ince s limits for confluent and double confluent heun equations we find pairs of solutions to a differential equation which is obtained as a special limit of a generalized spheroidal wave equation this is also known as confluent heun equation one solution in each pair is given by a series of hypergeometric functions and converges for any finite value of the independent variable z while the other is given by a series of modified bessel functions and converges for z z0 where z0 denotes a regular singularity for short the preceding limit is called ince s limit after ince who have used the same procedure to get the mathieu equations from the whittaker hill ones we find as well that when z0 tends to zero the ince limit of the generalized spheroidal wave equation turns out to be the ince limit of a double confluent heun equation for which solutions are provided finally we show that the schrodinger equation for inverse fourth and sixth power potentials reduces to peculiar cases of the double confluent heun equation and its ince s limit respectively

convergence and applications of some solutions of the confluent heun equation we study the convergence of a group of solutions in series of confluent hypergeometric functions for the confluent heun equation these solutions are expansions in two sided infinite series summation from minus to plus infinity which are interpreted as a modified version of expansions proposed by leaver 1986 we show that the two sided solutions yield two nonequivalent groups of one sided series solutions summation from zero to plus infinity in the second place we find that one sided solutions of one of these groups can be used to solve an equation which describes a time dependent two level system of quantum optics for this problem in addition to finite series solutions we obtain infinite series wavefunctions which are convergent and bounded for any value of the time t and vanish when t goes to infinity

comment on dirac equation in the background of the nutku helicoid metric j math phys 48 092301 2007 the dirac equation written on the boundary of the nutku helicoid space consists of a system of ordinary differential equations we tried to analyze this system and we found that it has a higher singularity than those of the heun equations which give the solutions of the dirac equation in the bulk we also lose an independent integral of motion on the boundary this facts explain why we could not find the solution of the system on the boundary in terms of known functions we make the stability analysis of the helicoid and catenoid cases and end up with an appendix which gives a new example wherein one encounters a form of the heun equation

a quantum computing primer for operator theorists this is an exposition of some of the aspects of quantum computation and quantum information that have connections with operator theory after a brief introduction we discuss quantum algorithms we outline basic properties of quantum channels or equivalently completely positive trace preserving maps the main theorems for quantum error detection and correction are presented and we conclude with a description of a particular passive method of quantum error correction

sequential quantum measurements a quantum effect is an operator a on a complex hilbert space h that satisfies 0 a i we denote the set of quantum effects by e h the set of self adjoint projection operators on h corresponds to sharp effects and is denoted by p h we define the sequential product of a b e h by a b a1 2ba1 2 the main purpose of this article is to study some of the algebraic properties of the sequential product many of our results show that algebraic conditions on a b imply that a and b commute for the usual operator product for example if a b satisfies certain distributive or associative laws then ab ba moreover if a b p h then ab ba and a b b or b a b if and only if ab ba b a natural definition of stochastic independence is introduced and briefly studied

fixed points of commutative luders operations this paper verifies a conjecture posed in a pair of papers on the fixed point sets for a class of quantum operations specifically it is proved that if a quantum operation has mutually commuting operation elements that are effects forming a resolution of the identity then the fixed point set of the quantum operation is exactly the commutant of the operation elements

integral relations for solutions of the confluent heun equation firstly we construct kernels for integral relations among solutions of the confluent heun equation che additional kernels are systematically generated by applying substitutions of variables secondly we establish integral relations between known solutions of the che that are power series and solutions that are series of special functions thirdly by using one of the integral relations as an integral transformation we obtain a new series solution of the ordinary spheroidal wave equation a particular che from this solution we construct new series solutions of the general che and show that these are suitable for solving the radial part of the two center problem in quantum mechanics

generalized confluent hypergeometric solutions of the heun confluent equation abstract we show that the heun confluent equation admits infinitely many solutions in terms of the confluent generalized hypergeometric functions for each of these solutions a characteristic exponent of a regular singularity of the heun confluent equation is a non zero integer and the accessory parameter obeys a polynomial equation each of the solutions can be written as a linear combination with constant coefficients of a finite number of either the kummer confluent hypergeometric functions or the bessel functions

on fixed points of luders operation in this paper we give a concrete example of a luders operation la with n 3 such that la b b does not imply that b commutes with all e1 e2 and e3 in a this example answers an open problem of professor gudder

fixed points of quantum operations quantum operations frequently occur in quantum measurement theory quantum probability quantum computation and quantum information theory if an operator a is invariant under a quantum operation we call a a fixed point physically the fixed points are the operators that are not disturbed by the action of our main purpose is to answer the following question if a is a fixed point is a compatible with the operation elements of we shall show in general that the answer is no and we shall give some sufficient conditions under which the answer is yes our results will follow from some general theorems concerning completely positive maps and injectivity of operator systems and von neumann algebras

analytical results for a modified trigonometric razavy potential we have presented analytical results for the modified trigonometric razavy potential with an additional relative phase between their components it is shown that the analytical solutions can be given in terms of the heun double confluent functions based on the properties of the heun double confluent functions the transfer matrix has been constructed to compute energy bands

on spectra of luders operations we show that all the eigenvalues of certain generalized luders operations are non negative real numbers in two cases of interest in particular given a commuting n tuple a a1 an consisting of positive operators on a hilbert space h satisfying j 1naj i we show that the spectrum of the luders operation a b h x j 1naj1 2xaj1 2 b h is contained in 0 so the only solution of the equation a x i x is the expected one x 12i

non disturbance for fuzzy quantum measurements we consider three criteria for describing non disturbance between quantum measurements while previous discussions of these criteria considered sharp measurements we shall treat more general measurements that may be unsharp or fuzzy it has been shown that in the sharp case the three criteria are equivalent to compatibility of the measurements we shall show that only the third criterion is equivalent to compatibility for the general fuzzy case moreover the first two criteria are not symmetric relative to the two measurements

examples of heun and mathieu functions as solutions of wave equations in curved spaces we give examples of where the heun function exists as solutions of wave equations encountered in general relativity as a new example we find that while the dirac equation written in the background of nutku helicoid metric yields mathieu functions as its solutions in four spacetime dimensions the trivial generalization to five dimensions results in the double confluent heun function we reduce this solution to the mathieu function with some transformations

fixed points of dual quantum operations abstract let a be a normal completely positive map on b h with kraus operators a i i 1 denote m the subset of normal completely positive maps by m a i 1 a i a i i i 1 a i a i i and a is normal in this note the relations between the fixed points of a and a are investigated we obtain that b k h a b b b k h a b b where k h is the set of all compact operators on h and a is the dual of a m in addition we show that the map a a is a bijection on m

stable adaptive fuzzy control of nonlinear systems a direct adaptive fuzzy controller that does not require an accurate mathematical model of the system under control is capable of incorporating fuzzy if then control rules directly into the controllers and guarantees the global stability of the resulting closed loop system in the sense that all signals involved are uniformly bounded is developed the specific formula for the bounds is provided so that controller designers can determine the bounds based on their requirements the direct adaptive fuzzy controller is used to regulate an unstable system to the origin and to control the duffing chaotic system to track a trajectory the simulation results show that the controller worked without using any fuzzy control rules and that after fuzzy control rules were incorporated the adaptation speed became much faster it is shown explicitly how the supervisory control forces the state to remain within the constraint set and how the adaptive fuzzy controller learns to regain control

text ad _ f g text ad _ f g and locally text ad _ f g invariant and controllability distributions in the study of nonlinear control systems the concepts of an invariant foliation and an invariant distribution play important roles in this paper we explore various forms of these concepts and show how they occur in the study of controllability observability and decoupling of nonlinear systems

transformations of heun s equation and its integral relations using the transformation theory for the heun equation we find substitutions of variables which preserve the form of the equation for the kernels of integral relations among solutions of the heun equation these transformations lead to new kernels for the heun equation given by single hypergeometric functions lambe ward type kernels and by products of two hypergeometric functions erd lyi type such kernels by a limiting process also afford new kernels for the confluent heun equation

solutions to a spheroidal wave equation fackerell crossman s solutions to a generalized spheroidal wave equation are extended for the case in which there is no free parameter in that equation the results together with leaver s expansions in series of coulomb wave functions are used to find the time dependence of massive scalar test fields in nonflat friedmann robertson walker models of universes with dust

fixed points of normal completely positive maps on b h abstract given a sequence of bounded operators a j on a hilbert space h with j 1 a j a j 1 j 1 a j a j we study the map defined on b h by x j 1 a j x a j and its restriction to the hilbert schmidt class c 2 h in the case when the sum j 1 a j a j is norm convergent we show in particular that the operator 1 is not invertible if and only if the c algebra a generated by a j j 1 has an amenable trace this is used to show that may have fixed points in b h which are not in the commutant a of a even in the case when the weak closure of a is injective however if a is abelian then all fixed points of are in a even if the operators a j are not positive

characterizations of fixed points of quantum operations let a be a general quantum operation an operator b is said to be a fixed point of a if a b b in this note we shall show conditions under which b a fixed point a implies that b is compatible with the operation element of a in particular we offer an extension of the generalized luders theorem

generalized spheroidal wave equation and limiting cases we find sets of solutions to the generalized spheroidal wave equation gswe or equivalently to the confluent heun equation each set is constituted by three solutions one given by a series of ascending powers of the independent variable and the others by series of regular and irregular confluent hypergeometric functions for a fixed set the solutions converge over different regions of the complex plane but present series coefficients proportional to each other these solutions for the gswe afford solutions to a double confluent heun equation by a taking limit process due to leaver e w leaver j math phys 27 1238 1986 another procedure called whittaker ince limit b d figueiredo j math phys 46 113503 2005 provides solutions in series of powers and bessel functions for two other equations with a different type of singularity at infinity in addition new solutions are obtained for the whittaker hill and mathieu equations f m arscott proc r soc edinburg a67 265 1967 b

confluent heun equations convergence of solutions in series of coulomb wavefunctions the leaver solutions in series of coulomb wavefunctions for the confluent heun equation are given by two sided infinite series that is by series where the summation index n runs from minus to plus infinity leaver 1986 j math phys 27 1238 first we show that in contrast to the d alembert test under certain conditions the raabe test ensures that the domains of convergence of these solutions include an additional singular point we also consider solutions for a limit of the confluent heun equation for both equations new solutions are generated by transformations of variables finally we discuss the time dependence of the klein gordon equation in two cosmological models and the spatial dependence of the schr dinger equation to a family of quasi exactly solvable potentials for a subfamily of these potentials we obtain infinite series solutions which converge and are bounded for all values of the independent variable in opposition to a common belief

normality and fixed points associated to commutative row contractions abstract let a a k k 1 n n is a positive integer or be a commutative row contraction on a complex hilbert space h and a the normal completely positive map associated with a we give some characterizations for a to be a normal sequence in the case that a is unital we show a is normal if either a is contained in a finite von neumann algebra or the set k h of all compact operators or k 1 n a k a k i moreover the fixed point set b h a of a is considered when a j i is convergent to a projection in strong operator topology

the multiplicative domain in quantum error correction we show that the multiplicative domain of a completely positive map yields a new class of quantum error correcting codes in the case of a unital quantum channel these are precisely the codes that do not require a measurement as part of the recovery process the so called unitarily correctable codes in the arbitrary not necessarily unital case they form a proper subset of unitarily correctable codes that can be computed from the properties of the channel as part of the analysis we derive a representation theoretic characterization of subsystem codes we also present a number of illustrative examples

a note on operator probability theory involving numerical ranges abstract in this note the relationships between the expectation and variance in operator probability theory and numerical range of operators are considered

a nodal link perceptron network with applications to control of a nonholonomic system a new perceptron neural network pnn for functional approximation and control of a general class of nonlinear systems is introduced the basic structure of the network along with the conditions for its exponential convergence under a suitable training law are derived a novel discrete time control strategy is formulated that employs the pnn for direct online estimation of the feedforward control input the developed controller can be applied to both discrete and continuous time plants unlike most of the existing direct adaptive or learning schemes the nonlinear plant is not assumed to be feedback linearizable the developed controller is then applied for tracking control of a nonholonomic free flying robot the simulation results of this application demonstrate a perfect tracking performance after the network is fully trained

fixed points for luders operations and commutators let b h be the algebra of all bounded linear operators on a complex hilbert space h and let tj j 1 b h be a sequence of operators such that j 1tj tj 1h in the weak operator topology let us define t b h b h by setting t y j 1t jytj for y b h we show that if x b h is an operator such that all the commutators xtj tjx as well as x itself are fixed points for t then xtj tjx 0 for all j 1 in the case when tj j 1 consists of commuting normal operators this shows that every fixed point for t commutes with all tj for j 1

gaussian networks for direct adaptive control a direct adaptive tracking control architecture is proposed and evaluated for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture uses a network of gaussian radial basis functions to adaptively compensate for the plant nonlinearities under mild assumptions about the degree of smoothness exhibit by the nonlinear functions the algorithm is proven to be globally stable with tracking errors converging to a neighborhood of zero a constructive procedure is detailed which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy a stable weight adjustment mechanism is determined using lyapunov theory the network construction and performance of the resulting controller are illustrated through simulations with example systems

on some solutions to generalized spheroidal wave equations and applications expansions in series of coulomb and hypergeometric functions for the solutions of the generalized spheroidal wave equations gswes are analysed and written together in pairs each pair consists of a solution in series of hypergeometric functions and another in series of coulomb wavefunctions and has the same recurrence relations for the series coefficients but the solutions inside it present different radii of convergence expansions without a phase parameter are derived by truncating the series with a phase parameter for the whittaker hill equation solutions are found by treating that equation as a particular case of gswe while for the confluent gswe solutions with and without a phase parameter are given as pairs of series of coulomb wavefunctions amongst the applications there are equations for the time dependence of dirac test fields in some nonflat friedmann robertson walker spacetimes the radial schrodinger equation for an electron in the field of two coulombian centres and the schrodinger equation for the razavy type quasi exactly solvable potentials for these problems it is possible to find wavefunctions in terms of infinite series regular and convergent over the entire range of the independent variable by matching expansions belonging to one or more of the above pairs the infinite series solutions for the razavy type potentials in addition to the polynomial ones suggest that the whole energy spectra may be determined without appealing to perturbation theory or semi classical methods of approximation

a limit of the confluent heun equation and the schrodinger equation for an inverted potential and for an electric dipole we re examine and extend a group of solutions in series of bessel functions for a limiting case of the confluent heun equation and then apply such solutions to the one dimensional schrodinger equation with an inverted quasiexactly solvable potential as well as to the angular equation for an electron in the field of a point electric dipole for the first problem we find finite and infinite series solutions which are convergent and bounded for any value of the independent variable for the angular equation we also find expansions in series of jacobi polynomials

solutions for confluent and double confluent heun equations this paper examines some solutions for confluent and double confluent heun equations in the first place we review two leaver s solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation leaver e w j math phys 27 1238 1986 and introduce an additional expansion in series of irregular confluent hypergeometric functions then we find the conditions under which one of these solutions can be written as a linear combination of the others in the second place by means of limiting procedures we generate solutions for the double confluent equation as well as for special limits of both the confluent and double confluent equations finally we present some problems which are ruled by one or the other of these four equations

toeplitz operators associated to commuting row contractions let h be a complex hilbert space and let tn n 1 be a sequence of commuting bounded operators on h such that n 1tntn ih let f t denote the space of all operators x in b h for which n 1tnxtn x and suppose that f t 0 we will show that there exists a triple k un n 1 where k is a hilbert space k h is a bounded operator and un n 1 b k is a sequence of commuting normal operators with n 1unun ik such that tn un for n 1 and for which the mapping y y is a complete isometry from the commutant of un n 1 onto the space f t moreover we show that the inverse of this mapping can be extended to a homomorphism c ih f t un n 1 from the unital c algebra generated by f t onto the commutant of un n 1 we also show that there exists a homomorphism c ih tn n 1 c ik un n 1 such that tn un for n 1 in the particular case when tn n 1 has only a finite number of non zero components it turns out that un n 1 is unitarily equivalent to the spherical unitary part of the standard commuting dilation of tn n 1

a noether theorem for stochastic operators on schatten classes abstract we show that a stochastic markov operator s acting on a schatten class c 1 satisfies the noether condition i e s a a and s a 2 a 2 where a c is a hermitian and bounded operator on a fixed separable and complex hilbert space h if and only if s e a g x e a g e a g s x e a g for any state x c 1 and all borel sets g r where e a g denotes the orthogonal projection coming from the spectral resolution a a z e a d z similar results are obtained for stochastic one parameter continuous semigroups

fixed points of commutative super operators in this paper we study fixed points of commutative super operators first under some circumstances we discuss the question of necessary and sufficient conditions for the set of fixed points of a super operator to be non trivial then we establish a structure lemma to completely characterize this set using this lemma we show that if a commutative super operator is also a generalized luders operator then its fixed points are precisely those operators intertwining the two sequences of operation elements associated with it specifically the main results of weihua and junde 2010 j phys a math theor 43 395206 are special cases of our results

asymptotic properties of quantum markov chains the asymptotic dynamics of discrete quantum markov chains generated by the most general physically relevant quantum operations is investigated it is shown that it is confined to an attractor space in which the resulting quantum markov chain is diagonalizable a construction procedure of a basis of this attractor space and its associated dual basis of 1 forms is presented it is applicable whenever a strictly positive quantum state exists which is contracted or left invariant by the generating quantum operation moreover algebraic relations between the attractor space and kraus operators involved in the definition of a quantum markov chain are derived this construction is not only expected to offer significant computational advantages in cases in which the dimension of the hilbert space is large and the dimension of the attractor space is small but it also sheds new light onto the relation between the asymptotic dynamics of discrete quantum markov chains and fixed points of their generating quantum operations finally we show that without any restriction our construction applies to all initial states whose support belongs to the so called recurrent subspace

quantum channels arising from abstract harmonic analysis we present a new application of harmonic analysis to quantum information by constructing intriguing classes of quantum channels stemming from specific representations of multiplier algebras over locally compact groups g beginning with a representation of the measure algebra m g we unify and elaborate on recent counter examples to a conjecture on the structure of fixed point subalgebras in infinite dimensions as well as present an application to the noiseless subsystems method of quantum error correction using a representation of the completely bounded fourier multiplier algebra mcba g we provide a new class of counter examples to the recently solved asymptotic quantum birkhoff conjecture along with a systematic method of producing the examples using a geometric representation of schur maps further properties of our channels including duality quantum capacity and entanglement preservation are discussed along with potential applications to additivity conjectures