products of projections and positive operators this article is devoted to the study of the set t of all products pa with p an orthogonal projection and a a positive semidefinite operator we describe this set and study optimal factorizations we also relate this factorization with the notion of compatibility and explore the polar decomposition of the operators in t

decomposition of matrices into three involutions abstract in this paper we generalize some results of c s ballantine concerning products of three n n complex involutions we prove that each n n complex matrix a with determinant 1 and dimker a n 2 for all c is the product of three involutions on the other hand we show that if an n n complex matrix a is the product of three involutions then m 2n r 3 and m 3n 4 where m dimker a and r dimker a 3 for any 0 and 4 1 we also completely characterize products of three 5 5 complex involutions

each n by n matrix with n 1 is a sum of 5 coninvolutory matrices abstract an n n complex matrix a is called coninvolutory if a a i n and skew coninvolutory if a a i n which implies that n is even we prove that each matrix of size n n with n 1 is a sum of 5 coninvolutory matrices and each matrix of size 2 m 2 m is a sum of 5 skew coninvolutory matrices we also prove that each square complex matrix is a sum of a coninvolutory matrix and a condiagonalizable matrix a matrix m is called condiagonalizable if m s 1 d s in which s is nonsingular and d is diagonal

grammian solutions and pfaffianization of a non isospectral and variable coefficient kadomtsev petviashvili equation abstract in this paper we first present the grammian determinant solutions to the non isospectral and variable coefficient kadomtsev petviashvili vckp equation then by using the pfaffianization procedure of hirota and ohta a new non isospectral and variable coefficient integrable coupled system is generated moreover gramm type pfaffian solutions of the pfaffianized system are proposed

decomposition of matrices into 2 involutions abstract we consider the group slnf of all n by n matrices with determinant 1 over a field let res a denote rank a i for any a in slnf a matrix a is called a k involution if a2 i and res a k we prove that every matrix a in slnf is the product of at most res a 1 2 2 2 involutions for n 3 if char f 2 and for n 4 if char f 2

when is every linear transformation a sum of two commuting invertible ones abstract a well known result of wolfson 7 and zelinsky 8 says that every linear transformation of a vector space v over a division ring d is a sum of two invertible linear transformations except when dim v 1 and d f 2 indeed many of these linear transformations satisfy a stronger property that they are sums of two commuting invertible linear transformations the goal of this note is to prove that every linear transformation of a vector space v over a division ring d is a sum of two commuting invertible ones if and only if d 3 and dim v as a consequence a sufficient and necessary condition is obtained for a semisimple module to have the property that every endomorphism is a sum of two commuting automorphisms

the sum of orthogonal matrices abstract let f r c h let u n f be the set of unitary matrices in m n f and let o n f be the set of orthogonal matrices in m n f suppose n 2 we show that every a m n f can be written as a sum of matrices in u n f and of matrices in o n f let a m n f be given and let k 2 be the least integer that is a least upper bound of the singular values of a when f c we show that a can be written as a sum of k matrices from u n f when f r we show that if k 3 then a can be written as a sum of 6 orthogonal matrices if k 4 we show that a can be written as a sum of k 2 orthogonal matrices

response to comment on on the coulomb potential in one dimension by fischer leschke and muller response to comment on on the coulomb potential in one dimension by w fisher h leschke and p muller

on sums of idempotent matrices over a field of positive characteristic we study which square matrices are sums of idempotents over a field of positive characteristic in particular we prove that any such matrix provided it is large enough is actually a sum of five idempotents and even of four when the field is a prime one

the functional analytic versus the functional integral approach to quantum hamiltonians the one dimensional hydrogen atoma the capabilities of the functional analytic and of the functional integral approach for the construction of the hamiltonian as a self adjoint operator on hilbert space are compared in the context of non relativistic quantum mechanics differences are worked out by taking the one dimensional hydrogen atom as an example that is a point mass on the euclidean line subjected to the inverse distance potential this particular choice is made with the intent to clarify a long lasting discussion about its spectral properties in fact for the four parameter family of possible hamiltonians the corresponding energy dependent green functions are derived in closed form the multiplicity of hamiltonians should be kept in mind when modeling certain experimental situations as for instance in quantum wires

an extension of the wronskian technique for the multicomponent wronskian solution to the vector nonlinear schrodinger equation in this paper the wronskian technique is applied to the vector nonlinear schrodinger equation with arbitrary m components which arises from some applications in the multimode fibers photorefractive materials and bose einstein condensates via the iterative algorithm based on the darboux transformation the m 1 component wronskian solution is generated from the zero solution the verification of the solution is finished by using the m 1 component wronskian notation and new determinantal identities with a set of n linearly independent solutions of the zero potential lax pair the m 1 component wronskian solution is found to be the representation of the bright n soliton solution which contains m 1 n parameters for characterizing the asymptotic behavior of the generic bright n soliton solution an algebraic procedure is derived to obtain the explicit expressions of asymptotic solitons as t

sums and products of quasi nilpotent operators it is proved that a bounded operator on hilbert space is the sum of two quasi nilpotent operators if and only if it is not a non zero scalar plus a compact operator necessary conditions and sufficient conditions for an operator to be the product of two quasi nilpotent operators are given

on the ϕj polar decomposition of matrices abstract we present new results on the j polar decomposition of matrices we show that every symplectic matrix may be written as a product of symplectic operation matrices we present a simple form attained under symplectic equivalence which makes it easy to determine if a matrix does not have a j polar decomposition we also determine the rank 4 matrices with j polar decomposition

the cartan dieudonne scherk theorems for complex s orthogonal matrices abstract let m n c be the set of all n by n matrices with complex entries let s n be the set of all nonsingular symmetric matrices in m n c let s n be the set of all nonsingular skew symmetric matrices in m n c and let s n s n s n let s s n be given an a m n c is called s orthogonal if a t s a s let o s be the set of all s orthogonal matrices in m n c an h o s is called a symmetry if rank h i 1 let h s be the set of all symmetries in o s we show that every q o s is a product of elements of h s if q i then q is a product of two elements of h s suppose that rank q i m 1 if s q i is not skew symmetric then q can be written as a product of m elements of h s and q cannot be written as a product of less than m elements of h s if s q i is skew symmetric and if s s n then q can be written as a product of m 2 elements of h s and q cannot be written as a product of less than m 2 elements of h s if s q i is skew symmetric and if s s n then q can be written as a product of m 1 elements of h s and q cannot be written as a product of less than m 1 elements of h s

remarks on the local time rescaling in path integration the local time transformation used in path integration is shown to be equivalent to standard procedures applied to schrodinger s equation

on the nonisospectral modified kadomtsev peviashvili equation abstract two types of grammian solutions to the nonisospectral modified kadomtsev peviashvili mkp equation are derived with the help of the pfaffian derivative formulae starting from the grammian solutions obtained we construct the nonisospectral mkp equation with self consistent sources by applying the source generalization procedure whose solutions can be given simultaneously

coulomb green s functions in an n dimensional euclidean space the h atom green s function is calculated in an n dimensional euclidean space following the feynman lagrangian formulation the use of generalized polar coordinates allows the expansion of the propagator into partial propagators and the separation of the angular and radial variables the angular part is shown to be a generalized legendre polynomial while the radial part may be transformed in that of the harmonic oscillator the h atom spectrum is given by the poles of the green s function

powers of 1 cyclic conjugacy classes in the groups gln f and sln f abstract let a gl n f where f is a field we say that a is 1 cyclic if a is similar to a matrix of the form a diag a 1 a 2 a k where a i gl l i f is cyclic for 1 i k l 1 0 1 and l i 2 for 2 i k it is shown that if a gl n f is 1 cyclic where n 2 and f 4 then every nonscalar matrix m gl n f whose determinant equals det a 4 is the product of four matrices which are similar to a under matrices of sl n f the problem of expressing a scalar matrix as a product of similar 1 cyclic matrices is also discussed the above result is applied to problems of factorizing matrices in the group sl n f into products of unipotent matrices of index 2 and into products of matrices of fixed finite order

commutators and companion matrices over rings of stable rank 1 abstract we consider the group gl n a of all invertible n by n matrices over a ring a satisfying the first bass stable range condition we prove that every matrix is similar to the product of a lower and upper triangular matrix and that it is also the product of two matrices each similar to a companion matrix we use this to show that when n 3 and a is commutative every matrix in sl n a is the product of two commutators

products of involutory matrices over rings abstract let gln a be a matrix over a commutative ring a with 1 such that det 2 1 if is cyclic it can be written as a product of at most three involutions when a satisfies the first bass stable range condition then can be written as a product of at most five involutions if in addition either n 3 or n 4 and det 1 then can be written as a product of at most four involutions when a is a dedekind ring of arithmetic type the number of involutions needed to express is uniformly bounded for any n 3 when a c x the number of involutions is unbounded for any n 2

contragredient equivalence a canonical form and some applications abstract let a and c be m by n complex matrices and let b and d be n by m complex matrices the pair a b is contragrediently equivalent to the pair c d if there are square nonsingular complex matrices x and y such that xay 1 c and ybx 1 d contragredient equivalence is a common generalization of four basic equivalence relations similarity consimilarity complex orthogonal equivalence and unitary equivalence we develop a complete set of invariants and an explicit canonical form for contragredient equivalence and show that a a t is contragrediently equivalent to c c t if and only if there are complex orthogonal matrices p and q such that c paq using this result we show that the following are equivalent for a given n by n complex matrix a 1 1 a qs for some complex orthogonal q and some complex symmetric s 2 2 a t a is similar to aa t 3 3 a a t is contragrediently equivalent to a t a 4 4 a q 1 a t q 2 for some complex orthogonal q 1 q 2 5 5 a pa t p for some complex orthogonal p we then consider a linear operator on n by n complex matrices that shares the following properties with transpose operators for every pair of n by n complex matrices a and b a preserves the spectrum of a b a a and c ab b a we show that a a is contragrediently similar to b b if and only if a x 1 bx 2 for some nonsingular x 1 x 2 that satisfy x 1 1 x 1 and x 1 2 x 2 we also consider a factorization of the form a xy where x 1 x and y y we use the canonical form for the contragredient equivalence relation to give a new proof of a theorem of flanders concerning the relative sizes of the nilpotent jordan blocks of ab and ba we present a sufficient condition for the existence of square roots of ab and ba and close with a canonical form for complex orthogonal equivalence

every 2n by 2n complex matrix is a sum of three symplectic matrices abstract let j 2 n 0 i n i n 0 an a m 2 n c is called symplectic if a t j 2 n a j 2 n if n 1 then we show that every matrix in m 2 n c is a sum of two symplectic matrices if n 1 then we show that every matrix in m 2 n c is a sum of three symplectic matrices moreover we show that some matrices cannot be written with less than three symplectic matrices we also show that for every a m 2 n c there exist symplectic p q m 2 n c and b c d m n c such that p a q b c 0 d if a is skew hamiltonian j 2 n 1 a t j 2 n a then we show that a is a sum of two symplectic matrices

products of nilpotent matrices we show that any complex singular square matrix t is a product of two nilpotent matrices a and b with rank a rank b rank t except when t is a 2 x 2 nilpotent matrix of rank one an n x n complex matrix t is nilpotent if t 0 it is easily seen that a product of finitely many nilpotent matrices must be singular the purpose of this note is to prove the converse

solving the kdv hierarchy with self consistent sources by inverse scattering method abstract the evolution of the eigenfunctions in the lax representation of the kdv hierarchy with self consistent sources possesses singularity by proposing a method to treat the singularity to determine the evolution of scattering data the kdv hierarchy with self consistent sources is integrated by the inverse scattering method the soliton solutions of these equations are obtained it is shown that the insertion of a source may cause the variation of the speed of soliton this approach can be applied to other 1 1 dimensional soliton hierarchies

on some factorizations of operators abstract given two subsets a and b of the algebra of bounded linear operators on a hilbert space h we denote by a b ab a a b b the goal of this article is to describe a b if a and b denote classes of projections partial isometries positive semidefinite operators etc moreover fixed t a b we shall describe a b t a b a b ab t p 1 a b t a a t ab for some b b and p 2 a b t b b t ab for some a a

the sum of two ϕs orthogonal matrices when s ts is normal and 1 σ s ts abstract let a nonsingular s m n c be given for a m n c set s a s 1 a t s we say that a is s symmetric if s a a we say that a is s orthogonal if a gl n and s a a 1 we say that a has a s polar decomposition if a u p for some s orthogonal u and s symmetric p suppose that s t s is normal and 1 s t s we determine conditions on a m n c so that a can be written as a sum of two s orthogonal matrices

the operator factorization problems we survey various results concerning operator factorization problems more precisely we consider the following setting let h be a complex hilbert space and let g h be the algebra of all bounded linear operators on h for a given subset of 3 h we are interested in the characterization of operators in q h which are expressible as a product of finitely many operators in v and for each such operator the minimal number of factors in a factorization the classes of operators we consider include normal operators involutions partial isometries together with their various subclasses and other miscellaneous classes of operators most of the known results are for operators on finite dimensional spaces or finite matrices the paper concludes with some applications due to hochwald concerning the uniqueness of the adjoint operation on operators

two singular potentials the space splitting effect the quantum mechanical problem of a particle moving in one dimension 1d under the influence of a singular potential is reexamined an attractive delta function potential of infinite strength and the 1d coulomb potential is considered in particular in both cases the singularity acts as an impenetrable barrier thus dividing space into two independent regions analysis of the scattering states gives results in agreement with this conclusion the implications that the bound states of the 1d coulomb potential cannot be classified according to parity even though the potential is symmetric is also discussed herein

treatment of some singular potentials by change of variables in wiener integrals using a regularization of the classical kepler problem we show that a change of process and a change of time in stochastic differential equations allow a treatment of singular potentials in wiener integrals

on the coulomb potential in one dimension a mathematically rigorous definition of the one dimensional schrodinger operator d 2 dx 2 gamma x is given it is proven that the domain of the operator is defined by the boundary conditions connecting the values of the function on the left and right half axes the investigated operator is compared with the schrodinger operator containing the coulomb potential gamma x

the length problem for a sum of idempotents abstract it was shown by r e hartwig and m s putcha and independently by p y wu that the complex square matrix t is a sum of finitely many idempotent matrices if and only if tr t is an integer and tr t rank t we determine the minimal number of required idempotents for such matrices in terms of their traces and sizes and we completely solve the problem for matrices t with size 2 3 4 or 5 also we give some sufficient necessary conditions for a complex matrix t to be the sum of three or more idempotents

products of cyclic similarity classes in the groups gln f abstract let g gln f be the group of all n n invertible matrices over a field f where f 4 and n 3 let j1 j2 be two prescribed cyclic elements of g in jordan canonical form and assume that all the eigenvalues of j1 or j2 lie in f it is shown that any nonscalar matrix in g can be written as a product of two matrices having the prescribed jordan canonical forms j1 j2 subject only to the obvious 0 determinant condition as a corollary we show that if f 4 the group psln f has a conjugacy class c such that c2 psln f another corollary states that every nonscalar matrix in g can be written as a product of two cyclic unipotent matrices

comment on on the coulomb potential in one dimension by p kurasov the mathematical analysis in 1996 j phys a math gen 29 1767 is not sufficient to decide whether in one dimension the singularities of the potentials and split the corresponding one particle quantum systems at the origin into two completely decoupled subsystems in fact it is argued that this question cannot be answered by mathematical considerations alone

the sums of symplectic hamiltonian and skew hamiltonian matrices abstract a complex 2 n 2 n matrix a is called skew hamiltonian hamiltonian and symplectic if a j a a j a and a j a 1 respectively in which j 0 i n i n 0 and a j j 1 a t j we prove that each 2 n 2 n matrix is a sum of type symplectic hamiltonian a 2 n 2 n matrix a is a sum of type symplectic symplectic if and only if a a j is similar to a j a a 2 n 2 n matrix a is a sum of type symplectic skew hamiltonian if and only if the jordan blocks of a a j with eigenvalue 2i and size k 2 come in pairs of the form j k 2 i j k 2 i and j k 2 i j k 1 2 i

multiplicative maps on matrices that preserve the spectrum abstract let m n denote the set of all n n matrices over the complex numbers n 2 let a n m n be either the set of all invertible matrices the set of all unitary matrices or a multiplicative semigroup containing the singular matrices theorem if a n m n is a spectrum preserving multiplicative map then there exists a matrix r in m n such that s r 1 sr for all s in a n

unified treatment of the coulomb and harmonic oscillator potentials in d dimensions quantum mechanical models and practical calculations often rely on some exactly solvable models like the coulomb and the harmonic oscillator potentials the d dimensional generalized coulomb potential contains these potentials as limiting cases thus it establishes a continuous link between the coulomb and harmonic oscillator potentials in various dimensions we present results which are necessary for the utilization of this potential as a model and practical reference problem for quantum mechanical calculations we define a hilbert space basis the generalized coulomb sturmian basis and calculate the green s operator on this basis and also present an su 1 1 algebra associated with it we formulate the problem for the one dimensional case too and point out that the complications arising due to the singularity of the one dimensional coulomb problem can be avoided with the use of the generalized coulomb potential

factorizations of upper triangular matrices if d is an integral domain in which every element can be factored as a product of irreducible elements then every element a tn d the semigroup of upper triangular matrices with nonzero determinant can be factored as a product of irreducible elements we classify both the irreducible elements and units of this cancellative semigroup having achieved this we provide a means of measuring the uniqueness or non uniqueness of these factorizations in terms of the uniqueness or non uniqueness of factorizations in d in addition to using well studied invariants from factorization theory of commutative semigroups we introduce a new tool weak transfer homomorphisms which are more appropriate for noncommutative settings

products of cyclic conjugacy classes in the groups psl n f abstract let g be the group psl n f where f is a field and n 3 if c is a conjugacy class of g denote c 1 c 1 c c the following is proved 1 if c1 c2 c3 are cyclic conjugacy classes of g then c1c2c3 g 1g 2 if f is algebraically closed and c1 c2 are cyclic conjugacy classes of g then c1c2 g if and only if c1 c 12 generalizations of these results concerning factorizations of a given nonscalar invertible matrix as a product of two or three cyclic matrices each lying in a prescribed conjugacy class of gl n f or sl n f are discussed

response to comment on penetrability of a one dimensional coulomb potential by roger g newton for original paper see ibid vol 26 p 2445 1993 by expressing in matrix form the one dimensional hamiltonians where the potentials are proportional to mod x mod 1 and x 1 i show that we are dealing with two very different problems thus in my viewpoint the criticism of r g newton of the paper mentioned in the title which is based on results relating to mod x mod 1 does not apply to a potential of the form x 1

factorizations of operator matrices abstract this note gives explicit factorizations of a 2 2 operator matrix as a product of an upper triangular operator matrix and an involutory unitary or j unitary operator matrix a pattern is given for construction of factorizations of this kind

each symplectic matrix is a product of four symplectic involutions abstract gustafson halmos and radjavi in 1973 proved that each matrix a with det a 1 is a product of four involutions we prove that these involutions can be taken to be symplectic if a is symplectic every symplectic matrix has unit determinant using this result we give an alternative proof of laffey s theorem that every nonsingular even size matrix is a product of skew symmetric matrices ballantine in 1978 proved that each matrix a with det a 1 is a product of four coninvolutions we prove that these coninvolutions can be taken to be symplectic if a is symplectic we also prove that each hamiltonian matrix is a sum of two square zero hamiltonian matrices

products of diagonalizable matrices over a field of characteristic two abstract it is shown that every square matrix over a characteristic two field with at least four elements is a product of two diagonalizable matrices

the use of backlund transformations in obtaining n soliton solutions in wronskian form the wronskian formulation of the n soliton solutions of various nonlinear evolution equations modified korteweg de vries with both zero and non zero asymptotic conditions sine gordon korteweg de vries kadomtsev petviashvili and boussinesq equations are obtained by inductive use of backlund transformations in hirota s 1980 bilinear notation

deficiency indices and singular boundary conditions in quantum mechanics we consider schrodinger operators h in l2 rn n n with countably infinitely many local singularities of the potential which are separated from each other by a positive distance it is proved that due to locality each singularity yields a separate contribution to the deficiency index of h in the special case where the singularities are pointlike and the potential exhibits certain symmetries near these points we give an explicit construction of self adjoint boundary conditions

the product of matrix subspaces abstract in factoring matrices into the product of two matrices operations are typically performed with elements restricted to matrix subspaces such modest structural assumptions are realistic for example in large scale computations this paper is concerned with analyzing associated matrix geometries curvature of the product of two matrix subspaces is assessed the case of vanishing curvature gives rise to the notion of factorizable matrix subspace this can be regarded as an analogue of the internal zappa szep product in group theory interpreted in this way several classical instances are encompassed by this structure the craig sakamoto theorem fits naturally into this framework

hamiltonian square roots of skew hamiltonian matrices revisited abstract recently h fassbender et al linear algebra appl 287 1999 125 proved the following theorem every real skew hamiltonian matrix w has a real hamiltonian square root h i e h 2 w we prove an analog of this theorem for complex matrices our approach may be of independent interest namely we use the polar decomposition of a nonsingular operator acting in a space with the symplectic inner product

decomposition of a scalar operator into a product of unitary operators with two points in spectrum abstract we consider products of unitary operators with at most two points in their spectra 1 and e i we prove that the scalar operator e i i is a product of k such operators if 1 1 k 3 k 1 1 k 3 for k 5 also we prove that for e i 1 only a countable number of scalar operators can be decomposed in a product of four operators from the mentioned class as a corollary we show that every unitary operator on an infinite dimensional space is a product of finitely many such operators

the one dimensional schrodinger coulomb operator with definite parity as a model for one dimensional hydrogenic systems the schrodinger operator with the singular potential x 1 is studied by analyzing it in its reducing subspaces corresponding to even and odd parity we derive spectral properties common to all self adjoint extensions of the reduced operators and extract the boundary conditions at x 0 specific to the different self adjoint extensions a transcendental equation for the eigenvalues is derived which leads to a construct that we call the spectral helix for the distinguished self adjoint extension with dirichlet boundary conditions virial and supersymmetry relations are established

pairs of alternating forms and products of two skew symmetric matrices let k be a field and let s t s1 t1 be skew symmetric matrices over k with s s1 both nonsingular if k has characteristic 2 a skew symmetric matrix is a symmetric one with zero diagonal it is shown that there exists a nonsingular matrix p over k with p sp s1 p tp t1 where p denotes the transpose of p if and only if s 1t and s 11t1 are similar it is also shown that a 2m 2m matrix over k can be factored as st with s t skew symmetric and s nonsingular if and only if a is similar to a matrix direct sum b b where b is an m m matrix over k this is equivalent to saying that all elementary divisors of a occur with even multiplicity an extension of this result giving necessary and sufficient conditions for a square matrix to be so expressible without assuming that either s or t is nonsingular is included

comment on penetrability of a one dimensional coulomb potential by m moshinsky see ibid vol 26 p 2445 1993 the analysis of the one dimensional schrodinger equation with the potential lambda x given is found to be incorrect contrary to the author s claim the origin is indeed impenetrable

products of symplectic normal matrices abstract a matrix a m 2 n c is symplectic if a t 0 i n i n 0 a 0 i n i n 0 we show that every symplectic matrix is a product of a symplectic unitary and a symplectic skew hermitian matrix we show that every symplectic matrix is a product of four symplectic skew hermitian matrices or a product of four symplectic hermitian matrices we give the possible jordan canonical forms of symplectic matrices which can be written as a product of a symplectic hermitian and a matrix which is either symplectic hermitian or symplectic skew hermitian

a modified kdv equation with self consistent sources in non uniform media and soliton dynamics two non isospectral modified kdv equations with self consistent sources are derived which correspond to the time dependent spectral parameter satisfying t and t 3 respectively gauge transformation between the first non isospectral equation corresponding to t and its isospectral counterpart is given from which exact solutions and conservation laws for the non isospectral one are easily listed besides solutions to the two nonisospectral modified kdv equations with self consistent sources are derived by means of the hirota method and the wronskian technique respectively nonisospectral dynamics and source effects including one soliton characteristics in non uniform media two solitons scattering and special behaviours related to sources for example the ghost solitons in the degenerate two soliton case are investigated analytically

connection conditions and the spectral family under singular potentials to describe a quantum system whose potential is divergent at one point one must provide proper connection conditions for the wavefunctions at the singularity generalizing the scheme used for point interactions in one dimension we present a set of connection conditions which are well defined even if the wavefunctions and or their derivatives are divergent at the singularity our generalized scheme covers the entire u 2 family of quantizations self adjoint hamiltonians admitted for the singular system we use this scheme to examine the spectra of the coulomb potential v x e2 x and the harmonic oscillator with square inverse potential v x m 2 2 x2 g x2 and thereby provide a general perspective for these models which have previously been treated with restrictive connection conditions resulting in conflicting spectra we further show that for any parity invariant singular potential v x v x the spectrum is determined solely by the eigenvalues of the characteristic matrix u u 2

soliton solutions to the 3rd nonisospectral akns system abstract bilinear form of the nonisospectral akns equation is given the n soliton solutions and double wronskian solution are obtained through hirota s direct method and wronskian technique respectively the nonisospectral mkdv equation and its multi soliton solutions are presented by reducing

changing dimension and time two well founded and practical techniques for path integration in quantum physics in a reasonably self contained presentation mathematical rigour is supplied to the important ideas of solving certain non gaussian path integrals by changes of dimension and or path dependent time transformations the resulting genuine path integral calculus neither requires discretization prescriptions nor sophisticated methods from the theory of stochastic differential equations the power of the calculus is illustrated by two standard quantum physics applications first the calculation of the time dependent propagator corresponding to a particle on the half line in a harmonic plus inverse square potential is shown to be a simple exercise second the first rigorous derivation of the energy dependent green function of the one dimensional morse system is given

time dependent propagator with point interaction we compute the time dependent schrodinger propagator with a point interaction in dimension n or 3 including the new cases of n 2 and the most general interaction supported by a point for n 1 we also give the small time asymptotics for n or 3 the case n 2 has the peculiarity of involving logarithmic terms in the expansion

products of positive semidefinite matrices abstract we characterize the complex square matrices which are expressible as the product of finitely many positive semidefinite matrices a matrix t can be expressed as such if and only if det t 0 moreover the number of factors can always be limited to five we also determine those matrices which can be expressed as the product of two or four positive semidefinite matrices these results are analogous to the ones obtained before by c s ballantine for products of positive definite matrices

new aspects of the path integrational treatment of the coulomb potential the well known treatment of the path integral for the coulomb potential by means of the kustaanheimo stiefel transformation and a time transformation is made more transparent by reducing the problem to the equality of two measures on the space of paths for this equality two proofs are given an elementary computational one and a short one recurring to general features of stochastic processes it is shown that the time transformation is a special case of the well known time change for a continuous local martingale by which the process is changed to a brownian motion and which is determined by its quadratic variation

products of diagonalizable matrices abstract let f denote a field such that char f 2 it is shown that every square matrix over f is expressible as a product of two diagonalizable matrices except when f gf 3 in which case three diagonalizable matrices are needed in general partial results for the case where char f 2 is also presented finally the extent to which the nullities of these factors can be prescribed is also investigated

multi soliton and pfaffian solutions of a 2 1 dimensional nonlinear evolution equation via the jaulent miodek hierarchy abstract under investigation in this paper is a 2 1 dimensional nonlinear evolution equation generated by the jaulent miodek hierarchy via the hirota bilinear method multi soliton solutions of such equation are obtained soliton propagation and interactions are analyzed and both the elastic and inelastic interactions are seen with some parameters selected furthermore it is proved that the equation possesses the n soliton solutions expressed in terms of gramm type pfaffian and can be written in the form of a pfaffian identity

the n soliton solutions for the non isospectral mkdv equation the bilinear form of the non isospectral mkdv equation is given and the exact n soliton solutions are obtained through hirota method and wronskian technique respectively c

note on the factorization of a square matrix into two hermitian or symmetric matrices this paper presents elementary proofs of the factorization of a square matrix into two hermitian or symmetric matrices

factorizations of a matrix into symmetric matrices with prescribed nullities abstract necessary and sufficient conditions are presented for a square matrix over an arbitrary field to be a product of k 2 symmetric matrices with prescribed nullities a generalization of the case k 2 to hermitian matrices is also presented

products of idempotent linear transformations in 1966 j m howie characterised the transformations of an arbitrary set that can be written as a product under composition of idempotent transformations of the same set in 1967 j a erdos considered the analogous problem for linear transformations of a finite dimensional vector space and in 1983 r j dawlings investigated the corresponding idea for bounded operators on a separable hilbert space in this paper we study the case of arbitrary vector spaces

exact wavefunctions for a time dependent coulomb potential the one dimensional schrodinger equation associated with a time dependent coulomb potential is studied the invariant operator method lewis and riesenfeld and unitary transformation approach are employed to derive quantum solutions of the system we obtain an ordinary second order differential equation whose analytical exact solution has been unknown it is confirmed that the form of this equation is similar to the radial schrodinger equation for the hydrogen atom in a arbitrary strong magnetic field the qualitative properties for the eigenstates spectrum are described separately for the different values of the parameter 0 appearing in the x2 term x being the position i e 0 0 0 0 and 0 0 for the 0 0 case the eigenvalue equation of invariant operator reduces to a solvable form and consequently we have provided exact eigenstates of the time dependent hamiltonian system