pinecone/core-2020-05-10-deduplication huggingface dataset:
An Estimate of the Gap of Spectrum of Schrödinger Operators which Generate Hyperbounded Semigroups

AbstractLogarithmic Sobolev inequalities with potential functions on loop spaces were proved by L. Gross (1991, J. Funct. Anal.102, 265–313), S. Aida (1996, in “Proceedings of the Fifth Gregynog Symposium, Stochastic Analysis and Applications,” pp. 1–19, World Scientific, Singapore), and F.-Z. Gong and Z.-M. Ma (1998, J. Funct. Anal., 599–623). The generators of the Dirichlet forms are Schrödinger operators on loop spaces and they generate hyperbounded semigroups. Recently, F.-Z. Gong, M. Röckner and L. M. Wu (2000, Poincaré inequality for weighted first order Sobolev spaces on loop spaces, submitted) proved the existence of the gap of spectrum at the lowest eigenvalue of the Schrödinger operators by using M. Hino's exponential decay estimates of the semigroup (2000, Osaka J. Math.37, 603–624) and the author's strong ergodicity property of the semi-group (S. Aida, 1998, J. Funct. Anal.158, 152–185). In this paper, we will give a lower bound on the gap by using a weak Poincaré inequality which was introduced by M. Röckner and F.-Y. Wang (2000, Weak Poincaré inequalities and L2-convergence rates of Markov semigroups, preprint). Also we will give estimates on the distribution function of ground states using the weak Poincaré inequality
Poincaré Inequality for Weighted First Order Sobolev Spaces on Loop Spaces

AbstractLet E be the loop space over a compact connected Riemannian manifold with a torsion skew symmetric connection. Let LD be the Ornstein–Uhlenbeck operator on a nonempty connected component D of the loop space E and let V:D→R be the restriction on D of the potential in the logarithmic Sobolev inequality found by L. Gross on the loop group by S. Aida and by F. Z. Gong and Z. M. Ma on the loop space, respectively. We prove that the Schrödinger operator −LV≔−LD+V always has a spectral gap at the bottom λ0(V) of its spectrum and thus has its ground state transformed operator φ−1(−LV−λ0(V))φ, where φ is the unique ground state of −LV. In particular, our result proves L. Gross's conjecture about the existence of a spectral gap for the ground state transform of the Schrödinger operator studied by him on the loop group. In addition, in all the above cases we identify the domain of the Dirichlet forms associated with the ground state transforms as weighted first order Sobolev spaces with weight given by φ2, thus establishing a Poincaré inequality for them. All these results are consequences from some new results in this paper on Dirichlet forms characterizing certain classes with spectral gaps and from results by S. Aida and M. Hino
Weak Poincaré Inequalities and L2-Convergence Rates of Markov Semigroups

AbstractIn order to describe L2-convergence rates slower than exponential, the weak Poincaré inequality is introduced. It is shown that the convergence rate of a Markov semigroup and the corresponding weak Poincaré inequality can be determined by each other. Conditions for the weak Poincaré inequality to hold are presented, which are easy to check and which hold in many applications. The weak Poincaré inequality is also studied by using isoperimetric inequalities for diffusion and jump processes. Some typical examples are given to illustrate the general results. In particular, our results are applied to the stochastic quantization of field theory in finite volume. Moreover, a sharp criterion of weak Poincaré inequalities is presented for Poisson measures on configuration spaces
Essential Self-Adjointness for Semi-bounded Magnetic Schrödinger Operators on Non-compact Manifolds

AbstractWe prove essential self-adjointness for semi-bounded below magnetic Schrödinger operators on complete Riemannian manifolds with a given positive smooth measure which is fixed independently of the metric. Some singularities of the scalar potential are allowed. This is an extension of the Povzner–Wienholtz–Simader theorem. The proof uses the scheme of Wienholtz but requires a refined invariant integration by parts technique, as well as the use of a family of cut-off functions which are constructed by a non-trivial smoothing procedure due to Karcher
Estimates for Periodic and Dirichlet Eigenvalues of the Schrödinger Operator with Singular Potentials

AbstractIn this paper, the periodic and the Dirichlet problems for the Schrödinger operator −(d2/dx2)+V are studied for singular, complex-valued potentials V in the Sobolev space H−αper[0, 1] (0⩽α<1). The following results are shown: (1)The periodic spectrum consists of a sequence (λk)k⩾0 of complex eigenvalues satisfying the asymptotics (for any ε>0)λ2n−1, λ2n=n2π2+V(0)±V(−2n)V(2n)+O(n3α/2−1/2+ε), where V(k) denote the Fourier coefficients of V. (2)The Dirichlet spectrum consists of a sequence (μn)n⩾1 of complex eigenvalues satisfying the asymptotics (for any ε>0)μn=n2π2+V(0)−V(−2n)+V(2n)2+O(n2α−1+ε)
On the Spectral Resolution of the Quantum Toda Lattice

AbstractIn this paper we study the solutions of the equation det(λ−L)ψ=0, where L is the Lax operator of the quantum Toda lattice. The solutions of the equation are determined by the eigenvectors of L, [formula]. In the classical case, there exists the canonical embedding of n-dimensional Toda lattice ↪n+1-dimensional Toda lattice. We show that the quantum analogue of this embedding exists. In the classical case, the Lax operator of the Toda lattice lies in sl(n). In the quantum case, this fact corresponds to the restriction of det(λ−L)ψ=0 to the hyperplane x1+…+xn=constant. We make clear the gap between the solution space of the restricted case and that of the non-restricted case. In the example of the 2-dimensional case, we show that the Bessel functions appear as the basis of the solution space of the above equation
Rarita–Schwinger Type Operators in Clifford Analysis

AbstractIn this paper we investigate a generalization of the classical Rarita–Schwinger equations for spin 3/2 fields to the case of functions taking values in irreducible representation spaces with weight k+1/2. These fields may be realised as functions taking values in spaces of spherical monogenics earlier considered in F. Sommen and N. Van Acker (1993, in “Clifford Algebras and Their Applications in Mathematical Physics,” Kluwer Academic, Dordrecht/Norwell, MA). In this paper we develop the main function theoretic results
Critical Point Theory on Partially Ordered Hilbert Spaces

AbstractWe develop some abstract critical point theory in order to prove that boundary value problems like the model problem[formula] on a bounded domain Ω⊂RN, 2<p<2N/(N−2) have infinitely many sign changing solutions ±uk, k∈N, which are not comparable, that is, uk−ul and uk+ul change sign for k≠l. We also show that there are no subsolutions u such that u<uk for some k and u is positive somewhere. A corresponding nonexistence result applies to supersolutions, Related results on the existence of sign-changing solutions hold for other classes of nonlinearities
On a Degenerate Heat Equation with a Singular Potential

AbstractUsing a new method, we generalize the blow up and existence result from P. Baras and J. A. Goldstein (1984, Trans. Amer. Math. Soc.284, 121–139) to heat equations on the Heisenberg group. In doing so we need to overcome the difficulty that the equation in this case is both degenerate and of variable coefficients. Comparing with the Euclidean case, an interesting new result is that solutions can blow up even when the singularity of the potential is weaker than the inverse square of the distance function
Central Gaussian Semigroups of Measures with Continuous Density

AbstractThis paper investigates the existence and properties of symmetric central Gaussian semigroups (μt)t>0 which are absolutely continuous and have a continuous density x↦μt(x), t>0, with respect to Haar measure on groups of the form Rn×K where K is compact connected locally connected and has a countable basis for its topology. We prove that there always exists a wealth of such Gaussian semigroups on any such group. For instance, if ψ is any positive function increasing to infinity, there exists a symmetric central Gaussian semigroup having a continuous density such that logμt(e)⩽log(1+1/t)ψ(1/t) as t tends to zero. Among other results of this type we give a necessary and sufficient condition on the structure of K for the existence of symmetric central Gaussian semigroups having a continuous density and such that tλlogμt(e) is bounded above and below by positive constants for t∈(0, 1) and some fixed λ>0. This condition is independent of λ. These results are proved by splitting any Gaussian semigroup (in a canonical way) into a semisimple part living on the commutator group G′ and an Abelian part living on A=G/G′. For symmetric central Gaussian semigroups, we show that many properties hold for (μt)t>0 if and only if they hold for both the semisimple and the Abelian parts. This splitting principle is one of the main new tools developed in this paper. It leads to a much better understanding of central Gaussian semigroups and related objects and lets us answer a number of open questions. For instance, to any Gaussian convolution semigroup are associated a harmonic sheaf H and a quasi-distance d on G. For symmetric central Gaussian semigroups on G=Rn×K, we show that H is a Brelot sheaf if and only if limt→0tlogμt(e)=0. A sufficient (but not necessary) condition is that the distance d is continuous. Together with a celebrated result of Bony, our results show that the compact group K is a Lie group if and only if any bi-invariant elliptic Bauer harmonic sheaf on G is a Brelot sheaf
Quasi-invariant Measures on the Group of Diffeomorphisms and Smooth Vectors of Unitary Representations

AbstractLet M be a smooth manifold and Diff0(M) the group of all smooth diffeomorphisms on M with compact support. Our main subject in this paper concerns the existence of certain quasi-invariant measures on groups of diffeomorphisms, and the denseness of C∞-vectors for a given unitary representation U of Diff*0(M), the connected component of the identity in Diff0(M). We first generalize some results of Shavgulidze on quasi-invariant measures on diffeomorphism groups. Then we prove the following result: Suppose that M is compact and U has the property that the action extends continuously to Diff*k(M), the group of Ck diffeomorphisms which are homotopic to the identity, for some finite k. Then U has a dense set of C∞-vectors. We also give an extension of our theorem to non-compact M
Quantum Ω-Semimartingales and Stochastic Evolutions

AbstractWe explore Ω-adaptedness, a variant of the usual notion of adaptedness found in stochastic calculus. It is shown that the (non-adapted) quantum stochastic integrals of bounded, Ω-adapted processes are themselves bounded and Ω-adapted, a fact that may be deduced from the Bismut–Clark–Ocone formula of Malliavin calculus. An algebra analogous to Attal's class S of regular quantum semimartingales is defined, and product and functional Itô formulae are given. We consider quantum stochastic differential equations with bounded, Ω-adapted coefficients that are time dependent and act on the whole Fock space. Solutions to such equations may be used to dilate quantum dynamical semigroups in a manner that generalises, and gives new insight into, that of R. Alicki and M. Fannes (1987, Comm. Math. Phys.108, 353–361); their unitarity condition is seen to be the usual condition of R. L. Hudson and K. R. Parthasarathy (1984, Comm. Math. Phys93, 301–323)
Simple C*-algebras with Locally Bounded Irreducible Representation

AbstractLet A and B be two unital separable simple nuclear C*-algebras with tracial topological rank zero. Suppose that both A and B have the local approximation property: for any finite subset F and ε>0, there is a C*-subalgebra C such that its dimensions of irreducible representations are bounded anddist(x, C)<εforallx∈F. Suppose also that ρA(K0(A)) is a divisible group, where ρA:K0(A)→Aff(T(A)) is the homomorphism from K0(A) to affine functions on the tracial space of A. Then A≅B if and only if(K0(A), K0(A)+, [1A], K1(A))≅(K0(B), K0(B)+, [1B], K1(B)). Applications to simple crossed products arising from smooth minimal dynamical systems are given. Let M1 and M2 be compact manifolds, hi:Mi→Mi be a minimal diffeomorphism (i=1, 2) and let Ai=C*(Z, Mi, hi). Suppose that TR(Ai)=0 and the range of K0(Ai) in Aff(T(Ai)) is divisible. Then A1≅A2 if and only if(K0(A1), K0(A1), [1A1], K0(A1)))≅(K0(A2), K0(A2)+, [1A2], K1(A2))
A Theorem of Brown–Halmos Type for Bergman Space Toeplitz Operators

AbstractWe study the analogues of the Brown–Halmos theorem for Toeplitz operators on the Bergman space. We show that for f and g harmonic, TfTg=Th only in the trivial case, provided that h is of class C2 with the invariant laplacian bounded. Here the trivial cases are f or g holomorphic. From this we conclude that the zero-product problem for harmonic symbols has only the trivial solution. Finally, we provide examples that show that the Brown–Halmos theorem fails for general symbols, even for symbols continuous up to the boundary
R-Cyclic Families of Matrices in Free Probability

AbstractWe introduce the concept of “R-cyclic family” of matrices with entries in a noncommutative probability space; the definition consists in asking that only the “cyclic” noncrossing cumulants of the entries of the matrices are allowed to be nonzero. Let A1, …, As be an R-cyclic family of d×d matrices over a noncommutative probability space (A, ϕ). We prove a convolution-type formula for the explicit computation of the joint distribution of A1, …, As (considered in Md(A) with the natural state), in terms of the joint distribution (considered in the original space (A, ϕ)) of the entries of the s matrices. Several important situations of families of matrices with tractable joint distributions arise by application of this formula. Moreover, let A1, …, As be a family of d×d matrices over a noncommutative probability space (A, ϕ), let D⊂Md(A) denote the algebra of scalar diagonal matrices, and let C be the subalgebra of Md(A) generated by {A1, …, As}∪D. We prove that the R-cyclicity of A1, …, As is equivalent to a property of C—namely that C is free from Md(C), with amalgamation over D
The Jacobson Radical for Analytic Crossed Products

AbstractWe characterise the Jacobson radical of an analytic crossed product C0(X)×φZ+, answering a question first raised by Arveson and Josephson in 1969. In fact, we characterise the Jacobson radical of analytic crossed products C0(X)×φZd+. This consists of all elements whose “Fourier coefficients” vanish on the recurrent points of the dynamical system (and the first one is zero). The multidimensional version requires a variation of the notion of recurrence, taking into account the various degrees of freedom
Bounded Mean Oscillation on the Bidisk and Operator BMO

AbstractWe identify several of the spaces in the inclusion chain of BMO spaces in two variables with certain BMO spaces of operator-valued functions in one variable and discuss various interesting consequences of this identification. Our main result is a previously unknown strict inclusion in the chain of BMO spaces in two variables that translates into the fact that there exists a function b on the bidisk such that the associated little Hankel operator γb=P⊥1P⊥2bP2P1 is bounded on products of normalized Szego&#x030B; kernels kz1⊗kz2, z1, z2∈D, but does not extend to a bounded linear operator on H2(T2). In other words, the well-known result that boundedness of Hankel operators in one variable can be tested on normalized Szego&#x030B; kernels does not extend to little Hankel operators in two variables. In the framework of operator BMO functions, this can be expressed as a new result about BMO spaces of Hankel operator-valued functions. We also study an interesting link between the celebrated Carleson counterexample (L. Carleson, Mittag–Leffler Report No. 7, 1974) for the bidisk and counterexamples to the Operator Carleson Embedding Theorem
Nonlinear Fredholm Operators with Noncompact Fibers and Applications to Elliptic Problems on RN

AbstractUnder suitable conditions, an equation F(x)=y between Banach spaces involving a nonlinear Fredholm mapping F of nonnegative index is shown to have a noncompact and hence infinite set of solutions for almost every y for which the equation is solvable. The proof of this nonuniqueness (but not existence) result relies on an entirely new line of arguments in which the concept of generalized critical value plays a central role. When F:W2, p(RN)→Lp(RN) is associated with a quasilinear elliptic PDE on RN with “constant coefficient,” it often happens that a crucial denseness hypothesis in the abstract theorem is equivalent to the existence of a nontrivial solution to the equation F(x)=F(0) and hence can be verified in practice. Generalizations exist for some classes of problems with nonconstant coefficients and for problems on exterior domains
Localization of the Riemann–Roch Character

AbstractWe present a K-theoretic approach to the Guillemin–Sternberg conjecture (V. Guillemin and S. Sternberg, Invent. Math.67 (1982), 515–538), about the commutativity of geometric quantization and symplectic reduction, which was proved by E. Meinrenken (J. Amer. Math. Soc.9 (1996), 373–389; Adv. Math.134, (1998), 240–277) and Tian–Zhang (Y. Tian and W. Zhang, Invent. Math.132 (1998), 229–259). Besides providing a new proof of this conjecture for the full non-Abelian group action case, our methods lead to a generalization for compact Lie group actions on manifolds that are not symplectic; these manifolds carry an invariant almost complex structure and an abstract moment map
The Dirac Operator of a Commuting d-Tuple

AbstractGiven a commuting d-tuple T=(T1, …, Td) of otherwise arbitrary operators on a Hilbert space, there is an associated Dirac operator DT. Significant attributes of the d-tuple are best expressed in terms of DT, including the Taylor spectrum and the notion of Fredholmness. In fact, all properties of T derive from its Dirac operator. We introduce a general notion of Dirac operator (in dimension d=1, 2, …) that is appropriate for multivariable operator theory. We show that every abstract Dirac operator is associated with a commuting d-tuple, and that two Dirac operators are isomorphic iff their associated operator d-tuples are unitarily equivalent. By relating the curvature invariant introduced in a previous paper to the index of a Dirac operator, we establish a stability result for the curvature invariant for pure d -contractions of finite rank. It is shown that for the subcategory of all such T that are (a) Fredholm and and (b) graded, the curvature invariant K(T) is stable under compact perturbations. We do not know if this stability persists when T is Fredholm but ungraded, although there is concrete evidence that it does
A Convergence Result for Nonautonomous Subgradient Evolution Equations and Its Application to the Steepest Descent Exponential Penalty Trajectory in Linear Programming

AbstractWe present a new result on the asymptotic behavior of nonautonomous subgradient evolution equations of the form u(t)∈−∂ϕt(u(t)), where {ϕt:t⩾0} is a family of closed proper convex functions. The result is used to study the flow generated by the family ϕt(x)=f(x, r(t)), where f(x, r)≔cTx+r∑exp[(Aix−bi)/r] is the exponential penalty approximation of the linear program min{cTx:Ax⩽b}, and r(t) is a positive function tending to 0 when t→∞. We prove that the trajectory u(t) converges to an optimal solution u∞ of the linear program, and we give conditions for the convergence of an associated dual trajectory μ(t) toward an optimal solution of the dual program
Some Qualitative Properties for the Total Variation Flow

AbstractWe prove the existence of a finite extinction time for the solutions of the Dirichlet problem for the total variation flow. For the Neumann problem, we prove that the solutions reach the average of its initial datum in finite time. The asymptotic profile of the solutions of the Dirichlet problem is also studied. It is shown that the profiles are nonzero solutions of an eigenvalue-type problem that seems to be unexplored in the previous literature. The propagation of the support is analyzed in the radial case showing a behaviour entirely different to the case of the problem associated with the p-Laplacian operator. Finally, the study of the radially symmetric case allows us to point out other qualitative properties that are peculiar of this special class of quasilinear equations
Graph C*-Algebras with Real Rank Zero

AbstractGiven a row-finite directed graph E, a universal C*-algebra C*(E) generated by a family of partial isometries and projections subject to the relations determined by E is associated to the graph E. The Cuntz–Krieger algebras are those graph C*-algebras associated to some finite graphs. We prove that a graph C*-algebra C*(E) has real rank zero in the sense that the set of invertible self-adjoint elements is dense in the set of all self-adjoint elements in C*(E) (or in the unitization of C*(E) if C*(E) is nonunital) if and only if E satisfies a loop condition (K) that is analogous to the condition for a finite {0, 1} matrix A under which Cuntz analyzed the ideal structure of the Cuntz–Krieger algebra OA
Large Deviations Asymptotics for Spherical Integrals

AbstractConsider the spherical integral I(β)N(DN, EN)≔∫exp{Ntr(UDNU*EN)}dmβN(U), where mβN denote the Haar measure on the orthogonal group ON when β=1 and on the unitary group UN when β=2, and DN, EN are diagonal real matrices whose spectral measures converge to μD, μE. In this paper we prove the existence and represent as solution to a variational problem the limit I(β)(μD, μE)≔limN−2logI(β)N(DN, EN). This limit appears in so-called “matrix models” but also in the evaluation of large deviations of the spectral measure of generalized Wishart matrices. Our technique is based on stochastic calculus, large deviations, and elements from free probability
Noncommutative Pressure and the Variational Principle in Cuntz–Krieger-type C*-Algebras

AbstractLet a be a self-adjoint element of an exact C*-algebra A and θ:A→A a contractive completely positive map. We define a notion of dynamical pressure Pθ(a) which adopts Voiculescu's approximation approach to noncommutative entropy and extends the Voiculescu–Brown topological entropy and Neshveyev and Størmer unital-nuclear pressure. A variational inequality bounding Pθ(a) below by the free energies hσ(θ)+σ(a) with respect to the Sauvageot–Thouvenot entropy hσ(θ) is established in two stages via the introduction of a local state approximation entropy, whose associated free energies function as an intermediate term. Pimsner C*-algebras furnish a framework for investigating the variational principle, which asserts the equality of Pθ(a) with the supremum of the free energies over all θ -invariant states. In one direction we extend Brown's result on the constancy of the Voiculescu–Brown entropy upon passing to the crossed product, and in another we show that the pressure of a self-adjoint element over the Markov subshift underlying the canonical map on the Cuntz–Krieger algebra OA is equal to its classical pressure. The latter result is extended to a more general setting comprising an expanded class of Cuntz–Krieger-type Pimsner algebras, leading to the variational principle for self-adjoint elements in a diagonal subalgebra. Equilibrium states are constructed from KMS states under certain conditions in the case of Cuntz–Krieger algebras