exact solutions to the schrodinger equation for the potential in two dimensions making use of an ansatz for the eigenfunctions we obtain an exact closed form solution to the non relativistic schrodinger equation with the anharmonic potential in two dimensions where the parameters of the potential a b and c satisfy some constraints

large order perturbation expansions for the charged harmonic oscillator the charged oscillator defined by the hamiltonian h d2 dr2 r2 lambda r in the domain 0 infinity is a particular case of the family of spike oscillators which does not behave as a supersingular hamiltonian this problem is analysed around the three regions lambda to infinity lambda to 0 and lambda to infinity by using rayleigh ritz large order perturbative expansions a path is found to connect the large lambda regions with the small lambda region by means of the renormalization of the series expansions in lambda finally the riccati pade method is used to construct an implicit expansion around lambda to 0 which extends to very large values of mod lambda mod

perturbed factorization of the singular anharmonic oscillator eigenequation the perturbed ladder operator method is applied to the solution of the perturbed eigenequation d2 dx2 m m 1 x2 b2x2 v x ix 0 where v x b1 1 x 2 b2 1 x 4 is a singular perturbation this method which is the extension of the schrodinger infeld hull factorization method within the perturbation scheme provides closed form expressions of the perturbed eigenvalues and ladder functions by means of algebraic manipulations as an illustrative application an analytical solution of the spiked harmonic oscillator eigenequation d2 dx2 b2x2 x4 e x 0 is worked out up to the second order of the perturbation by considering specifically adapted m and dependent perturbing and unperturbed potentials in order to tentatively avoid the known difficulties of convergence of the perturbation series closed form expressions of the x4 anharmonic oscillator energies are obtained in terms of the coupling constant and the quantum number v results following from these expressions are compared wit

universality of 2 qudit ternary reversible gates in this paper we investigate the synthesis of ternary reversible circuits in the absence of ancilla bits we demonstrated that 2 qudit ternary swap not and 1 controlled not gates are universal for realization of arbitrary ternary n qudit reversible circuits without ancilla qudits and all even ternary n qudit reversible circuits can be constructed by ternary not and ternary 1 controlled not gates without ancilla qudits the realization approach is constructive this result is significantly different from the binary case

nonperturbative treatment of singular potentials a nonperturbative but absolutely convergent algorithm is applied to the determination of the eigenfunctions of hamiltonian with a singular potential of the form h d2 dx2 x2 x in the domain 0 which obey dirichlet boundary conditons the formal structure of the algorithm is identical to that of the lanczos algorithm when it is formally extended to self adjoint operators

ultraviolet divergence and scaling in a class of singular potentials for the class of singular potentials r2 r the perturbation theory in has an ultraviolet divergence for a range of which causes the perturbation series to be ordered by z where z is a fraction less than unity we use a scaling argument to find z in any dimension n

existence of conditional shape invariance symmetry for singular power potential v r ar2 br 4 cr 6 we examine the solvability conditions of the singular power potential v r ar2 br 4 cr 6 a 0 c 0 in the light of shape invariance we prove that one energy level can be obtained algebraically if a corresponding constraint condition is satisfied by the parameters we show that energies of at most two levels can be obtained algebraically for a severely restricted set of parameters resulting from the satisfaction of two such constraint conditions leading to the existence of conditional shape invariance

perturbation expansions for a class of singular potentials harrell s modified perturbation theory ann phys n y 105 379 1977 is applied and extended to obtain nonpower perturbation expansions for a class of singular hamiltonians h d2 dx2 x2 a x2 x a 0 2 known as generalized spiked harmonic oscillators the perturbation expansions developed here are valid for small values of the coupling 0 and they extend the results which harrell obtained for the spiked harmonic oscillator a 0 formulas for the excited states are also developed

matrix elements for a generalized spiked harmonic oscillator closed form expressions for the singular potential integrals m x n are obtained with respect to the gol dman and krivchenkov eigenfunctions for the singular potential bx2 a x2 b 0 a 0 the formulas obtained are generalizations of those found earlier by use of the odd solutions of the schrodinger equation with the harmonic oscillator potential aguilera navarro et al j math phys 31 99 1990

a generalization of tight closure and multiplier ideals we introduce a new variant of tight closure associated to any fixed ideal a which we call tight closure and study various properties thereof in our theory the annihilator ideal r a of all tight closure relations which is a generalization of the test ideal in the usual tight closure theory plays a particularly important role we prove the correspondence of the ideal a and the multiplier ideal associated to a or the adjoint of a in lipman s sense in normal q gorenstein rings reduced from characteristic zero to characteristic p 0 also in fixed prime characteristic we establish some properties of r a similar to those of multiplier ideals e g a briancon skoda type theorem subadditivity etc with considerably simple proofs and study the relationship between the ideal and the f rationality of rees algebras

variational analysis for a generalized spiked harmonic oscillator a variational analysis is presented for the generalized spiked harmonic oscillator hamiltonian operator d2 dx 2 bx 2 a x 2 x where is a real positive parameter the formalism makes use of a basis provided by exact solutions of schrodinger s equation for the gol dman and krivchenkov hamiltonian and the corresponding matrix elements that were previously found for all the discrete eigenvalues the method provides bounds which improve as the dimension d of the basis set is increased extension to the n dimensional case in arbitrary angular momentum subspaces is also presented by minimizing over the free parameter a we are able to reduce substantially the number of basis functions needed for a given accuracy

eigenvalues of schrodinger operators with definite and indefinite weights in this paper computational algorithms are presented to compute the eigenvalues of schrodinger operators with definite and indefinite weights the algorithms are based on titchmarsh weyl s theory and can be applied to solve a wide class of problems in quantum mechanics explicit examples are presented some of which recover the known exact solutions finally some analytical bounds have been reviewed for definite and indefinite singular problems weyl s spectral bisection algorithm has been proposed to solve definite problems as algebraic problems open problems have been presented based on the results of the presented algorithms

study of the singular anharmonic potentials by means of the analytic continuation method here we study the singular anharmonic potentials by applying the analytic continuation method of holubec and stauffer 1985 in order to do that we have developed several approximations to the problem because this method cannot be applied when the solution has essential singularities at any point of its domain all the options here shown have the same precision giving us the eigenvalues correct to all the decimal places provided by the computer

a different approach for solving the nonlinear fredholm integral equations of the second kind in this paper we proposed a method for finding a approximate solution for the nonlinear fredholm integral equations of the second kind in this approach the nonlinearity of the kernel has no serious effect on the convergence of the solution first we convert the problem to an optimal control problem by introducing an artificial control function then by using some concepts of measure theory we make a metamorphosis in the space of the problem and finally we will get a linear programming whose solution gives rise to the approximate solution of the integral equation

initial ideals veronese subrings and rates of algebras abstract let s be a polynomial ring over an infinite field and let i be a homogeneous ideal of s let t d be a polynomial ring whose variables correspond to the monomials of degree d in s we study the initial ideals of the ideals v d i t d that define the veronese subrings of s i in suitable orders they are easily deduced from the initial ideal of i we show that in v d i is generated in degree max reg i d 2 where reg i is the regularity of the ideal i in other words the d th veronese subrings of any commutative graded ring s i has a grobner basis of degree max i d 2 we also give bounds on the regularity of i in terms of the degrees of the generators of in i and some combinatorial data this implies a version of backelin s theorem that high veronese subrings of any ring are homogeneous koszul algebras in the sense of priddy trans amer math soc 152 1970 39 60 we also give a general obstruction for a homogeneous ideal i s to have an initial ideal in i that is generated by quadrics beyond the obvious requirement that i itself should be generated by quadrics and the stronger statement that s i is koszul we use the obstruction to show that in certain dimensions a generic complete intersection of quadrics cannot have an initial ideal that is generated by quadrics for the application to backelin s theorem we require a result of backelin whose proof has never appeared we give a simple proof of a sharpened version bounding the rate of growth of the degrees of generators for syzygies of any multihomogenous module over a polynomial ring modulo an ideal generated by monomials following a method of bruns and herzog

singular anharmonicities and the analytic continued fractions the potential v r 2r2 2r 4 is investigated and its bound states are constructed by a generalized hill determinant method it is shown that the binding energy e and another free parameter in the underlying laurent series representation of the wave function may be obtained from a coupled pair of hill determinant equations

multiple valued logic its status and its future multiple valued logic in which the number of discrete logic levels is not confined to two has been the subject of much research over many years the practical objective of this work has been to increase the information content of the digital signals in a system to a higher value than that provided by binary operation in this tutorial survey paper we will review the historical developments in this field both in circuit realizations and in methods of handling multiple valued design data and consider the present state of the art and future expectations

a basis for variational calculations in d dimensions in this paper we derive expressions for matrix elements i hj for the hamiltonian h qa q rq in d 2 dimensions the basis functions in each angular momentum subspace are of the form the matrix elements are given in terms of the gamma function for all d the significance of the parameters t and p and scale s are discussed applications to a variety of potentials are presented including potentials with singular repulsive terms of the form r 0 perturbed coulomb potentials d r br ar2 and potentials with weak repulsive terms such as r2 r4 0

arcs on determinantal varieties we study arc spaces and jet schemes of generic determinantal varieties using the natural group action we decompose the arc spaces into orbits and analyze their structure this allows us to compute the number of irreducible components of jet schemes log canonical thresholds and topological zeta functions

algebras defined by powers of determinantal ideals let x be an m x n matrix of indeterminates over a field k a k x and i z x the ideal generated by the t minors of x the main objects of this article are the rees algebra 2 y these characteristics will be called non exceptional we show how this result generalizes to an arbitrary integral domain of coefficients the intersection of primary ideals which gives i in non exceptional characteristics always is the integral closure of i it follows immediately from the primary decomposition that the powers i are integrally closed in non exceptional characteristics therefore is a normal domain and the primary decomposition of i turns out easy giving some insight into the structure of y an interesting observation the primary decomposition of the ideals i can be computed very quickly if one knows in advance that all these ideals are integrally closed the best results are obtained in characteristic 0 since one has a multiplicity free action of the linearly reductive group gl m k x gl n k on k x under which z is stable cf dep or bv sect 111 applying the theory of u invariants kraft kr one shows that g has rational singularities in particular and hence 9 are cohen macaulay rings we have no doubt that and 3 are cohen macaulay in arbitrary non exceptional characteristic it seems however that in exceptional characteristic they are as far as possible from this property the case t 2

spiked harmonic oscillators a complete variational treatment is provided for a family of spiked harmonic oscillator hamiltonians h d2 dx2 bx2 x b 0 0 for arbitrary 0 a compact topological proof is presented that the set s n of known exact solutions for 2 constitutes an orthonormal basis of the hilbert space l2 0 closed form expressions are derived for the matrix elements of h with respect to s these analytical results and the inclusion of a further free parameter facilitate optimized variational estimation of the eigenvalues of h to high accuracy

quantum states of a sextic potential hidden symmetry and quantum monodromy a known terminating polynomial ansatz for a finite sub set of states of the sextic central potential v a b c r ar2 br4 cr6 c 0 subject to integer related parameter constraints is confirmed to apply in two as well as three dimensions a hidden symmetry relating the eigenstates of v a b c r to those of v a b c r is used to establish the boundary between the ansatz sub set and the infinite set of higher states connections between the radial wavefunctions rn a b c r and rn a b c r also provide semiclassical insight into the origin of the parameter constraint finally the ansatz eigenvalue dispositions are related to the phenomenon of quantum monodromy

singular anharmonicities and the analytic continued fractions ii the potentials v r ar2 br 4 cr 6 the c 0 results of paper i j math phys 30 23 1989 are extended in spite of the presence of an additional coupling constant the laurent series solutions of the schrodinger equation that are obtained remain similar to mathieu functions indeed the recurrences for coefficients preserve their three term character their analytic continued fraction solutions still converge etc the formulas become even slightly simpler for c 0 due to a certain symmetry of the equations to be solved an acceleration of convergence is better understood and a few numerical illustrations of efficiency are also delivered

majority based reversible logic gates reversible logic plays an important role in the synthesis of circuits for quantum computing in this paper we introduce families of reversible gates based on the majority boolean function mbf and we prove their properties in reversible circuit synthesis these gates can be used to synthesize reversible circuits of minimum scratchpad register width for arbitrary reversible functions we show that given a mbf f with 2k 1 inputs f can be implemented by a reversible logic gate with 2k 1 inputs and 2k 1 outputs i e without any constant inputs we also demonstrate new gates from this family with very efficient quantum realizations for majority based applications they can be used to synthesize any reversible function of the same width in conjunction with inverters and feynman 2 qubit controlled not gates the gate universality problem is formulated in terms of elementary group theory and solved using the algebraic software gap

a simple numerical method for singular potentials a one parameter family of coordinate transformations is shown to lead to a simple finite difference method which gives highly accurate energies and expectation values for the schrodinger equation in which the potential consists of a smooth term plus a perturbing term which is singular at the origin the method is effective down to very small values of the perturbation parameter and supplements the previously reported perturbation approach which is valuable for large values

on the two dimensional stationary schrodinger equation with a singular potential abstract we study the equation u x y x y u x y 0 when the potential has the following expression r a r 2 where r are the polar coordinates in r 2 and a is a 2 periodic function of class c k with k 1 we obtain series representations for solutions in a full neighborhood of the singular point and we also give representations in terms of pseudoanalytic formal powers in sectors having the singular point as a vertex the results are obtained with the aid of the reduction of the stationary schrodinger equation to a vekua equation of a special form and by using recent developments in pseudoanalytic function theory

on the elementary schrodinger bound states and their multiplets the problem of the existence of elementary bound states is discussed a trivial observation that every elementary wave function i r is an exact bound state for an appropriate potential v r v i r r is shown to lead to a very transparent form of the quasiexact qe solvability condition v i v j for doublets and multiplets of the s in this sense the particular class of elementary ansatze i r r polynomial r2 exp r polynomial r2 also defines the particular class of qe solvable potentials they have an elementary nonpolynomial rational form possibly also with a strongly singular repulsive core at the origin the properties of these forces are discussed in detail

direct expectation value calculations a new application of the hellmann feynman theorem is shown to allow accurate calculation of expectation values along with energies for the schrodinger equation for variable boundary position and boundary conditions power series and finite difference versions of the method are developed and applied to perturbed oscillator and perturbed hydrogen atom problems

on the eigenvalues of the s state radial equation of a spiked harmonic oscillator eigenvalues of the ground state of the radial schrodinger equation for a spiked harmonic oscillator potential have been evaluated employing two methods numerically via the lanczos grid technique and by means of standard pade approximants constructed from an expansion of large coupling parameter series for the energy numerical results are compared for several values of the parameters characterizing the spiked singular potential

nonsingular spiked harmonic oscillator a perturbative study of a class of nonsingular spiked harmonic oscillators defined by the hamiltonian h d2 dr 2 r 2 r in the domain 0 is carried out in the two extremes of a weak coupling and a strong coupling regimes a path has been found to connect both expansions for near 2

part of the d dimensional spiked harmonic oscillator spectra the pseudoperturbative shifted l expansion technique pslet is generalized for states with an arbitrary number of nodal zeros interdimensional degeneracies emerging from the isomorphism between the angular momentum and the dimensionality of the central force schrodinger equation are used to construct part of the d dimensional spiked harmonic oscillator bound state spectra pslet results are found to compare excellently with those from direct numerical integration and generalized variational methods

realizing ternary quantum switching networks without ancilla bits this paper investigates the synthesis of quantum networks built to realize ternary switching circuits in the absence of ancilla bits the results we established are twofold the first shows that ternary swap ternary not and ternary toffoli gates are universal for the realization of arbitrary n n ternary quantum switching networks without ancilla bits the second result proves that all n n quantum ternary networks can be generated by not controlled not multiply two and toffoli gates our approach is constructive

energy bounds for a class of singular potentials and some related series perturbation expansions up to third order for the generalized spiked harmonic oscillator hamiltonians h d2 dx2 x2 a x2 x a 0 2 1 1 2 1 4a and small values of the coupling 0 are developed upper and lower bounds for the eigenvalues are computed by the procedure of burrows et al 1987 j phys a math gen 20 889 97 for assessing the accuracy of a truncated perturbation expansion closed form sums for some related perturbation double infinite series then immediately follow as a result of this investigation

supersymmetry and the relationship between a class of singular potentials in arbitrary dimensions the eigenvalues of the potentials v1 r a1 r a2 r2 a3 r3 a4 r4 and v2 r b1r2 b2 r2 b3 r4 b4 r6 and of the special cases of these potentials such as kratzer and goldman krivchenkov potentials are obtained in n dimensional space the explicit dependence of these potentials in higher dimensional space is discussed which has not been previously covered

study of anharmonic singular potentials a simple and efficient variational method is introduced to accelerate the convergence of the eigenenergy computations for a hamiltonian h with singular potentials closed form analytic expressions in n dimensions are obtained for the matrix elements of h with respect to the eigenfunctions of a soluble singular problem with two free parameters a and b the matrix eigenvalues are then optimized with respect to a and b for a given n applications convergence rates and comparisons with earlier work are discussed in detail

strategies for designing geometric transformations on quantum images three design strategies for constructing new geometric transformations on quantum images from other transformations are proposed the strategies focus on the affected areas in the images the separability and smoothness of the transformations by exploiting a representation of images on quantum computers extensively the complexity in terms of the number of basic gates and the depth of the corresponding circuits are analyzed to understand the advantages and disadvantages of each strategy examples to demonstrate the applicability of the proposed strategies are presented the strategies provide high level tools to explore and analyze transformations which are necessary to build practical image processing applications on quantum computers

how do supersingular perturbations generate non taylor series we show that a suitable choice of the zero order problem can lead directly to the emergence of non taylor expansions in for the ground state energies of the hamiltonian family h 2 r2 r in regions 3 the discussion includes the role of dimension and the right order parameter the effect of choosing a more general potential of the form r r is also considered

the schrodinger equation with position dependent mass it is shown that the traditional hartree second order shooting method for eigenvalues can be modified to give accurate energy levels for a schrodinger equation involving a position dependent mass some numerical applications are given to demonstrate the effectiveness of the method

unified qlm for regular and arbitrary singular potentials a parameterless numerical implementation of the quasilinearization method qlm is constructed and tested to 21 25 digits precision to give quadratically convergent energies e of the klauder effect exhibiting spiked harmonic oscillator with the r or exp r type spikes in a riccati reformulation of the schrodinger equation the radial solution is uniformly quadratic convergent to the same precision as e except in the small minorization interval where the self correcting property of qlm assures geometric convergence like in the picard algorithm to about 12 16 digits sufficient not to affect the convergence of e confirming what is expected on physical grounds it was shown before that for regular potentials immediate onset of quadratic convergence is guaranteed by the initial iteration of the wkb form and that for quadratic convergence of e for power type spikes it suffices to augment this by a nonlinear integration point distribution and by minorization of negative solution values the form of the riccati equation used allows the minorization function to easily be formally defined over the entire interval without the need for a cutoff radius of application and dependence on its scale factor is plateau like and negligible

implementing an arbitrary reversible logic gate the 2w reversible logic gates of width w i e reversible logic gates with w inputs and w outputs together with the action of cascading form a group r we define a subgroup k consisting of the switched gates there are 2w 1 2 such gates they partition the group r into 2w 1 1 double cosets this allows us to decompose an arbitrary reversible gate into the cascade of three gates a switched gate an upside down simple control gate and a second switched gate we present an algorithm to perform this factorization and thus provide a method of implementing an arbitrary reversible gate into hardware the algorithm can be used to automate the implementation of a reversible function in future c mos technologies realizing low cost computing

spiked oscillators exact solution a procedure to obtain the eigenenergies and eigenfunctions of a quantum spiked oscillator is presented the originality of the method lies in an adequate use of asymptotic expansions of wronskians of algebraic solutions of the schrodinger equation the procedure is applied to three familiar examples of spiked oscillators

chain models and their effective hamiltonians we develop a new formalism suitable for diagonalization of hamiltonians with 2t 1 nonzero diagonals chain models in a systematic perturbation like way we get the expansions of green and or wave functions they are generated by solution of a sequence of fixed point quadratic equations for t t dimensional matrices for t 1 and t 2 this solution is feasible by elementary means so that the respective tridiagonal and pentadiagonal chain models may be considered exactly solvable in this context in an alternative first order variational perturbation formulation the method may provide the simultaneous upper and lower energy bounds for any t

generating the group of reversible logic gates reversible logic plays a fundamental role both in ultra low power electronics and in quantum computing it is therefore important to have an insight into the structure of the group formed by the reversible logic gates and their cascading into reversible circuits such insight is gained from constructing chains of maximal subgroups the subgroup of control gates plays a prominent role as it is a sylow 2 subgroup

the decomposition of an arbitrary reversible logic circuit the 2w reversible logic circuits of width w i e reversible logic circuits with w inputs and w outputs together with the action of cascading form a group g isomorphic to the symmetric group we define two conjugate subgroups g1 and g2 together they partition the group g into 2w 1 1 double cosets these allow us to decompose an arbitrary member of g into a cascade of three simpler members this decomposition is a far relative of the well known lu decomposition of a square matrix

on f pure thresholds using the frobenius map we introduce a new invariant for a pair r a of a ring r and an ideal a r which we call the f pure threshold c a of a and study its properties we see that the f pure threshold characterizes several ring theoretic properties by virtue of hara and yoshida s result trans amer math soc 355 2003 3143 the f pure threshold c a in characteristic zero corresponds to the log canonical threshold lc a which is an important invariant in birational geometry using the f pure threshold we prove some ring theoretic properties of three dimensional terminal singularities of characteristic zero also in fixed prime characteristic we establish several properties of the f pure threshold similar to those of the log canonical threshold with quite simple proofs

r universal reversible logic gates reversible logic plays a fundamental role both in ultra low power electronics and in quantum computing it is therefore important to know which reversible logic gates can be used as building block for the reversible implementation of an arbitrary boolean function and which cannot

the generalized continued fractions and potentials of the lennard jones type for a broad class of the strongly singular potentials v r which are defined as superpositions of separate power law components the general solution of the corresponding schrodinger differential equation is constructed as an analog of mathieu functions the analogy is supported by the use of the generalized continued fractions the questions of convergence are analyzed in detail

realization and synthesis of reversible functions reversible circuits play an important role in quantum computing this paper studies the realization problem of reversible circuits for any n bit reversible function we present a constructive synthesis algorithm given any n bit reversible function there are n distinct input patterns different from their corresponding outputs where email protected 2 n and the other 2 n n input patterns will be the same as their outputs we show that this circuit can be synthesized by at most email protected n n 1 cnot gates and 4n email protected n not gates the time and space complexities of the algorithm are w email protected 4 n and w email protected 2 n respectively the computational complexity of our synthesis algorithm is exponentially lower than that of breadth first search based synthesis algorithms

on the symmetry of logic embedding logical operations in non dissipative physical processes requires the use of reversible logic following feynman s 1986 approach the sixteen distinct truth tables of classical logic are shown to be contained in the 8 reversible logic operations covered by the symmetric group s8 which permute the eight values of three logical variables small subgroups of s8 are shown to cover respectively reversible logic reversible switching and reversible arithmetic a new universal primitive is found which generates a covering group of reversible logic it is shown that the octahedral group in four dimensions covers both reversible logic and switching and hence that the orthogonal group o 4 provides a covering group for quantum gates

elementary bound states for the power law potentials the author shows how the exact polynomial exp polynomial bound state solutions to the radial schrodinger equation may be constructed for any potential of the form v r sigma delta in d gdelta rdelta where d is a finite set of arbitrary rational numbers and some of the couplings gdelta are not independent

test multiplier and invariant ideals abstract this paper gives an explicit formula for the multiplier ideals and consequently for the log canonical thresholds of any gl v gl w invariant ideal in s sym v w where v and w are vector spaces over a field of characteristic 0 this characterization is done in terms of a polytope constructed from the set of young diagrams corresponding to the schur modules generating the ideal our approach consists in computing the test ideals of some invariant ideals of s in positive characteristic namely we compute the test ideals and so the f pure thresholds of any sum of products of determinantal ideals not all the invariant ideals are as the latter not even in characteristic 0 but they are up to integral closure and this is enough to reach our goals the results concerning the test ideals are obtained as a consequence of general results holding true in a special situation within such framework fall determinantal objects of a generic matrix as well as of a symmetric matrix and of a skew symmetric one similar results are thus deduced for the gl v invariant ideals in sym sym 2 v and in sym 2 v monomial ideals also fall in this framework thus we recover howald s formula for their multiplier ideals and more generally hara yoshida s formula for their test ideals in the proof we introduce the notion of floating test ideals a property that in a sense is satisfied by ideals defining schemes with the nicest possible singularities as will be shown products of determinantal ideals and by passing to characteristic 0 ideals generated by a single schur module have this property

variational and perturbative schemes for a spiked harmonic oscillator a variational analysis of the spiked harmonic oscillator hamiltonian operator d2 dx2 x2 l l 1 x2 x where is a real positive parameter is reported in this work the formalism makes use of the functional space spanned by the solutions of the schrodinger equation for the linear harmonic oscillator hamiltonian supplemented by a dirichlet boundary condition and a standard procedure for diagonalizing symmetric matrices the eigenvalues obtained by increasing the dimension of the basis set provide accurate approximations for the ground state energy of the model system valid for positive and relatively large values of the coupling parameter additionally a large coupling perturbative expansion is carried out and the contributions up to fourth order to the ground state energy are explicitly evaluated numerical results are compared for the special case 5 2

a classification of the maximal subgroups of the finite alternating and symmetric groups following the classification of finite simple groups one of the major problems in finite group theory today is the determination of the maximal subgroups of the almost simple groups that is of groups x such that x0 u x aut x for some finite non abelian simple group x0 the problem has been solved for most sporadic groups and groups of lie type of low rank see 19 sects 3 43 for discussion and references this paper is a contribution to the case where x0 is an alternating group a so that for n 6 x is a or s the maximal subgroups of a and s are known for several classes of degrees n