent works we have used quantum tools in the analysis of the time evolution of
several macroscopic systems. The main ingredient in our approach is the self-adjoint
Hamiltonian H of the system S. This Hamiltonian quite often, and in particular for
systems with a finite number of degrees of freedom, gives rise to reversible and
oscillatory dynamics. Sometimes this is not what physical reasons suggest. We discuss
here how to use non self-adjoint Hamiltonians to overcome this difficulty: the time
evolution we obtain out of them show a preferable arrow of time, and it is not reversible.
Several applications are constructed, in particular in connection to information dynamics
A dynamical approach to compatible and incompatible questions
Full text linkWe propose a natural strategy to deal with compatible and incompatible binary questions, and with their time evolution. The strategy is based on the simplest, non-commutative, Hilbert space H=C 2 , and on the (commuting or not) operators on it. As in ordinary Quantum Mechanics, the dynamics is driven by a suitable operator, the Hamiltonian of the system. We discuss a rather general situation, and analyse the resulting dynamics if the Hamiltonian is a simple Hermitian matrix
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