Ground states and associated path measures in the renormalized Nelson model

We prove the existence, uniqueness, and strict positivity of ground states of the possibly massless renormalized Nelson operator under an infrared regularity condition and for Kato decomposable electrostatic potentials fulfilling a binding condition. If the infrared regularity condition is violated, then we show non-existence of ground states of the massless renormalized Nelson operator with an arbitrary Kato decomposable potential. Furthermore, we prove the existence, uniqueness, and strict positivity of ground states of the massless renormalized Nelson operator in a non-Fock representation where the infrared condition is unnecessary. Exponential and superexponential estimates on the pointwise spatial decay and the decay with respect to the boson number for elements of spectral subspaces below localization thresholds are provided. Moreover, some continuity properties of ground state eigenvectors are discussed. Byproducts of our analysis are a hypercontractivity bound for the semigroup and a new remark on Nelson’s operator theoretic renormalization procedure. Finally, we construct path measures associated with ground states of the renormalized Nelson operator. Their analysis entails improved boson number decay estimates for ground state eigenvectors, as well as upper and lower bounds on the Gaussian localization with respect to the field variables in the ground state. As our results on uniqueness, positivity, and path measures exploit the ergodicity of the semigroup, we restrict our attention to one matter particle. All results are non-perturbative.

Regularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equation

In this paper, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to a semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all orders. Because we cannot prove u ( t , x ) ∈ D ∞ u(t,x)\in \mathbb {D}^\infty for measure-valued initial data, we need a localized version of Malliavin calculus. Furthermore, we prove that the (joint) density is strictly positive in the interior of the support of the law, where we allow both measure-valued initial data and unbounded diffusion coefficient. The criteria introduced by Bally and Pardoux are no longer applicable for the parabolic Anderson model. We have extended their criteria to a localized version. Our general framework includes the parabolic Anderson model as a special case.

Nonnegative and Strictly Positive Linearization of Jacobi and Generalized Chebyshev Polynomials

In the theory of orthogonal polynomials, as well as in its intersection with harmonic analysis, it is an important problem to decide whether a given orthogonal polynomial sequence $$(P_n(x))_{n\in \mathbb {N}_0}$$ ( P n ( x ) ) n ∈ N 0 satisfies nonnegative linearization of products, i.e., the product of any two $$P_m(x),P_n(x)$$ P m ( x ) , P n ( x ) is a conical combination of the polynomials $$P_{|m-n|}(x),\ldots ,P_{m+n}(x)$$ P | m - n | ( x ) , … , P m + n ( x ) . Since the coefficients in the arising expansions are often of cumbersome structure or not explicitly available, such considerations are generally very nontrivial. Gasper (Can J Math 22:582–593, 1970) was able to determine the set V of all pairs $$(\alpha ,\beta )\in (-1,\infty )^2$$ ( α , β ) ∈ ( - 1 , ∞ ) 2 for which the corresponding Jacobi polynomials $$(R_n^{(\alpha ,\beta )}(x))_{n\in \mathbb {N}_0}$$ ( R n ( α , β ) ( x ) ) n ∈ N 0 , normalized by $$R_n^{(\alpha ,\beta )}(1)\equiv 1$$ R n ( α , β ) ( 1 ) ≡ 1 , satisfy nonnegative linearization of products. Szwarc (Inzell Lectures on Orthogonal Polynomials, Adv. Theory Spec. Funct. Orthogonal Polynomials, vol 2, Nova Sci. Publ., Hauppauge, NY pp 103–139, 2005) asked to solve the analogous problem for the generalized Chebyshev polynomials $$(T_n^{(\alpha ,\beta )}(x))_{n\in \mathbb {N}_0}$$ ( T n ( α , β ) ( x ) ) n ∈ N 0 , which are the quadratic transformations of the Jacobi polynomials and orthogonal w.r.t. the measure $$(1-x^2)^{\alpha }|x|^{2\beta +1}\chi _{(-1,1)}(x)\,\mathrm {d}x$$ ( 1 - x 2 ) α | x | 2 β + 1 χ ( - 1 , 1 ) ( x ) d x . In this paper, we give the solution and show that $$(T_n^{(\alpha ,\beta )}(x))_{n\in \mathbb {N}_0}$$ ( T n ( α , β ) ( x ) ) n ∈ N 0 satisfies nonnegative linearization of products if and only if $$(\alpha ,\beta )\in V$$ ( α , β ) ∈ V , so the generalized Chebyshev polynomials share this property with the Jacobi polynomials. Moreover, we reconsider the Jacobi polynomials themselves, simplify Gasper’s original proof and characterize strict positivity of the linearization coefficients. Our results can also be regarded as sharpenings of Gasper’s one.

Strict Positivity for the Principal Eigenfunction of Elliptic Operators with Various Boundary Conditions

We consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain {\Omega\subset\mathbb{R}^{d}} and show that the first eigenfunction v satisfies {v(x)\geq\delta>0} for all {x\in\overline{\Omega}}, even if the boundary {\partial\Omega} is only Lipschitz continuous. Under such weak regularity assumptions the Hopf–Oleĭnik boundary lemma is not available; instead we use a new approach based on an abstract positivity improving condition for semigroups that map {L_{p}(\Omega)} into {C(\overline{\Omega})}. The same tool also yields corresponding results for Dirichlet or mixed boundary conditions. Finally, we show that our results can be used to derive strong minimum and maximum principles for parabolic and elliptic equations.

Strict Positivity of Operators and Inflated Schur Products

In this paper we provide a characterization of strictly positive matrices of operators and a factorization of their inverses. Consequently, we provide a test of strict positivity of matrices in . We give equivalent conditions for the inequality . We prove a theorem involving inflated Schur products [4, P. 153] of positive matrices of operators with invertible elements in the main diagonal which extends the results [3, P. 479, Theorem 7.5.3 (b), (c)]. We also discuss strictly completely positive linear maps in the paper.

On the Passivity and Positivity Properties in Dynamic Systems: Their Achievement under Control Laws and Their Maintenance under Parameterizations Switching

This paper is devoted to discuss certain aspects of passivity results in dynamic systems and the characterization of the regenerative systems counterparts. In particular, the various concepts of passivity as standard passivity, strict input passivity, strict output passivity, and very strict passivity (i.e., joint strict input and output passivity) are given and related to the existence of a storage function and a dissipation function. Later on, the obtained results are related to external positivity of systems and positivity or strict positivity of the transfer matrices and transfer functions in the time-invariant case. On the other hand, how to achieve or how eventually to increase the passivity effects via linear feedback by the synthesis of the appropriate feed-forward or feedback controllers or, simply, by adding a positive parallel direct input-output matrix interconnection gain is discussed.