Spiros Konstantogiannis

Among the one-dimensional, real and analytic polynomial potentials, the sextic anharmonic oscillator is the only one that can be quasi-exactly solved, if it is properly parametrized. In this work, we present a new method to quasi-exactly solve the sextic anharmonic oscillator and apply it to derive specific solutions. Our approach is based on the i...

Considering the stationary Schrödinger equation for a general pseudo-Coulomb potential as the normal form of the associated Laguerre equation, we review, in one and three dimensions, the bound-state solutions for the potential, when the inverse-square-term coupling is not less than a negative critical value. We show that, as a consequence of the in...

We show how two fundamental sequential criteria for real functions, namely the sequential criterion for continuity and the sequential criterion for uniform continuity, are relaxed. Further, we derive a global continuity criterion, as an immediate consequence of the relaxed sequential criterion for continuity, and a fundamental property of uniformly...

The aim of this paper is to examine infinite unions and intersections of sets in a general metric space, with a view to explaining when an infinite intersection of open sets is an open set and when an infinite union of closed sets is a closed set.

We show that every equation that is derived by the application of the Euclidean algorithm to two coprime positive integers can be written as a modular inverse switching formula, and this allows us to provide a backward recurrence relation for computing modular inverses. Using the recurrence relation, we then give an alternative proof of a theorem d...

Applying a modular inverse switching formula to the Euclidean algorithm, we derive an inverse recurrence relation for computing modular inverses that uses only the sequence of remainders in the algorithm.

The additivity of the Riemann integral, i.e. the property that the integral is an additive function of the interval of integration, is among the most important properties of Riemann integration, as it is widely used in the manipulation and simplification of integrals that appear not only in specific calculations, but also in cornerstone theorems, s...

In an attempt to address the need for an alternative presentation of the quantum mechanical position and momentum spaces, we provide a presentation that is more constructive and less calculative than those found in literature. Our approach is based on a simple, intuitively-understood relation that expresses the physical equivalence of the quantum m...

The purpose of this work is to present a different, sequential approach to Riemann integration. To this end, from Riemann's definition of the integral, we derive a sequential if-and-only-if criterion for Riemann integrability that is expressed in terms of Cauchy sequences and that allows us to give a sequential definition of the integral. In this f...

We use the completeness of the real numbers along with the definition and the sequential criterion of continuity to prove a lemma from which Bolzano's theorem follows easily.

We present a method of constructing PT-symmetric sextic oscillators using quotient polynomials and show that the reality of the energy spectrum of the oscillators is directly related to the PT symmetry of the respective quotient polynomials. We then apply the method to derive sextic oscillators from real quotient polynomials and demonstrate that th...

Using the Frobenius method, we find all polynomial and non-polynomial terminating series solutions of the associated Laguerre differential equation and its special case, the Laguerre differential equation.

Plugging the closed-form expression of the associated Laguerre polynomials into their orthogonality relation, the latter reduces to a factorial identity that takes a simple, non-trivial form for even-degree polynomials.

The purpose of this work is to introduce, in a simple, intuitive way, the coherent and squeezed states of the quantum harmonic oscillator (QHO), through a series of exercises, which are solved in detail. Starting from the application of a spatial translation to the ground state of the QHO, we introduce the spatial and momentum translations, focusin...

Considering a particle moving in a general, three-dimensional complex potential, we show the following: - The anti-Hermitian term of the Hamiltonian, i.e. the imaginary part of the potential, destroys the time-independence of the state-vector norm, which becomes time-dependent, in general. - The total probability, which is equal to the square of th...

Starting with a general rational wave function, we search for potentials admitting it as a bound energy eigenfunction. We thus derive singular and regular potentials asymptotically decaying as the inverse of x squared, with the latter being simple or multiple volcanoes having a finite number of bound eigenstates. We present specific examples and ex...

For one-dimensional potentials having, at most, finite discontinuities and simple poles at which the wave functions have simple zeros, we give an algebraic – i.e. operator-based – proof that an eigenfunction having no other zeros is a minimum-energy eigenfunction, and thus it describes the ground state.

Contents: - The infinite Riemann sheets of the complex argument resulting from the degeneracy of the complex numbers - Definitions - The polar form of the complex numbers as a means of doing rotations on the complex plane - Examples of multi-valued functions - Branch point at infinity - General examples

Using a length scale, we construct an n-independent, one-dimensional shifted Coulomb potential, which, with the addition of a delta potential with n-dependent coupling, forms a quasi-exactly solvable model. Making a polynomial ansatz for the closed-form eigenfunctions, we obtain a three-term recursion relation, from which the known energies are der...

Making an intuitive assumption, and using the completeness of the position and momentum eigenstates, along with the postulates of quantum mechanics, we provide a geometric presentation of the position and momentum representations in quantum mechanics, in the hope of offering a perspective complementary to those given in standard textbooks.

Exactly solvable rational extensions of the harmonic oscillator have been constructed as supersymmetric partner potentials of the harmonic oscillator [1] as well as using the so-called prepotential approach [2]. In this work, we use the factorization property of the energy eigenfunctions of the harmonic oscillator and a simple integrability conditi...

The harmonic oscillator is one of the most important elementary systems in both classical and quantum physics. The present eBook is a – hopefully successful – attempt to present some of the many important aspects of the one-dimensional quantum harmonic oscillator (QHO), through a series of non-trivial exercises, which are solved in detail. In each...