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AppliedMath

On a Schrödinger Equation in the Complex Space Variable

Manuel Leote Tavares Ingles Esquivel, Nadezhda P. Krasii, Philippe L. Didier

Published Dec 19, 2024• 0 citations
Publisher

Abstract

We study a separable Hilbert space of smooth curves taking values in the Segal–Bergmann space of analytic functions in the complex plane, and two of its subspaces that are the domains of unbounded non self-adjoint linear partial differential operators of the first and second order. We show how to build a Hilbert basis for this space. We study these first- and second-order partial derivation non-self-adjoint operators defined on this space, showing that these operators are defined on dense subspaces of the initial space of smooth curves; we determine their respective adjoints, compute their respective commutators, determine their eigenvalues and, under some normalisation conditions on the eigenvectors, we present examples of a discrete set of eigenvalues. Using these derivation operators, we study a Schrödinger-type equation, building particular solutions given by their representation as smooth curves on the Segal–Bergmann space, and we show the existence of general solutions using an Fourier–Hilbert base of the space of smooth curves. We point out the existence of self-adjoint operators in the space of smooth curves that are obtained by the composition of the partial derivation operators with multiplication operators, showing that these operators admit simple sequences of eigenvalues and eigenvectors. We present two applications of the Schrödinger-type equation studied. In the first one, we consider a wave associated with an object having the mass of an electron, showing that two waves, when considered as having only a free real space variable, are entangled, in the sense that the probability densities in the real variable are almost perfectly correlated. In the second application, after postulating that a usual package of information may have a mass of the order of magnitude of the neutron’s mass attributed to it—and so well into the domain of possible quantisation—we explore some consequences of the model.

DOI
10.3390/appliedmath4040083

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We study a separable Hilbert space of smooth curves taking values in the Segal–Bergmann space of analytic functions in the complex plane, and two of its subspaces that are the domains of unbounded non self-adjoint linear partial differential operators of the first and second order. We show how to build a Hilbert basis for this space. We study these first- and second-order partial derivation non-self-adjoint operators defined on this space, showing that these operators are defined on dense subspaces of the initial space of smooth curves; we determine their respective adjoints, compute their respective commutators, determine their eigenvalues and, under some normalisation conditions on the eigenvectors, we present examples of a discrete set of eigenvalues. Using these derivation operators, we study a Schrödinger-type equation, building particular solutions given by their representation as smooth curves on the Segal–Bergmann space, and we show the existence of general solutions using an Fourier–Hilbert base of the space of smooth curves. We point out the existence of self-adjoint operators in the space of smooth curves that are obtained by the composition of the partial derivation operators with multiplication operators, showing that these operators admit simple sequences of eigenvalues and eigenvectors. We present two applications of the Schrödinger-type equation studied. In the first one, we consider a wave associated with an object having the mass of an electron, showing that two waves, when considered as having only a free real space variable, are entangled, in the sense that the probability densities in the real variable are almost perfectly correlated. In the second application, after postulating that a usual package of information may have a mass of the order of magnitude of the neutron’s mass attributed to it—and so well into the domain of possible quantisation—we explore some consequences of the model.
This paper derives the non-relativistic time-dependent Schrödinger equation as the logically unavoidable dynamical law for de Broglie matter waves, extending the reconstruction begun in the companion work “The Logical Origin of the de Broglie Matter Wave and its Physical Meaning (the Born Rule)” (Chan, 2026). Starting from the same minimal premises—continuous position in real space, Fourier conjugacy (and the logical uncertainty principle), translation invariance via Noether’s theorem, and probability conservation via statistical energy flux—the Schrödinger equation follows deductively once norm preservation and classical non-relativistic correspondence are enforced. The derivation proceeds in clear stages: Plane-wave solutions for free particles and constant potentials, Quadratic dispersion relation from classical kinematics, Exact local plane-wave structure (defined via phase gradients) that must hold at every point for any smooth wave function, The unique linear differential operator of order ≤ 2 that consistently “glues” these local structures across smooth external potentials V(x,t) while preserving the L²-norm (unitarity). No canonical quantization, operator promotion, commutation relations, or postulated wave equations are used. The Born rule is taken as already established from the prior paper and serves only to constrain unitary evolution. The Schrödinger equation thus emerges not as an empirical postulate or convenient ansatz, but as the mathematically and symmetrically forced description of indivisible particles propagating in continuous, translationally invariant (or smoothly perturbed) space. Domain of validity and natural limitations (non-relativistic kinematics, smooth potentials, single-particle/mean-field regime, closed-system unitarity) are explicitly delineated.
In this paper, we propose a rational and almost obliged path leading us from the most natural assumption regarding the SchrÅNodinger wave functions (wave functions classically belonging to the domain of the classic Schrödinger equation) to the space of tempered distributions defined upon the Minkowski space time. It’s extremely natural to consider the space E of complex valued smooth functions as the natural domain of the Schrödinger equation, as it was the obvious implicit assumption of any partial differential equation with constant coefficients at the time of Schrödinger himself. The orthodox Born interpretation of the normalized wave functions and the intervention of von Neumann (a Hilbert‘s pupil) have deviated the straightforward understanding of that domain towards the unnecessary and unnatural Hilbert space L2. The Hilbert space of square integrable functions is clearly incompatible with any partial differential equation, which would require at least Sobolev Spaces to live in; unfortunately, the inner product of the Sobolev space H2 is not good for quantum mechanics and the de Broglie solutions of the Schrödinger equation do not belong to L2. We show in this article that moving inside the very good space E - and finally inside the huge tempered distribution space S′ - represents the first framework in which any desirable property of the key characters and actors of basic Quantum Mechanics is satisfied.
A method for solution of the Schrödinger equation in momentum space is described. If the Schrodinger equation in momentum space is represented by Oχ=0, where O is an integral operator and χ is the momentum function, the method involves two steps: (i) the representation of χ as a rational function of P and (ii) expansion of Oχ in a power series in some appropriately chosen variable t (‖t‖≤1) which is a function of P. Equating to zero the coefficients of the first N powers of t (which determines the eigenvalues) makes Oχ differ from zero by terms of order tN. As N increases the eigenvalues approach the correct limiting values if the method converges. Within its circle of convergence a power series converges for both real and complex values of the variable and hence the Schrödinger equation, an integral equation, is forced to hold for both real and complex values of the variable. This is in contrast to variational methods which involve only real values of the independent variable. To test the method the motion of a particle in the field of a Yukawa potential is studied. Only the lowest energy state is considered. Two extreme cases, involving linear combinations of rational basis sets, are studied. Both basis sets are complete but the power series method converges only for one basis set. For the other set the power series method provides an eigenvalue which oscillates around the correct value without approaching a limit but a variation method using the same basis set converges rapidly to the correct eigenvalue. The variation method succeeds because the function is defined only on the real axis while the power series method fails because, in the complex plane, the basis set can only represent a meromorphic function (one whose only finite singular points are poles) and the momentum eigenfunction cannot be meromorphic. Thus the choice of a wrong basis set in a variation method can completely conceal the true analytic nature of an eigenfunction. It is shown that momentum eigenfunctions are not meromorphic because they possess branch points. This is true not only for the Yukawa potential but for several one-electron models including the Hartree–Fock equations.
Although• quantum mechanics has achieved great success in the explanation of physical phenomena, the mathematics used in quantum mechanics is not rigorous.Unrigorous treatments occur in the case of continuous spectrum.In a Hilbert • space, let A be a self-adjoint operator.When we define the eigenvalue and the eigenelement by the ' ' relation Af=J.f, (1) ' ' h d -. -f(q)=).f(q)2m dq has solutions 2,ri f(q)=ce"Aq for any ;.., But these are not quadratically integrable.E. Hellinger and H. Weyi< 0 have already investigated the problem of continuous spectrum and introduced the " eigendifferential."But they are concerned with a special problem, i. e. bounded quadratic forms and solutions of differential equation.In respect of generality, they do not satisfy.< 2 > E. Fues< 3 > and some other writers<" have used this conception " eigendifferential " in quantum mechanics.But the treatments are either complicated or not rigorous.J. v. Neumann< 5 > has proved that in Hilbert space, for any selfadjoint operator A, there exists a resolution of identity E(;..) such that Af= J:and he has applied this theory to quantum mechanics.Although his theory gave a rigorous mathematical foundation to quantum mechanics, it did not give any interpretation to the representation theory of Dirac's form.Recently I investigated the "theory of vector-valued set functions."This theory removes the difficulties in the treatement of continuous spectrum, and gives a rigorous interpretation to the representation theory.< 5 > The vector-valued set function q(U) is a function of set U, and its functional value is an element (vector) in the Hilbert ,space, (1) E. Hellinger [l], 231-271.H. Weyl [1], 238-251.(2) T. Carlemann used this conception "eigendifferential" in the treatise of singular integral equations.Cf.T. Carlemann [1].(3) E. Fues [1], 295-303.(4) For example, L. de Broglie [2], 140-159.(5) J. v. Neumann [1], 53-121.(6) F. Maeda [2], [5], [6].A summary of the theory of vector-valued set functions is given in the first part Qf the present paper.(1) G. Temple has formulated a principle of quantum theory in which projective operators are given priority over self-adjoint operators.(C(.G. Temple [1)).But. the .whole discussion differs from that of my present paper.
A general formulation of the eigenvalue condition is pointed out for a one-dimensional stationary part of a separable Schrodinger equation with finite bounds, i.e. where the wavefunction vanishes at two arbitrary values of the independent variable. The procedure is general and applies equally to other eigenvalue problems of second-order, homogeneous differential equations. Where appropriate a continuous density of eigenvalues can be defined and evaluated without first solving the eigenvalue problem. Illustrative examples are provided, including gravitational and harmonic oscillator potentials.