Field Theoretic Functional Calculus for the Anharmonic Oscillator in Low Approximations

1968 ◽  
Vol 23 (12) ◽  
pp. 1869-1887
Author(s):  
Wolfgang Schuler
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Test Problem ◽  
Infinite System ◽  
Anharmonic Oscillator ◽  
Explicit Representation ◽  
System Of Equations ◽  
Quantum Field ◽  
Functional Formulation ◽  
Technical Details

The theory of solution for quantum field functional equations as developped in II and III for a suitable test problem of quantum mechanics is investigated in low approximations. In Sect. 1 the functional formulation of the anharmonic oscillator is once more given and in Sect. 2 general translational equivalent functional equations. The expansion of the physical state functional into series of unsymmetrical and symmetrical base functionals and the representation of the functional equations for such expansions are discussed in Sect. 3. In the next Sect. 4 the unsymmetrical DYSON representation is investigated and the explicit representation of the smeared out functional equation by an infinite system of equations is derived. Then in Sect. 5 and 6 the system of equations is truncated for N = 3 and the corresponding eigenvalue equation is considered. The same is done in Sect. 7 and 8 for the HERWITTE representation. In the following Sect. 9 the original functional equation in a not smeared out form is treated in the DYSON representation and the corresponding system of unsymmetrized equations is given. Furthermore in Sect. 10 the N = 3 approximation together with other possibilities is investigated again. Finally the numerical results of our calculations for eigenvalues are stated and discussed. In the appendices technical details are derived.

scholarly journals On the Field Theoretic Functional Calculus for the Anharmonic Oscillator III

1968 ◽  
Vol 23 (6) ◽  
pp. 902-917 ◽  
Author(s):  
W. Schuler ◽  
H. Stumpf
Keyword(s):  
Quantum Mechanics ◽  
Functional Equations ◽  
Physical State ◽  
Anharmonic Oscillator ◽  
Explicit Representation ◽  
Necessary Condition ◽  
Quantum Field ◽  
Functional Formulation ◽  
Technical Details ◽  
Lowest Eigenvalue

The theory of solution for quantum field functional equations is developped for a suitable testproblem of quantum mechanics. In Sect. 1 the functional formulation of the anharmonic oscillator in its spinorial representation is given, and in Sect. 2 translational equivalent functional equations are discussed. The expansion of the physical state functionals into series of basefunctionals and the symmetrical representation of the functional equations for such an expansion is discussed in Sect. 3. In the following Sect. 4 the special symmetric orthogonal Hermitean functionals are used and the explicit representation is derived. In Sect. 5 the functionals are approximated by expansions with only a finite number of terms and the resulting equations are prepared for integration and in Sect. 6 a necessary condition of stationarity is considered. In Sect. 7 the simplest equation for N=1 is discussed in detail and the lowest eigenvalue is obtained. In the appendices technical details are derived.

On the Field Theoretic Functional Calculus for the Anharmonic Oscillator. II

1967 ◽  
Vol 22 (12) ◽  
pp. 1842-1865 ◽  
Author(s):  
W. Schuler ◽  
H. Stumpf
Keyword(s):  
Functional Equation ◽  
Quantum Mechanics ◽  
Finite Number ◽  
Functional Calculus ◽  
Functional Equations ◽  
Anharmonic Oscillator ◽  
Quantum Field ◽  
Operator Representation ◽  
Physical Conditions ◽  
Technical Details

The theory of solution for quantum field functional equations is developed for a suitable testproblem of quantum mechanics. In Sect. 1 the anharmonic oscillator is described in a field theoretic fashion. In Sect. 2 its functional equations are derived and in Sect. 3 these equations are symmetrized due to physical conditions. In Sect. 4, 5 the expansion of the physical functionals into series of base functionals is discussed and a convenient notation for the operator representation is introduced. In Sect. 6 the representation of the functional equation for an expansion into Dyson base functionals is given. In Sect. 7 and 8 functionals are approximated by expansions with only a finite number of terms and the resulting equations are prepared for integration. In Sect. 9, 10 the integration of the resulting equations for N = 2 and N = 4 is discussed in detail so that one finally obtains eigenvalue equations which contain only integrals to be solved. In the appendices technical details are derived.

On the Quantum Theory of the Anharmonic Oscillator in Functional Space

1969 ◽  
Vol 24 (2) ◽  
pp. 188-197 ◽  
Author(s):  
H. Stumpf
Keyword(s):  
Field Theory ◽  
Quantum Theory ◽  
Functional Equations ◽  
Scalar Product ◽  
Anharmonic Oscillator ◽  
Functional Space ◽  
Solution Procedure ◽  
Field Theories ◽  
Quantum Field ◽  
Product Definition

Dynamics of quantum field theory can be formulated by functional equations. For strong inter­action nonperturbative solutions of these functional equations are required. For the investigation of solution procedures the model of an anharmonic oscillator is used, because of its structural equi­valence with dressed one- and two-particel states of field theory. To perform a variational solution procedure a scalar product for the state functionals is introduced and its existence is proven. The scalar product definition admits a mapping of the physical Hilbert space on the functional space. Therefore a “functional” quantum theory seems to be possible. The whole procedure can be transferred to relativistic invariant field theories, provided these theories are regularized to give finite results at all.

Zur Konvergenz der einzeitigen Tamm-Dancoff-Methode beim anharmonischen Oszillator

1964 ◽  
Vol 19 (11) ◽  
pp. 1254-1267 ◽  
Author(s):  
H. Stumpf ◽  
F. Wagner ◽  
F. Wahl
Keyword(s):  
Infinite System ◽  
Anharmonic Oscillator ◽  
Approximation Procedure ◽  
Quantum Field ◽  
Matrix Elements ◽  
Eigenvalue Spectrum ◽  
The Matrix ◽  
Usual Quantum ◽  
Infinite Set ◽  
Limiting Value

As in the usual quantum field theory, the states, and therefore also the eigenvalue spectrum of an anharmonic oscillator can be characterized by means of the so-called τ-functions, that is the matrix element of the type 〈0 | qn | j〉. For the calculation of these matrix elements, the equation of motion of the anharmonic oscillator can be used to obtain an infinite set of equations, which define an eigenvalue problem. To solve it a new set of functions, the so-called φ-functions, are introduced by means of a transformation, whose matrix corresponds formally to the WICK rule. An analysis of this infinite system of φ-equations shows that a convergent secular polynomial can be obtained, which exists as a limiting value of the polynomials for the truncated N φ -equationsystems in the limit N → ∞ . It is therefore permissible to calculate the eigenvalues of the infinite system in an approximate way from the truncated systems. Such an approximation procedure is the essential content of the so-called TAMM-DANCOFF method. The above mentioned convergence of the determinants therefore provides its justification. The convergence of the eigenvalues of the truncated systems to the exact oscillator values is numerically examined up to N= 20. The results are satisfactory.

Functional Quantum Theory of Scattering Processes. II

1970 ◽  
Vol 25 (6) ◽  
pp. 795-803 ◽  
Author(s):  
H. Stumpf
Keyword(s):  
Quantum Theory ◽  
Particle Physics ◽  
Functional Equations ◽  
Potential Scattering ◽  
Quantum Field ◽  
Elementary Particle Physics ◽  
Calculational Method ◽  
S Matrix ◽  
Physical Information ◽  
Technical Details

Abstract Dynamics of quantum field theory can be formulated by functional equations. To develop a complete functional quantum theory one has to describe the physical information by functional operations only. The most important physical information of elementary particle physics is the S'-matrix. In this paper the functional S'-matrix is constructed for relativistic spin 1/2 fermion scattering in nonlinear spinortheory with noncanonical relativistic Heisenberg quantization. With appropriate modifications the procedure runs on the same pattern as in the case of nonrelativistic potential scattering treated in I. Furthermore a calculational method for scattering functionals is proposed. In the appendices technical details are discussed.

Solutions and Stability of Generalized Mixed Type QCA-Functional Equations in Random Normed Spaces

2013 ◽  
Vol 59 (2) ◽  
pp. 299-320
Author(s):  
M. Eshaghi Gordji ◽  
Y.J. Cho ◽  
H. Khodaei ◽  
M. Ghanifard
Keyword(s):  
Functional Equation ◽  
General Solution ◽  
Functional Equations ◽  
Mixed Type ◽  
Additive Functional ◽  
Normed Spaces ◽  
Additive Functional Equation ◽  
Probabilistic Normed Spaces ◽  
Random Normed Spaces ◽  
Generalized Stability

Abstract In this paper, we investigate the general solution and the generalized stability for the quartic, cubic and additive functional equation (briefly, QCA-functional equation) for any k∈ℤ-{0,±1} in Menger probabilistic normed spaces.

scholarly journals Stability of a generalized n-variable mixed-type functional equation in fuzzy modular spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Murali Ramdoss ◽  
Divyakumari Pachaiyappan ◽  
Choonkil Park ◽  
Jung Rye Lee
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
General Solution ◽  
Functional Equations ◽  
Mixed Type ◽  
Research Paper ◽  
Fixed Point Method ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Modular Spaces

AbstractThis research paper deals with general solution and the Hyers–Ulam stability of a new generalized n-variable mixed type of additive and quadratic functional equations in fuzzy modular spaces by using the fixed point method.

scholarly journals The new investigation of the stability of mixed type additive-quartic functional equations in non-Archimedean spaces

2020 ◽  
Vol 53 (1) ◽  
pp. 174-192
Author(s):  
Anurak Thanyacharoen ◽  
Wutiphol Sintunavarat
Keyword(s):  
Functional Equation ◽  
Normed Space ◽  
Functional Equations ◽  
Mixed Type ◽  
Additive Group ◽  
Ulam Stability ◽  
Quartic Functional Equation ◽  
The Stability

AbstractIn this article, we prove the generalized Hyers-Ulam stability for the following additive-quartic functional equation:f(x+3y)+f(x-3y)+f(x+2y)+f(x-2y)+22f(x)+24f(y)=13{[}f(x+y)+f(x-y)]+12f(2y),where f maps from an additive group to a complete non-Archimedean normed space.

scholarly journals On alienation of two functional equations of quadratic type

2021 ◽  
Author(s):  
Roman Ger
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
Real Numbers ◽  
Alpha Beta ◽  
Beta 2 ◽  
Special Case

Abstract  We deal with an alienation problem for an Euler–Lagrange type functional equation $$\begin{aligned} f(\alpha x + \beta y) + f(\alpha x - \beta y) = 2\alpha ^2f(x) + 2\beta ^2f(y) \end{aligned}$$ f ( α x + β y ) + f ( α x - β y ) = 2 α 2 f ( x ) + 2 β 2 f ( y ) assumed for fixed nonzero real numbers $$\alpha ,\beta ,\, 1 \ne \alpha ^2 \ne \beta ^2$$ α , β , 1 ≠ α 2 ≠ β 2 , and the classic quadratic functional equation $$\begin{aligned} g(x+y) + g(x-y) = 2g(x) + 2g(y). \end{aligned}$$ g ( x + y ) + g ( x - y ) = 2 g ( x ) + 2 g ( y ) . We were inspired by papers of Kim et al. (Abstract and applied analysis, vol. 2013, Hindawi Publishing Corporation, 2013) and Gordji and Khodaei (Abstract and applied analysis, vol. 2009, Hindawi Publishing Corporation, 2009), where the special case $$g = \gamma f$$ g = γ f was examined.

scholarly journals Algebraic functional equations and completely faithful Selmer groups

2015 ◽  
Vol 11 (04) ◽  
pp. 1233-1257
Author(s):  
Tibor Backhausz ◽  
Gergely Zábrádi
Keyword(s):  
Functional Equation ◽  
Elliptic Curve ◽  
Galois Group ◽  
Number Field ◽  
Functional Equations ◽  
Complex Multiplication ◽  
Selmer Groups ◽  
Selmer Group ◽  
Torsion Module ◽  
Ordinary Reduction

Let E be an elliptic curve — defined over a number field K — without complex multiplication and with good ordinary reduction at all the primes above a rational prime p ≥ 5. We construct a pairing on the dual p∞-Selmer group of E over any strongly admissible p-adic Lie extension K∞/K under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group G = Gal(K∞/K). Under some mild additional hypotheses, this gives an algebraic functional equation of the conjectured p-adic L-function. As an application, we construct completely faithful Selmer groups in case the p-adic Lie extension is obtained by adjoining the p-power division points of another non-CM elliptic curve A.

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