Erratum: Dirichlet Forms and Dirichlet Operators for Infinite Particle Systems: Essential Self-adjointness

We reprove the essential self-adjointness of the Dirichlet operators of Dirchlet forms for infinite particle systems with superstable and sub-exponentially decreasing interactions.Comment: This is an erratum to the work appeared in J. Math. Phys. 39(12), 6509-6536 (1998

Phase transitions and quantum effects in anharmonic crystals

The most important recent results in the theory of phase transitions and quantum effects in quantum anharmonic crystals are presented and discussed. In particular, necessary and sufficient conditions for a phase transition to occur at some temperature are given in the form of simple inequalities involving the interaction strength and the parameters describing a single oscillator. The main characteristic feature of the theory is that both mentioned phenomena are described in one and the same setting, in which thermodynamic phases of the model appear as probability measures on path spaces. Then the possibility of a phase transition to occur is related to the existence of multiple phases at the same values of the relevant parameters. Other definitions of phase transitions, based on the non-differentiability of the free energy density and on the appearance of ordering, are also discussed

An explicitly solvable model of the spontaneous PT-symmetry breaking

We contemplate the pair of the purely imaginary delta-function potentials on a finite interval with Dirichlet boundary conditions. The two parameter model exhibits nicely the expected quantitative features of the unavoided level crossing and of a "phase-transition" complexification of the energies. Combining analytic and numerical techniques we investigate strength- and position-dependence of its spectrum.Comment: presented in the int. conference "Pseudo-Hermitian Hamiltonians in Quantum Physics III" (Instanbul, Koc University, June 20 - 22, 2005). accepted in Czechoslovak J. Phy

Boundary Conditions for Singular Perturbations of Self-Adjoint Operators

Let A:D(A)\subseteq\H\to\H be an injective self-adjoint operator and let \tau:D(A)\to\X, X a Banach space, be a surjective linear map such that \|\tau\phi\|_\X\le c \|A\phi\|_\H. Supposing that \text{\rm Range} (\tau')\cap\H' =\{0\}, we define a family $A^\tau_\Theta$ of self-adjoint operators which are extensions of the symmetric operator $A_{|\{\tau=0\}.}$. Any $\phi$ in the operator domain $D(A^\tau_\Theta)$ is characterized by a sort of boundary conditions on its univocally defined regular component \phireg, which belongs to the completion of D(A) w.r.t. the norm \|A\phi\|_\H. These boundary conditions are written in terms of the map $\tau$, playing the role of a trace (restriction) operator, as \tau\phireg=\Theta Q_\phi, the extension parameter $\Theta$ being a self-adjoint operator from X' to X. The self-adjoint extension is then simply defined by A^\tau_\Theta\phi:=A \phireg. The case in which $A\phi=T*\phi$ is a convolution operator on LD, T a distribution with compact support, is studied in detail.Comment: Revised version. To appear in Operator Theory: Advances and Applications, vol. 13

Non Commutative Arens Algebras and their Derivations

Given a von Neumann algebra $M$ with a faithful normal semi-finite trace $\tau,$ we consider the non commutative Arens algebra $L^{\omega}(M, \tau)=\bigcap\limits_{p\geq1}L^{p}(M, \tau)$ and the related algebras $L^{\omega}_2(M, \tau)=\bigcap\limits_{p\geq2}L^{p}(M, \tau)$ and $M+L^{\omega}_2(M, \tau)$ which are proved to be complete metrizable locally convex *-algebras. The main purpose of the present paper is to prove that any derivation of the algebra $M+L^{\omega}_2(M, \tau)$ is inner and all derivations of the algebras $L^{\omega}(M,\tau)$ and $L^{\omega}_2(M, \tau)$ are spatial and implemented by elements of $M+L^{\omega}_2(M, \tau).$Comment: 19 pages. Submitted to Journal of Functional analysi