scholarly journals PATH INTEGRAL REPRESENTATION FOR SCHRÖDINGER OPERATORS WITH BERNSTEIN FUNCTIONS OF THE LAPLACIAN

2012 ◽  
Vol 24 (06) ◽  
pp. 1250013 ◽  
Author(s):  
FUMIO HIROSHIMA ◽  
TAKASHI ICHINOSE ◽  
JÓZSEF LŐRINCZI
Keyword(s):  
Brownian Motion ◽  
Path Integral ◽  
Schrödinger Operators ◽  
Integral Representations ◽  
Schrodinger Operators ◽  
Kato Class ◽  
Path Integral Representation ◽  
Bernstein Functions ◽  
Subordinate Brownian Motion ◽  
The One

Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman–Kac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an Lp-Lq bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

PARTITION FUNCTIONS OF THE SUPERCONFORMAL GHOST THIRRING MODEL

1989 ◽  
Vol 04 (20) ◽  
pp. 5553-5574 ◽  
Author(s):  
D. Z. FREEDMAN ◽  
K. PILCH
Keyword(s):  
Energy Spectrum ◽  
Path Integral ◽  
Formal Series ◽  
Open String ◽  
Integral Representations ◽  
Thirring Model ◽  
Partition Functions ◽  
First Order ◽  
Integral Methods ◽  
The One

The one-loop partition functions of the superconformal Thirring model for first order b − c and β − γ ghost fields are studied for both closed and open string boundary conditions. Bosonized partition functions are given by formal series which usually diverge because the energy spectrum of the theory is unbounded below as a correlate of nonunitarity. However, the same partition functions are then calculated by path integral methods directly in the fermionic formulation, and well-defined (convergent) integral representations are obtained. A formal series expansion of those integrals reproduces the bosonized partition functions.

SCHRÖDINGER OPERATORS ON LOCAL FIELDS: SELF-ADJOINTNESS AND PATH INTEGRAL REPRESENTATIONS FOR PROPAGATORS

2008 ◽  
Vol 11 (04) ◽  
pp. 495-512 ◽  
Author(s):  
TROND DIGERNES ◽  
V. S. VARADARAJAN ◽  
D. E. WEISBART
Keyword(s):  
Path Integral ◽  
Adjoint Operator ◽  
Hamiltonian Operator ◽  
Internal Symmetry ◽  
Integral Representations ◽  
Local Fields ◽  
Path Integral Representation ◽  
Finite Dimensional ◽  
Space Of Paths ◽  
Complex Coefficients

We consider quantum systems that have as their configuration spaces finite-dimensional vector spaces over local fields. The quantum Hilbert space is taken to be a space with complex coefficients and we include in our model particles with internal symmetry. The Hamiltonian operator is a pseudo-differential operator that is initially only formally defined. For a wide class of potentials we prove that this Hamiltonian is well-defined as an unbounded self-adjoint operator. The free part of the operator gives rise to a measure on the Skorokhod space of paths, D [0, ∞), and with respect to this measure there is a path integral representation for the semigroup associated to the Hamiltonian. We prove this Feynman–Kac formula in the local field setting as a consequence of the Hille–Yosida theory of semigroups.

scholarly journals Brownian motion and finite approximations of quantum systems over local fields

2017 ◽  
Vol 29 (05) ◽  
pp. 1750016 ◽  
Author(s):  
Erik Makino Bakken ◽  
Trond Digernes ◽  
David Weisbart
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Local Field ◽  
Schrödinger Operators ◽  
Quantum Systems ◽  
Local Fields ◽  
Schrodinger Operators ◽  
Finite Systems ◽  
Finite Approximations ◽  
Finite Approximability

We give a stochastic proof of the finite approximability of a class of Schrödinger operators over a local field, thereby completing a program of establishing in a non-Archimedean setting corresponding results and methods from the Archimedean (real) setting. A key ingredient of our proof is to show that Brownian motion over a local field can be obtained as a limit of random walks over finite grids. Also, we prove a Feynman–Kac formula for the finite systems, and show that the propagator at the finite level converges to the propagator at the infinite level.

scholarly journals PATH INTEGRAL REPRESENTATION FOR INTERFACE STATES OF THE ANISOTROPIC HEISENBERG MODEL

2000 ◽  
Vol 12 (10) ◽  
pp. 1325-1344 ◽  
Author(s):  
OSCAR BOLINA ◽  
PIERLUIGI CONTUCCI ◽  
BRUNO NACHTERGAELE
Keyword(s):  
Integral Representation ◽  
Path Integral ◽  
Two Dimensions ◽  
Integral Model ◽  
Partition Functions ◽  
Translation Invariant ◽  
Xxz Model ◽  
Path Integral Representation ◽  
Ferromagnetic Chain ◽  
The One

We develop a geometric representation for the ground state of the spin-1/2 quantum XXZ ferromagnetic chain in terms of suitably weighted random walks in a two-dimensional lattice. The path integral model so obtained admits a genuine classical statistical mechanics interpretation with a translation invariant Hamiltonian. This new representation is used to study the interface ground states of the XXZ model. We prove that the probability of having a number of down spins in the up phase decays exponentially with the sum of their distances to the interface plus the square of the number of down spins. As an application of this bound, we prove that the total third component of the spin in a large interval of even length centered on the interface does not fluctuate, i.e. has zero variance. We also show how to construct a path integral representation in higher dimensions and obtain a reduction formula for the partition functions in two dimensions in terms of the partition function of the one-dimensional model.

THE ROTATION NUMBER APPROACH TO EIGENVALUES OF THE ONE-DIMENSIONAL p-LAPLACIAN WITH PERIODIC POTENTIALS

2001 ◽  
Vol 64 (1) ◽  
pp. 125-143 ◽  
Author(s):  
MEIRONG ZHANG
Keyword(s):  
Dirichlet Problem ◽  
Rotation Number ◽  
Periodic Potential ◽  
Schrödinger Operators ◽  
Negative Integer ◽  
Schrodinger Operators ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Periodic Potentials ◽  
The One

The paper studies the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic potential. After a rotation number function ρ(λ) has been introduced, it is proved that for any non-negative integer n, the endpoints of the interval ρ−1(n/2) in ℝ yield the corresponding periodic or anti-periodic eigenvalues. However, as in the Dirichlet problem of the higher dimensional p-Laplacian, it remains open if these eigenvalues represent all periodic and anti-periodic eigenvalues. The result obtained is a partial generalization of the spectrum theory of the one-dimensional Schrödinger operators with periodic potentials.

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