eigenvalue estimates for schrodinger operators with complex potentials we discuss properties of eigenvalues of non self adjoint schrodinger operators with complex valued potential v among our results are estimates of the sum of powers of imaginary parts of eigenvalue

self adjoint elliptic operators with boundary conditions on not closed hypersurfaces abstract the theory of self adjoint extensions of symmetric operators is used to construct self adjoint realizations of a second order elliptic differential operator on r n with linear boundary conditions on a relatively open part of a compact hypersurface our approach allows to obtain kre n like resolvent formulae where the reference operator coincides with the free operator with domain h 2 r n this provides an useful tool for the scattering problem from a hypersurface concrete examples of this construction are developed in connection with the standard boundary conditions dirichlet neumann robin and type assigned either on a n 1 dimensional compact boundary or on a relatively open part schatten von neumann estimates for the difference of the powers of resolvents of the free and the perturbed operators are also proven these give existence and completeness of the wave operators of the associated scattering systems

on the discrete spectrum of non selfadjoint operators abstract we prove quantitative bounds on the eigenvalues of non selfadjoint unbounded operators obtained from selfadjoint operators by a perturbation that is relatively schatten these bounds are applied to obtain new results on the distribution of eigenvalues of schrodinger operators with complex potentials

non symmetric perturbations of self adjoint operators abstract we investigate the effect of non symmetric relatively bounded perturbations on the spectrum of self adjoint operators in particular we establish stability theorems for one or infinitely many spectral gaps along with corresponding resolvent estimates these results extend and improve classical perturbation results by kato and by gohberg kre n further we study essential spectral gaps and perturbations exhibiting additional structure with respect to the unperturbed operator in the latter case we can even allow for perturbations with relative bound 1 the generality of our results is illustrated by several applications massive and massless dirac operators point coupled periodic systems and two channel hamiltonians with dissipation

bounds on complex eigenvalues and resonances we obtain bounds on the complex eigenvalues of non self adjoint schrodinger operators with complex potentials and also on the complex resonances of self adjoint schrodinger operators our bounds are compared with numerical results and are seen to provide useful information

spectral theory of semibounded sturm liouville operators with local interactions on a discrete set we study the hamiltonians hx q with type point interactions at the centers xk on the positive half line in terms of energy forms we establish analogs of some classical results on operators hq d2 dx2 q with locally integrable potentials q lloc1 r in particular we prove that the hamiltonian hx q is self adjoint if it is lower semibounded this result completes the previous results of brasche perturbation of schrodinger hamiltonians by measures selfadjointness and semiboundedness j math phys 26 621 1985 on lower semiboundedness also we prove the analog of molchanov s discreteness criteria birman s result on stability of a continuous spectrum and investigate discreteness of a negative spectrum in the recent paper kostenko a and malamud m 1 d schrodinger operators with local point interactions on a discrete set j differ equations 249 253 2010 it was shown that the spectral properties of hx hx 0 correlate with the corresponding spectral properties of a certain clas

grid refinement for lattice bgk models lattice boltzmann models proposed at the end of the 1980s as the noise free version of lattice gas models are based on gas kinetic representation of fluid flow their recent modifications the lattice bgk models provide especially simple effective and stable algorithms for the solution of hydrodynamical problems a local second order grid refinement scheme for the lattice bgk model is proposed in this work the refinement scheme and a boundary fitting scheme for complicated geometries are applied to simulate a benchmark problem of flow past a cylinder in a channel with small and moderate reynolds numbers

non self adjoint graphs on finite metric graphs we consider laplace operators subject to various classes of non self adjoint boundary conditions imposed at graph vertices we investigate spectral properties existence of a riesz basis of projectors and similarity transforms to self adjoint laplacians among other things we describe a simple way to relate the similarity transforms between laplacians on certain graphs with elementary similarity transforms between matrices defining the boundary conditions

an efficient framework for fluid structure interaction using the lattice boltzmann method and immersed moving boundaries this paper presents a serial computational framework that hydrodynamically couples the lattice boltzmann method lbm and the discrete element method dem for the solution of particle suspension problems in two and three dimensions the single relaxation time bhatnagar gross krook lbgk form of the lattice boltzmann equation is employed with an immersed moving boundary method for the fluid structure interaction similar algorithms have been previously reported in the literature however this work deliberately utilizes solution options that minimize the computational overheads of the framework to facilitate simulations of multibody structural fields in large fluid domains in particular mixed boundary conditions are employed which combine the simple bounce back technique with the immersed moving boundary method and the relatively inexpensive d3q15 lattice is employed for 3d solutions the fundamentals of the lbm are briefly discussed followed by a review of the coupling techniques available for fsi using the lbm options for mapping solid obstacles to the lbm grid are presented and an algorithm for automatic dynamic subcycling of the two explicit solution schemes is outlined the lbm dem framework is then validated and benchmarked against previously published lbm results with comments made where appropriate on the comparative accuracy and convergence characteristics finally a multi particle suspension simulation is presented to qualitatively assess the performance of the framework when a large number of dynamic contacts exist

weakly coupled bound state of 2 d schrodinger operator with potential measure abstract we consider a self adjoint two dimensional schrodinger operator h which corresponds to the formal differential expression where is a finite compactly supported positive radon measure on r 2 from the generalized kato class and 0 is the coupling constant it was proven earlier that ess h 0 we show that for sufficiently small the condition d h 1 holds and that the corresponding unique eigenvalue has the asymptotic expansion c o 1 exp 4 r 2 0 with a certain constant c 0 we also obtain a formula for the computation of c the asymptotic expansion of the corresponding eigenfunction is provided the statements of this paper extend the results of simon 41 to the case of potentials measures also for regular potentials our results are partially new

eigenvalue bounds for dirac and fractional schrodinger operators with complex potentials abstract we prove lieb thirring type bounds for fractional schrodinger operators and dirac operators with complex valued potentials the main new ingredient is a resolvent bound in schatten spaces for the unperturbed operator in the spirit of frank and sabin 11

quasi boundary triples and semibounded self adjoint extensions in this note semibounded self adjoint extensions of symmetric operators are investigated with the help of the abstract notion of quasi boundary triples and their weyl functions the main purpose is to provide new sufficient conditions on the parameters in the boundary space to induce self adjoint realizations and to relate the decay of the weyl function to estimates on the lower bound of the spectrum the abstract results are illustrated with uniformly elliptic second order pdes on domains with non compact boundaries

extension theory and kreĭn type resolvent formulas for nonsmooth boundary value problems for a strongly elliptic second order operator a on a bounded domainr n it has been known for many years how to interpret the general closed l2 realizations of a as representing boundary conditions generally nonlocal when the domain and coe cients are smooth the purpose of the present paper is to extend this representation to nonsmooth domains and coecients including the case of hol der c 3 2 smoothness in such a way that pseudodierential methods are still avail able for resolvent construc tions and ellipticity considerations we show how it can be done for domains with

asymptotic spectral analysis in colliding leaky quantum layers abstract we consider the schrodinger operator with a complex delta interaction supported by two parallel hypersurfaces in the euclidean space of any dimension we analyse spectral properties of the system in the limit when the distance between the hypersurfaces tends to zero we establish the norm resolvent convergence to a limiting operator and derive first order corrections for the corresponding eigenvalues

spectral enclosures for non self adjoint extensions of symmetric operators abstract the spectral properties of non self adjoint extensions a b of a symmetric operator in a hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding weyl functions these extensions are given in terms of abstract boundary conditions involving an in general non symmetric boundary operator b in the abstract part of this paper sufficient conditions for sectoriality and m sectoriality as well as sufficient conditions for a b to have a non empty resolvent set are provided in terms of the parameter b and the weyl function special attention is paid to weyl functions that decay along the negative real line or inside some sector in the complex plane and spectral enclosures for a b are proved in this situation the abstract results are applied to elliptic differential operators with local and non local robin boundary conditions on unbounded domains to schrodinger operators with potentials of complex strengths supported on unbounded hypersurfaces or infinitely many points on the real line and to quantum graphs with non self adjoint vertex couplings

inverse spectral theory for symmetric operators with several gaps scalar type weyl functions let s be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non zero deficiency indices let j be an open subset of r if there exists a self adjoint extension s0 of s such that j is contained in the resolvent set of s0 and the associated weyl function of the pair s s0 is monotone with respect to j then for any self adjoint operator r there exists a self adjoint extension s such that the spectral parts s j and rj are unitarily equivalent it is shown that for any extension s of s the absolutely continuous spectrum of s0 is contained in that one of s moreover for a wide class of extensions the absolutely continuous parts of s and s are even unitarily equivalent

growth of weil petersson volumes and random hyperbolic surfaces of large genus in this paper we investigate the geometric properties of random hyperbolic surfaces of large genus we describe the relationship between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces first we study the asymptotic behavior of weil petersson volume vg n of the moduli spaces of hyperbolic surfaces of genus g with n punctures as g then we discuss basic geometric properties of a random hyperbolic surface of genus g with respect to the weil petersson measure as g

a remark on schatten von neumann properties of resolvent differences of generalized robin laplacians on bounded domains in this note we investigate the asymptotic behavior of the s numbers of the resolvent difference of two generalized self adjoint maximal dissipative or maximal accumulative robin laplacians on a bounded domain with smooth boundary for this we apply the recently introduced abstract notion of quasi boundary triples and weyl functions from extension theory of symmetric operators together with krein type resolvent formulae and well known eigenvalue asymptotics of the laplace beltrami operator on it is shown that the resolvent difference of two generalized robin laplacians belongs to the schatten von neumann class of any order p for which

some progress in lattice boltzmann method part i nonuniform mesh grids a new lattice boltzmann algorithm is proposed to simulate the navier stokes equation on arbitrary nonuniform mesh grids the new algorithm retains the advantages of the lattice boltzmann method parallel of algorithm ease of programming and ability to incorporate microscopic interactions a simulation of flow in a two dimensional symmetric channel with sudden expansion is carried out using the new algorithm on a nonuniform mesh the results of the simulation are in excellent agreement with previous experimental and numerical results

boundary value problems for elliptic partial differential operators on bounded domains for a symmetric operator or relation a with infinite deficiency indices in a hilbert space we develop an abstract framework for the description of symmetric and self adjoint extensions a of a as restrictions of an operator or relation t which is a core of the adjoint a this concept is applied to second order elliptic partial differential operators on smooth bounded domains and a class of elliptic problems with eigenvalue dependent boundary conditions is investigated

generalized resolvents and the boundary value problems for hermitian operators with gaps a hermitian operator a with gaps j j 1 j m is studied the self adjoint extensions which put exactly kj eigenvalues into each gap j j in particular for kj 0 1 j m the extensions preserving the gaps are described in terms of boundary conditions the generalized resolvents of the extensions with the indicated properties are described also a solvability criterion and description of all the solutions of the hamburger moment problem with supports in r j 1m j j are obtained in terms of the nevanlinna matrix

boundary relations and their weyl families the concepts of boundary relations and the corresponding weyl families are introduced let s be a closed symmetric linear operator or more generally a closed symmetric relation in a hilbert space h let h be an auxiliary hilbert space let graphics and let jh be defined analogously a unitary relation g from the krein space h 2 j h to the kre in space h 2 j h is called a boundary relation for the adjoint s if ker gamma s the corresponding weyl family m lambda is de ned as the family of images of the defect subspaces n over cap lambda lambda is an element of c r under gamma here gamma need not be surjective and is even allowed to be multi valued while this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex hilbert space h and the class of unitary relations gamma h 2 j h h 2 j h it also generalizes the notion of so called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems moreover these new notions yield for instance the following realization theorem every h valued maximal dissipative for lambda is an element of c holomorphic family of linear relations is the weyl family of a boundary relation which is unique up to unitary equivalence if certain minimality conditions are satisfied further connections between analytic and spectral theoretical properties of weyl families and geometric properties of boundary relations are investigated and some applications are given

spectral analysis of selfadjoint elliptic differential operators dirichlet to neumann maps and abstract weyl functions abstract the spectrum of a selfadjoint second order elliptic differential operator in l 2 r n is described in terms of the limiting behavior of dirichlet to neumann maps which arise in a multi dimensional glazman decomposition and correspond to an interior and an exterior boundary value problem this leads to pde analogs of renowned facts in spectral theory of odes the main results in this paper are first derived in the more abstract context of extension theory of symmetric operators and corresponding weyl functions and are applied to the pde setting afterwards

schrodinger operators with nonlocal point interactions schrodinger operators with nonlocal point interactions are considered as new solvable models with point interactions examples in one and three dimensions are discussed corresponding direct and inverse scattering problems in one dimension are also discussed

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

riesz bases for p subordinate perturbations of normal operators abstract for a class of unbounded perturbations of unbounded normal operators the change of the spectrum is studied and spectral criteria for the existence of a riesz basis with parentheses of root vectors are established a riesz basis without parentheses is obtained under an additional a priori assumption on the spectrum of the perturbed operator the results are applied to two classes of block operator matrices

self adjoint extensions of the laplace beltrami operator and unitaries at the boundary abstract we construct in this article a class of closed semi bounded quadratic forms on the space of square integrable functions over a smooth riemannian manifold with smooth compact boundary each of these quadratic forms specifies a semi bounded self adjoint extension of the laplace beltrami operator these quadratic forms are based on the lagrange boundary form on the manifold and a family of domains parametrized by a suitable class of unitary operators on the boundary that will be called admissible the corresponding quadratic forms are semi bounded below and closable finally the representing operators correspond to semi bounded self adjoint extensions of the laplace beltrami operator this family of extensions is compared with results existing in the literature and various examples and applications are discussed

planar stochastic hyperbolic triangulations pursuing the approach of angel and ray ann probab 2015 we introduce and study a family of random infinite triangulations of the full plane that satisfy a natural spatial markov property these new random lattices naturally generalize angel and schramm s uniform infinite planar triangulation uipt and are hyperbolic in flavor we prove that they exhibit a sharp exponential volume growth are non liouville and that the simple random walk on them has positive speed almost surely we conjecture that these infinite triangulations are the local limits of uniform triangulations whose genus is proportional to the size an artistic representation of a random 3 connected triangulation of the plane with hyperbolic flavor

interpolated boundary condition for lattice boltzmann simulations of flows in narrow gaps several different interpolation schemes have been proposed for improving the accuracy of lattice boltzmann simulations in the vicinity of a solid boundary however these methods require at least two or three fluid nodes between nearby solid surfaces a condition that may not be fulfilled in dense suspensions or porous media for example here we propose an interpolation of the equilibrium distribution which leads to a velocity field that is both second order accurate in space and independent of viscosity the equilibrium interpolation rule infers population densities on the boundary itself to reduce the span of nodes needed for interpolation it requires a minimum of one grid spacing between the nodes by contrast the linear interpolation rule requires two fluid nodes in the gap and leads to a viscosity dependent slip velocity while the multireflection rule is viscosity independent but requires a minimum of three fluid nodes

lattice boltzmann method for 3 d flows with curved boundary in this work we investigate two issues that are important to computational efficiency and reliability in fluid dynamic applications of the lattice boltzmann equation lbe 1 computational stability and accuracy of different lattice boltzmann models and 2 the treatment of the boundary conditions on curved solid boundaries and their 3 d implementations three athermal 3 d lbe models q15d3 q19d3 and q27d3 are studied and compared in terms of efficiency accuracy and robustness the boundary treatment recently developed by filippova and hanel 1998 j comp phys 147 219 and mei et al 1999 j comp phys 155 307 in 2 d is extended to and implemented for 3 d the convergence stability and computational efficiency of the 3 d lbe models with the boundary treatment for curved boundaries were tested in simulations of four 3 d flows 1 fully developed flows in a square duct 2 flow in a 3 d lid driven cavity 3 fully developed flows in a circular pipe and 4 a uniform flow over a sphere we found that while the 15 velocity 3 d q15d3 model is more prone to numerical instability and the q27d3 is more computationally intensive the q19d3 model provides a balance between computational reliability and efficiency through numerical simulations we demonstrated that the boundary treatment for 3 d arbitrary curved geometry has second order accuracy and possesses satisfactory stability characteristics

1 d schrodinger operators with local point interactions on a discrete set abstract spectral properties of 1 d schrodinger operators h x d 2 d x 2 x n x n x x n with local point interactions on a discrete set x x n n 1 are well studied when d inf n k n x n x k 0 our paper is devoted to the case d 0 we consider h x in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding weyl functions we show that the spectral properties of h x like self adjointness discreteness and lower semiboundedness correlate with the corresponding spectral properties of certain classes of jacobi matrices based on this connection we obtain necessary and sufficient conditions for the operators h x to be self adjoint lower semibounded and discrete in the case d 0 the operators with type interactions are investigated too the obtained results demonstrate that in the case d 0 as distinguished from the case d 0 the spectral properties of the operators with and type interactions are substantially different

the dirichlet to neumann operator on rough domains abstract we consider a bounded connected open set r d whose boundary has a finite d 1 dimensional hausdorff measure then we define the dirichlet to neumann operator d 0 on l 2 by form methods the operator d 0 is self adjoint and generates a contractive c 0 semigroup s s t t 0 on l 2 we show that the asymptotic behaviour of s t as t is related to properties of the trace of functions in h 1 which may or may not have

generalized interactions supported on hypersurfaces we analyze a family of singular schrodinger operators with local singular interactions supported by a hypersurface n n 2 being the boundary of a lipschitz domain bounded or unbounded not necessarily connected at each point of the interaction is characterized by four real parameters the earlier studied case of and interactions being particular cases we discuss spectral properties of these operators and derive operator inequalities between those referring to the same hypersurface but different couplings and describe their implications for spectral properties

poisson dirichlet distribution for random belyi surfaces brooks and makover introduced an approach to studying the global geometric quantities in particular the first eigenvalue of the laplacian injectivity radius and diameter of a typical compact riemann surface of large genus based on compactifying finite area riemann surfaces associated with random cubic graphs by a theorem of belyi these are dense in the space of compact riemann surfaces the question as to how these surfaces are distributed in the teichmuller spaces depends on the study of oriented cycles in random cubic graphs with random orientation brooks and makover conjectured that asymptotically normalized cycle lengths follow poisson dirichlet distribution we present a proof of this conjecture using representation theory of the symmetric group

the dirichlet to neumann operator via hidden compactness abstract we show that to each symmetric elliptic operator of the form a k a k l l c on a bounded lipschitz domain r d one can associate a self adjoint dirichlet to neumann operator on l 2 which may be multi valued if 0 is in the dirichlet spectrum of a to overcome the lack of coerciveness in this case we employ a new version of the lax milgram lemma based on an indirect ellipticity property that we call hidden compactness we then establish uniform resolvent convergence of a sequence of dirichlet to neumann operators whenever the underlying coefficients converge uniformly and the second order limit operator in l 2 has the unique continuation property we also consider semigroup convergence

gravity in a lattice boltzmann model in this paper we consider the introduction of a body force in the incompressible limit into the lattice boltzmann model a number of methods are considered and their suitability to our objectives determined when there is no density variation across the fluid gravity can be introduced in the form of an altered pressure gradient this method correctly satisfies the navier stokes equation however if there is a non negligible density variation present produced by the body force or otherwise this method becomes less accurate as the density variation increases and the constant density approximation becomes less valid three other methods are also considered for application when there is a non negligible density variation the equations of motion satisfied by these models are found up to second order in the knudsen number and it is seen that only one of these methods satisfies the true navier stokes equation numerical simulations are performed to compare the different models and to assess the range of application of each

on the unitary equivalence of absolutely continuous parts of self adjoint extensions the classical weyl von neumann theorem states that for any self adjoint operator a0 in a separable hilbert space h there exists a non unique hilbert schmidt operator c c such that the perturbed operator a0 c has purely point spectrum we are interesting whether this result remains valid for non additive perturbations by considering the set exta of self adjoint extensions of a given densely defined symmetric operator a in h and some fixed a0 a0 exta we show that the ac parts a ac and a0ac of a a exta and a0 are unitarily equivalent provided that the resolvent difference ka a i 1 a0 i 1 is compact and the weyl function m of the pair a a0 admits weak boundary limits m t w limy 0m t iy for a e t r this result generalizes the classical kato rosenblum theorem moreover it demonstrates that for such pairs a a0 the weyl von neumann theorem is in general not true in the class exta

variations on a theme of jost and pais abstract we explore the extent to which a variant of a celebrated formula due to jost and pais which reduces the fredholm perturbation determinant associated with the schrodinger operator on a half line to a simple wronski determinant of appropriate distributional solutions of the underlying schrodinger equation generalizes to higher dimensions in this multi dimensional extension the half line is replaced by an open set r n n n n 2 where has a compact nonempty boundary satisfying certain regularity conditions our variant involves ratios of perturbation determinants corresponding to dirichlet and neumann boundary conditions on and invokes the corresponding dirichlet to neumann map as a result we succeed in reducing a certain ratio of modified fredholm perturbation determinants associated with operators in l 2 d n x n n to modified fredholm determinants associated with operators in l 2 d n 1 n 2 applications involving the birman schwinger principle and eigenvalue counting functions are discussed