upper bounds on wavepacket spreading for random jacobi matrices a method is presented for proving upper bounds on the moments of the position operator when the dynamics of quantum wavepackets is governed by a random possibly correlated jacobi matrix as an application one obtains sharp upper bounds on the diffusion exponents for random polymer models coinciding with the lower bounds obtained in a prior work the second application is an elementary argument not using multiscale analysis or the aizenman molchanov method showing that under the condition of uniformly positive lyapunov exponents the moments of the position operator grow at most logarithmically in time

the absolutely continuous spectrum of jacobi matrices i explore some consequences of a groundbreaking result of brei messer and pearson on the absolutely continuous spectrum of one dimensional schrodinger operators these include an oracle theorem that predicts the potential and rather general results on the approach to certain limit potentials in particular we prove a denisov rakhmanov type theorem for the general nite gap case the main theme is the following it is extremely dicult to produce abso lutely continuous spectrum in one space dimension and thus its existence has strong implications

on inverse spectral theory for self adjoint extensions mixed types of spectra abstract lethbe a symmetric operator in a separable hilbert space h suppose thathhas some gapj we shall investigate the question about what spectral properties the self adjoint extensions ofhcan have inside the gapjand provide methods on how to construct self adjoint extensions ofhwith prescribed spectral properties insidej under some weak assumptions about the operatorhwhich are satisfied e g provided the deficiency indices ofhare infinite and the operator h 1is compact for one regular point ofh we shall show that for every auxiliary self adjoint operatorm in the hilbert space h and every open subsetj0of the gapjofhthere exists a self adjoint extensionhofhsuch that insidejthe self adjoint extensionhofhhas the same absolutely continuous and the same point spectrum as the given operatorm and the singular continuous spectrum ofhinjequals the closure ofj0inj moreover we shall present a method of how to construct such a self adjoint extensionh via our methods it is possible to construct new kinds of self adjoint realizations of the laplacian on a bounded domain in r d d 1 with spectral properties very different from the spectral properties of the self adjoint realizations known before mathematics subject classification 1991 47a10 47a60 47b25 47e05 47f05

integrability of the 2d lotka volterra system via polynomial first integrals and polynomial inverse integrating factors we present new first integrals of the two dimensional lotka volterra systems which have a polynomial inverse integrating factor moreover we characterize all the polynomial first integrals of the two dimensional lotka volterra systems

eigenfunctions transfer matrices and absolutely continuous spectrum of one dimensional schrodinger operators on l r and its half line problem h on l2 0 with u 0 0 boundary conditions we will focus on a new approach to the absolutely continuous spectrum ac h and more generally ac h the essential support of the a c part of the spectral measures what is new in our approach is that it relies on estimates on the transfer matrix that is the 2 2 matrix te n m which takes u m 1 u m to u n 1 u n for solutions u of hu eu in the continuum case use u x u x instead of u x 1 u x we let te n te n 0 for example we will prove the following

a probabilistic approach to one dimensional schrodinger operators with sparse potentials we consider the one dimensional schrodinger equation with sparse potential v i e mainly v 0 it is shown that the asymptotics of the solutions corresponding to positive energies e can be approximately described by an infinite sum of independent random variables using results from probability theory we can then determine the spectral properties of the operators under consideration we prove absolute continuity for a general class of potentials and we also have examples with singular continuous spectrum

p dissipative operator in this paper the authors prove that the generalized positive p selfadjoint gpps operators in banach space satisfy the generalized schwarz inequality solve the maximal dissipative extension representation of p dissipative operators in banach space by using the inequality and introducing the generalized indefinite inner product giip space and apply the result to a certain type of schrodinger operator

polynomial first integrals of quadratic vector fields abstract we classify all quadratic polynomial differential systems having a polynomial first integral and provide explicit normal forms for such systems and for their first integrals

linguistic modelling of imperfect spatial information as a basis for simplifying spatial analysis spatial analysis is at the basis of several decision processes in fields such as urban planning and environmental management in order to carry it out effectively more flexible geographic information systems are needed to be able to represent and manage the imperfection that invariably affects geographic information starting from the consideration that experts usually deal with imperfect spatial information through linguistic terms e g they identify the approximate position of a phenomenon on a map and classify spatial properties through linguistic labels this contribution proposes the use of linguistic granule of information to represent and manage imperfect spatial information a fuzzy object based data model is proposed as a tool for supporting spatial analysis based on the management of linguistic granule in particular the problem of defining methods to manage imperfect information depending on the type of imperfection is discussed finally an example of spatial analysis applied to support a decision problem in environmental impact assessment is described

fuzzy association rules and the extended mining algorithms this paper focuses on the notion of fuzzy association rules that are of the form x y where either x or y is a collection of fuzzy sets the work stems from two observations first in generalized association rule mining the taxonomies concerned may not be crisp but fuzzy e g tomato could be regarded as both fruit and vegetable each at a different degree second managers often refer to decision rules in terms of linguistic expressions that may or may not be the nodes of the taxonomies e g very expensive cloth tropical fruit the paper deals with the fuzziness based upon fuzzy taxonomies that reflect partial belongings among itemsets as well as upon the extended settings for the degree of support and the degree of confidence apriori algorithms are extended accordingly to discover association rules across the higher level taxonomic nodes which are fuzzy sets in general as a result the discovered rules are fuzzy rules furthermore linguistic hedges are also incorporated in mining fuzzy rules to express more meaningful knowledge moreover the extended algorithms are tested with the synthetic data revealing similar computational complexities to that of the classical algorithm finally the extended algorithms are applied to a real data set with an explanation of the semantics of discovered fuzzy rules

a model for describing and composing direction relations between overlapping and contained regions current models of direction relations are not designed to describe direction information inside the minimum bounding rectangle mbr of a reference region thus the direction relations between overlapping and contained regions cannot be effectively described and derived to resolve this problem a new model of direction relations namely interior boundary direction relations is proposed in this study to describe direction concepts relative to the interior or boundary of a region such as east part of a region west border of a region a line goes through east part of a region and etc by combining the interior and exterior direction relations three types of compositions of direction relations are investigated 1 composing two interior direction relations which can be used to derive the interior or exterior direction relations between two regions with the same parent region 2 composing an interior direction relation with an exterior direction relation and 3 composing an interior with an exterior direction relation the results indicate that the new interior boundary direction relations and its compositions with exterior direction relations are powerful in describing and deriving direction relations between overlapped and contained regions

the lyapunov exponents for schrodinger operators with slowly oscillating potentials abstract by studying the integrated density of states we prove the existence of lyapunov exponents and the thouless formula for the schrodinger operator d 2 dx 2 cos x with 0 l 2 0 this yields an explicit formula for these lyapunov exponents by applying rank one perturbation theory we also obtain some spectral consequences

iconic indexing by 2 d strings in this paper we describe a new way of representing a symbolic picture by a two dimensional string a picture query can also be specified as a 2 d string the problem of pictorial information retrieval then becomes a problem of 2 d subsequence matching we present algorithms for encoding a symbolic picture into its 2 d string representation reconstructing a picture from its 2 d string representation and matching a 2 d string with another 2 d string we also prove the necessary and sufficient conditions to characterize ambiguous pictures for reduced 2 d strings as well as normal 2 d strings this approach thus allows an efficient and natural way to construct iconic indexes for pictures

autonomous self similar ordinary differential equations and the painleve connection we demonstrate an intimate connection between nonlinear higher order ordinary differential equations possessing the two symmetries of autonomy and self similarity and the leading order behaviour and resonances determined in the application of the painleve test similar behaviour is seen for systems of first order differential equations several examples illustrate the theory in an integrable case of the abc system the singularity analysis reveals a positive and a negative resonance and the method of leading order behaviour leads naturally to a laurent expansion containing both

on subordinacy and analysis of the spectrum of schrodinger operators with two singular endpoints the theory of subordinacy is extended to all one dimensional schrodinger operatorsfor which the corresponding differential expression l d 2 dr 2 v r is in the limit point case at both ends of an interval a b with v r locally integrable this enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions of lu xu x e as r a and r b the result provides a rigorous but straightforward method of direct spectral analysis which has very general application and somefurther properties of the spectrum are deduced from the underlying theory

polynomial first integrals via the poincare series in appl math 28 2001 17 a method to compute the poincare liapunov constants for an arbitrary analytic differential system which has a linear center at the origin in function of the coefficients of the system was given this method also computes the coefficients of the poincare series in function of the same coefficients in this work we use this method to determine polynomial differential systems which have a polynomial first integral

singular continuous spectrum is generic in a variety of contexts we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators those with singular spectrum are a dense g delta

invariant algebraic surfaces of the lorenz system in this paper we provide a complete classification of the darboux invariants of the irreducible darboux polynomials of the rational first integrals and of the algebraic integrability for the classical lorenz system s y x rx y xz z bz xy in the proofs we use the weight homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations

darbouxian first integrals and invariants for real quadratic systems having an invariant conic we apply the darboux theory to study the integrability of real quadratic differential systems having an invariant conic the fact that two intersecting straight lines or two parallel straight lines are particular cases of conics allows us to study simultaneously the integrability of quadratic systems having at least two invariant straight lines real or complex

mining spatial association rules in image databases in this paper we propose a novel spatial mining algorithm called 9dlt miner to mine the spatial association rules from an image database where every image is represented by the 9dlt representation the proposed method consists of two phases first we find all frequent patterns of length one next we use frequent k patterns k 1 to generate all candidate k 1 patterns for each candidate pattern generated we scan the database to count the pattern s support and check if it is frequent the steps in the second phase are repeated until no more frequent patterns can be found since our proposed algorithm prunes most of impossible candidates it is more efficient than the apriori algorithm the experiment results show that 9dlt miner runs 2 5 times faster than the apriori algorithm

singular continuous spectrum under rank one perturbations and localization for random hamiltonians we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic for a in terms of the spectral measure d a we give necessary and sufficient conditions for a p to have empty singular continuous spectrum or to have only point spectrum for a e we apply these results to questions of localization in the one and multi dimensional anderson models

pure point spectrum under 1 parameter perturbations and instability of anderson localization we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic fora we study the set of all eigenvalues of the operatora t a tp t that belong to its essential spectrum which does not depend on the parametert we prove that this set is empty for a dense set of values oft then we apply this result or its idea to questions of anderson localization for 1 dimensional schrodinger operators discrete and continuous

polynomial first integrals for weight homogeneous planar polynomial differential systems of weight degree 3 our main result is the classification of all weight homogeneous planar polynomial differential systems of weight degree 3 having a polynomial first integral

special nonuniform lattice snul orthogonal polynomials on discrete dense sets of points difference calculus compatible with polynomials i e such that the divided difference operator of first order applied to any polynomial must yield a polynomial of lower degree can only be made on special lattices well known in contemporary q calculus orthogonal polynomials satisfying difference relations on such lattices are presented in particular lattices which are dense on intervals q 1 are considered

on subordinacy and analysis of the spectrum of one dimensional schrodinger operators abstract by considering the behaviour as n of the ratio of l2 0 n norms of solutions of d 2 dr 2 v r u xu 0 r x r a characterisation of the absolutely continuous and singular spectra of one dimensional schrodinger operators is deduced the analysis is applicable to all operators for which l d 2 dr 2 v r is regular at 0 and in the limit point case at infinity with v r locally integrable

operators with singular continuous spectrum i general operators the baire category theorem implies that the family f of dense sets g in fixed metric space x is a candidate for generic sets since it is closed under countable intersections and if x is perfect has no isolated point then a f has uncountable intersections with any open ball in x

an association based case reduction technique for case based reasoning case based reasoning cbr is a type of problem solving technique which uses previous cases to solve new unseen and different problems although a larger number of cases in the memory can improve the coverage of the problem space the retrieval efficiency will be downgraded if the size of the case base grows to an unacceptable level in cbr systems the tradeoff between the number of cases stored in the case base and the retrieval efficiency is a critical issue this paper addresses the problem of case base maintenance by developing a new technique the association based case reduction technique acrt to reduce the size of the case base in order to enhance the efficiency while maintaining or even improving the accuracy of the cbr the experiments on 12 uci datasets and an actual case from taiwan s hospital have shown superior generalization accuracy for cbr with acrt cbr acrt as well as a greater solving efficiency

on the polynomial first integrals of the ital a ital b ital c lotka endash volterra system using elementary differential algebraic techniques we prove that the 3d lotka endash volterra dynamical system has no other nontrivial polynomial first integrals than the previously known ones copyright ital 1996 american institute of physics

operators with singular continuous spectrum ii rank one operators for an operator a with cyclic vector we studya p wherep is the rank one projection onto multiples of if spec a anda has no a c spectrum we prove thata p has purely singular continuous spectrum on for a denseg of s

exponential factors and darbouxian first integrals of the lorenz system in this article we characterize all the exponential factors and the darbouxian first integrals for the lorenz system s y x rx y xz z bz xy in the proofs we use the weight homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations

on the absolutely continuous spectrum of the perturbed hill s equation on considere l equation de hill y q x y y avec x et complexe ou q x est reel continu par morceaux et periodique de periode 1 et l equation augmentee perturbee par une fonction l 1 y q x p x y y ou p x est reel et absolument integrable le long de l axe reel

modified prufer and efgp transforms and deterministic models with dense point spectrum we provide a new proof of the theorem of simon and zhu that in the region e for a e energies d 2 dx 2 cos x 0 1 has lyapunov behavior with a quasi classical formula for the lyapunov exponent we also prove lyapunov behavior for a e e 2 2 for the discrete model with v j 2 e j v n 0 if n 1 4 9 the arguments depend on a direct analysis of the equations for the norm of a solution

boundary conditions of sturm liouville operators with mixed spectra we study bounds on averages of spectral functions corresponding to sturm liouville operators on the half line for different boundary conditions as a consequence constraints are obtained which imply existence of singular spectrum embedded in a c spectrum for sets of boundary conditions with positive measure and potentials vanishing in an interval 0 n these constraints are related to estimates on the measure of sets where the spectral density is positive

looking into the seeds of time discovering temporal patterns in large transaction sets this paper studies the problem of mining frequent itemsets along with their temporal patterns from large transaction sets a model is proposed in which users define a large set of temporal patterns that are interesting or meaningful to them a temporal pattern defines the set of time points where the user expects a discovered itemset to be frequent the model is general in that i no constraints are placed on the interesting patterns given by the users and ii two measures inclusiveness and exclusiveness are used to capture how well the temporal patterns match the time points given by the discovered itemsets intuitively these measures indicate to what extent a discovered itemset is frequent at time points included in a temporal pattern p but not at time points not in p using these two measures one is able to model many temporal data mining problems appeared in the literature as well as those that have not been studied by exploiting the relationship within and between itemset space and pattern space simultaneously a series of pruning techniques are developed to speed up the mining process experiments show that these pruning techniques allow one to obtain performance benefits up to 100 times over a direct extension of non temporal data mining algorithms

trace class perturbations and the absence of absolutely continuous spectra we show that various hamiltonians and jacobi matrices have no absolutely continuous spectrum by showing that under a trace class perturbation they become a direct sum of finite matrices

critical coupling constants and eigenvalue asymptotics of perturbed periodic sturm liouville operators perturbations of asymptotic decay c r 2 arise in the partial wave analysis of rotationally symmetric partial differential operators we show that for each end point 0 of the spectral bands of a perturbed periodic sturm liouville operator there is a critical coupling constant c crit such that eigenvalues in the spectral gap accumulate at 0 if and only if c c crit 1 the oscillation theoretic method used in the proof also yields the asymptotic distribution of the eigenvalues near 0

bounded solutions and absolute continuity of sturm liouville operators this work is a contribution to the theory of the absolutely continuous spectrum of one dimensional schriidinger operators i e sturm liouville operators generated by differential expressions of the form ru u qu in proofs of absolute continuity for such operators and also for the closely related dirac systems always asymptotic properties of solutions of zu lu for real or complex values of 1 have played an important role see for example 18 19 8 4 9 lo in these works and a number of others particular classes of potentials q were treated using properties of solutions in the course of the proof of absolute continuity somewhat more recently the connection between asymptotic properties of solutions and absolute continuity was given a theory i e results of the following form were proved let the solutions of zu lu satisfy some specific asymptotic property for a varying in some subset of r no or almost no a priori assumptions on q then operators generated by z are absolutely continuous in this set a first result of this type was given by carmona 3 who proved that local uniform boundedness of solutions together with their derivatives in some open a set implies pure absolute continuity in this set here we will mostly rely on the concept of non or racy of solutions which was shown by gilbert and pearson 7 12 61 to be a powerful tool in the investigation of the continuous spectrum of sturm liouville operators in section 2 we give some consequences of their results whi ch are useful in proofs of absolute continuity these consequences range from non existence of imbedded eigenvalues and the existence of absolutely continuous parts to pure absolute continuity and the existence of transient 210 0022 247x 92 5 00

operators with singular continuous spectrum iii almost periodic schrodinger operators we prove that one dimensional schrodinger operators with even almost periodic potential have no point spectrum for a dense g in the hull this implies purely singular continuous spectrum for the almost mathieu equation for coupling larger than 2 and a dense g in even if the frequency is an irrational with good diophantine properties

one dimensional perturbations of unbounded selfadjoint operators with empty spectrum abstract we study spectral properties of one dimensional singular nonselfadjoint perturbations of an unbounded selfadjoint operator and give criteria for the possibility to remove the whole spectrum by a perturbation of this type a counterpart of our results for the case of bounded operators provides a complete description of compact selfadjoint operators whose rank one perturbation is a volterra operator

wkb asymptotic behavior of almost all generalized eigenfunctions for one dimensional schrodinger operators with slowly decaying potentials we prove the wkb asymptotic behavior of solutions of the differential equation d2u dx2 v x u eu for a e e a where v v1 v2 v1 lp r and v2 is bounded from above with a lim supx v x while v 2 x lp r 1 p 2 these results imply that schrodinger operators with such potentials have absolutely continuous spectrum on a we also establish wkb asymptotic behavior of solutions for some energy dependent potentials

deriving two stage learning sequences from knowledge in fuzzy sequential pattern mining a fuzzy sequential pattern consisting of several fuzzy sets represents a frequently occurring behavior related to time and can be discovered from transaction bases an example is that large purchase amounts of one product were bought by customers after these consumers had bought small purchase amounts of another product recently hu et al 2003 proposed a fuzzy data mining method to discover fuzzy sequential patterns in this method consumers products preferences and consumers product buying orders related to purchase behaviors can be found in the fuzzy sequential pattern mining since for each decision problem there is a competence set consisting of ideas knowledge information and skills for solving that problem we consider knowledge found in fuzzy sequential pattern mining as a needed competence set for solving one decision problem this paper uses a known competence set expansion method the minimum spanning table method to find appropriate two stage learning sequences that can effectively acquire individual fuzzy knowledge sets found in the fuzzy sequential pattern mining a numerical example is used to show the usefulness of the proposed method

stability of singular spectral types under decaying perturbations we look at invariance of a e boundary condition spectral behavior under perturbations w of half line continuum or discrete schrodinger operators we extend the results of del rio simon stolz from compactly supported w s to suitable short range w we also discuss invariance of the local hausdorff dimension of spectral measures under such perturbations

an order clique based approach for mining maximal co locations most algorithms for mining spatial co locations adopt an apriori like approach to generate size k prevalence co locations after size k 1 prevalence co locations however generating and storing the co locations and table instances is costly a novel order clique based approach for mining maximal co locations is proposed in this paper the efficiency of the approach is achieved by two techniques 1 the spatial neighbor relationships and the size 2 prevalence co locations are compressed into extended prefix tree structures which allows the order clique based approach to mine candidate maximal co locations and co location instances and 2 the co location instances do not need to be stored after computing some characteristics of the corresponding co location which significantly reduces the execution time and space required for mining maximal co locations the performance study shows that the new method is efficient for mining both long and short co location patterns and is faster than some other methods in particular the join based method and the join less method

spectral analysis on infinite sierpinski gaskets we study the spectral properties of the laplacian on infinite sierpin ski gaskets we prove that the laplacian with the neumann boundary condition has pure point spectrum moreover the set of eigenfunctions with compact support is complete the same is true if the infinite sierpin ski gasket has no boundary but is false for the laplacian with the dirichlet boundary condition in all these cases we describe the spectrum of the laplacian and all the eigenfunctions with compact support to obtain these results first we prove certain new formulas for eigenprojectors of the laplacian on finite sierpin ski pre gaskets then we prove that the spectrum of the discrete laplacian on a sierpin ski lattice is pure point and the eigenfunctions are localized 1998 academic press

modified prufer and efgp transforms and the spectral analysis of one dimensional schrodinger operators using control of the growth of the transfer matrices wediscuss the spectral analysis of continuum and discrete half line schrodinger operators with slowly decaying potentials among our results we show if where w has compact support and then h has purely a c resp purely s c spectrum on 0 if for n 1 2 a n potentials where a n are independent identically distributed random variables with e a n 0 e a 2 n 1 and 2 we find singular continuous spectrum with explicitly computable fractional hausdorff dimension

probabilistic averages of jacobi operators i study the lyapunov exponent and the integrated density of states for general jacobi operators the main result is that questions about these can be reduced to questions about ergodic jacobi operators i use this to show that for finite gap jacobi operators regularity implies that they are in the cesaro nevai class proving a conjecture of barry simon furthermore i use this to study jacobi operators with coefficients a n 1 and b n f n mod 1 for 0 not an integer

absolutely continuous spectrum of one dimensional schrodinger operators and jacobi matrices with slowly decreasing potentials we prove that for any one dimensional schrodinger operator with potentialv x satisfying decay condition v x cx 3 4 e the absolutely continuous spectrum fills the whole positive semi axis the description of the set in on which the singular part of the spectral measure might be supported is also given analogous results hold for jacobi matrices

one dimensional schrodinger operators with random or deterministic potentials new spectral types abstract we prove sufficient conditions involving only potential asymptotic near one of the infinities in order to have purely absolutely continuous components in the spectrum these deterministic results are then applied to random cases and exhibit classes of models for which with probability one one component of the spectrum is purely absolutely continuous and the rest is dense pure point with exponentially decaying eigenfunctions

darboux integrability for 3d lotka volterra systems we describe the improved darboux theory of integrability for polynomial ordinary differential equations in three dimensions using this theory and computer algebra we study the existence of first integrals for the three dimensional lotka volterra systems only working up to degree two with the invariant algebraic surfaces and the exponential factors we find the major part of the known first integrals for such systems and in addition we find three new classes of integrability the method used is of general interest and can be applied to any polynomial ordinary differential equations in arbitrary dimension

instability of dense point spectrum under finite rank perturbations we prove that if a is a self adjoint operator and r a non negative finite rank operator whose range is cyclic for a then for a generic in the baire sense t in each eigenvalue of the operator has a neighborhood containing no other eigenvalues

continuous spectrum and uniform localization for ergodic schrodinger operators abstract in this paper we study localization for ergodic families of discrete schrodinger operators we prove that instability of pure point spectrum implies absence of uniform localization

transient and recurrent spectrum abstract we deal primarily with spectral analysis of an abstract self adjoint operator h on a hilbert space h we propose a further refinement of the absolutely continuous subspace h ac into the transient absolutely continuous subspace h tac which is the closure of those with e ith o t n for all n and the recurrent absolutely continuous subspace h rac h ac h tac we discuss general features of this breakup in a subsequent paper we construct analytic almost periodic functions v on so that h d 2 dx 2 v x on l 2 has only recurrent absolutely continuous spectrum in the sense that h rac h

on the polynomial first integrals of the a b c lotka volterra system using elementary differential algebraic techniques we prove that the 3 d lotka volterra dynamical system has no other nontrivial polynomial first integrals than the previously known ones

operators with singular continuous spectrum vii examples with borderline time decay we construct one dimensional potentials v x so that if h d 2 dx 2 v x on l 2 then h has purely singular spectrum but for a dense set d d implies that ith c t 1 2 in t for t 2 this implies the spectral measures have hausdorff dimension one and also following an idea of malozemov molchanov provides counterexamples to the direct extension of the theorem of simon spencer on one dimensional infinity high barriers

singular continuous spectrum for the period doubling hamiltonian on a set of full measure we consider the discrete one dimensional schrodinger operator with potential generated by the period doubling substitution we show that for almost every element in the hull with respect to the unique ergodic measure on there are no eigenvalues combining this with a result proven by kotani we establish purely singular continuous spectrum on a set of full measure

some schrodinger operators with power decaying potentials and pure point spectrum we construct deterministic potentials such that the schrodinger equation on has dense pure point spectrum in for almost all boundary conditions at as a by product we also obtain power decaying potentials for which the spectrum is purely singular continuous on for all boundary conditions