Perturbation theory series in quantum mechanics: Phase transition and exact asymptotic forms for the expansion coefficients

We consider the model of a harmonic oscillator with a power-law potential and derive new asymptotic formulas for the coefficients of the perturbation theory series in powers of the coupling constant in the case of a power-law perturbing potential |x|^p, p > 0. We prove the existence of a critical value p ₀ and compute it. It is a threshold in the sense that the asymptotic forms of the studied coefficients for 0 < p < p ₀ and for p > p ₀ differ qualitatively. We note that the considered physical system undergoes a phase transition at p = p ₀ . The analysis uses the Laplace method for functional integrals with Gaussian measures.     

Hyperbolic quantum mechanics

We start to develop the quantization formalism in a hyperbolic Hilbert space. Generalizing Born’s probability interpretation, we found that unitary transformations in such a Hilbert space represent a new class of transformations of probabilities which describe a kind of hyperbolic interference. The most interesting problem which prompted by our investigation is to find experimental evidence of hyperbolic interference. The hyperbolic quantum formalism can also be interesting as a new theory of probability waves that can be developed in parallel with the standard quantum theory. Comparative analysis of these two wave theories could be useful for understanding of the role of various structures of the standard quantum formalism. In particular, one of distinguishing feature of the hyperbolic quantum formalism is the restricted validity of the superposition principle.     

p-Adic model of quantum mechanics and quantum channels

A p-adic realization of the standard statistical model of quantum mechanics is constructed. Within this realization, a p-adic linear bosonic channel is defined, and its properties are analyzed. In particular, a criterion for the existence of a linear Gaussian bosonic channel is obtained, and its explicit construction is described. It is shown that the p-adic Gaussian bosonic channels possess an additivity property.     

Stochastic model of quantum mechanics

Moscow Radio Engineering Institute, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 82, No. 2, pp. 208–215, February, 1990.     

Measurement and probability in quantum mechanics

Measurement-theoretical foundations of quantum probabilities are investigated in the form of measurement statistics and a statistical ensemble interpretation of quantum mechanics.     

Nicolai map in supersymmetric quantum mechanics

Joint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matermaticheskaya Fizika, Vol. 86, No. 1, pp. 74–80, January, 1991.     

On point interaction in quantum mechanics

For the Schrödinger operator corresponding to the point interaction, a direct definition is given in terms of a singular perturbation.     

Self-consistent relaxation-time models in quantum mechanics

ABSTRACT. This paper is concerned with the relaxation-time von Neumann- Poisson (or quantum Liouville-Poisson) equation in three spatial dimensions which describes the self-consistent time evolution of an open quantum me- chanical system that includes some relaxation mechanism. This model and the equivalent relaxation-time Wigner-Poisson system play an important role in the simulation of quantum semiconductor devices. For initial density matrices with finite kinetic energy, we prove that this problem, formulated in the space of Hermitian trace class operators, admits a unique global strong solution. A key ingredient for our analysis is a new generalization of the Lieb-Thirring inequality for density matrix operators.     

Perturbation theory for eigenvalues and resonances of Schrödinger hamiltonians

Suppose that e^2_?|x|V ∈ ReL^P(R³) for some p > 2 and for g ∈ R, H(g) = ? Δ + gV, H(g) = ?Δ + gV. The main result, Theorem 3, uses Puiseaux expansions of the eigenvalues and resonances of H(g) to study the behavior of eigenvalues λ(g) as they are absorbed by the continuous spectrum, that is $\text{λ(g) ↗6 0}\text{as}\text{g ↘5 g}{}_0\text{> 0}$. We find a series expansion in powers of $\text{(g ? g}{}_0\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{, λ(g) = ∑}{}_{\mathrm{n\; =\; 2}}{}^{\mathrm{\infty }}\text{a}{}_{\mathrm{n}}\text{(g ? g}{}_0\text{)}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}$ whose values for g < g₀ correspond to resonances near the origin. These resonances can be viewed as the traces left by the just absorbed eigenvalues.