XI Americas Conference on Differential Equations and Nonlinear Analysis Construction of a Fractional Schrödinger Equation and Some First Predictions on Quantum Mechanical Problems ä Demian Nahuel Goos1,2,3

1 Departamento de Matemática, FCEIA, Universidad Nacional de Rosario, Argentina 2 CONICET, FCEIA, Universidad Nacional de Rosario 3 Instituto de Estudios Nucleares y Radiaciones Ionizantes, FCEIA, Universidad Nacional de Rosario

Introduction Particle in an infiinite potential well (independent of 푡) 0, 푥 ∈ −푎, 푎 Quantum Mechanics is a powerful theory, which revolutionized the The FSE with 푉 푥 = ቊ reads ∞, 푒푙푠푒 understanding of phenomena at small scales together with the study of the energy levels in these systems. Its consistency has ℏ2훼 휕훼 휕훼 훹 푥, 푡 + ε훼 − 푉(푥) 훹 푥, 푡 = 0, experimentally been proven in a huge range of applications. 2휇훼 휕푥훼 휕푥훼 Nevertheless, there are still different problems which are not with either even or odd solutions sufficiently well described by these postulates. 훼 훼 Within this theory, the Schrödinger equation plays a fundamental 퐴푠푖푛훼 푘 −푥 , 푥 ∈ −푎, 0 훼 훼 role, since it describes the changes over time of a physical system in 휓 푥 = ൞ −퐴푠푖푛훼 푘 푥 , 푥 ∈ 0, 푎 which quantum effects are significant. A simple way to construct this 0, 푒푙푠푒. equation is by combining the momentum 푝ො and energy 퐸෠ operators 퐴 serves as normalization constant and 푘 can be obtained with the and their general interpretations. continuity condition, 훼 훼 If we now consider fractional generalizations of these operators, 푠푖푛훼 푘 푎 = 0. 휕 휕훼 휕 휕훼 This also implies that the energy states are quantized, but unlike the 푝ො = −푖ℏ ⟶ 푝ො = −푖ℏ훼 퐸෠ = 푖ℏ ⟶ 퐸෠ = 푖ℏ훼 휕푥 훼 휕푥훼 휕푡 훼 휕푡훼 classical approach, there are finite many energy states, depending which can be obtained by replacing the regular plane wave on 훼 (훼 can be fixed in order to obtain as many states as desired). Ψ 푥, 푡 = 푒푖(푘푥−휔푡) by a fractional approximation, For a potential 푊 푥 = 푉 푥 − 푐 the exact same result is obtained by 훼 훼 훼 훼 Ψ 푥, 푡 = 퐸훼,1 푖푘 푥 퐸훼,1 −푖휔 푡 , where shifting the center of derivation. ∞ 푠푗 퐸 푠 = ෍ , 훼,훽 Γ(훼푗 + 훽) 휅 푗=0 Particle in a finite potential well the resulting Fractional Schrödinger 푉 훼, 푥 ∈ −푎, 푎 훼 훼 훼 훼 0 Equation (FSE) is given by Fig. 1: Fractional wave Ψ 푥, 푡 = 퐸훼,1 푖푘 푥 퐸훼,1 −푖휔 푡 Fig. 2: Modulus of the fractional wave Ψ 푥, 푡 The FSE with 푉 푥 = ቊ reads 휕훼 ℏ2훼 휕훼 휕훼 0, 푒푙푠푒 훼 2훼 훼 훼 푖ℏ 훼 훹 푥, 푡 = − 훼 훼 훼 + 푉(푥) 훹 푥, 푡 . ℏ 휕 휕 휕푡 2휇 휕푥 휕푥 훹 푥, 푡 + ε훼 − 푉(푥) 훹 푥, 푡 = 0. 2휇훼 휕푥훼 휕푥훼 For ε훼 < 푉 훼 there are either even or odd Fractional analysis 0 solutions Fig. 3: Solution to the particle in an infinite potential well Fig. 4: Solution to the particle in a finite potential well 훼훼 훼훼 Definition: Caputo Fractional Derivative 퐸퐸훼훼 푘푘 ((−−푥푥)) ,, 푥 ∈ (−∞, −푎) + 훼 훼 훼 훼 The left and right CFD of order 훼 ∈ ℝ and starting point 푎 of 푓 are −퐴−푠푖푛퐴푠푖푛훼 휅훼 휅−푥−푥 , , 푥 ∈ −푎, 0 푏 훼 훼 훼 휓 푥 = 훼 훼 훼 훼 −1 2훼 2휇 푉0 +휀 퐴푠푖푛퐴훼푠푖푛휅 훼푥휅 ,푥 , 푥 ∈ 0, 푎 퐶 훼 −훼 ′ 휕훼 휅 = 퐷푠 퐷푏 푓 푠 = න 푧 − 푠 푓 푧 푑푧, 훹 푥, 푡 = .퐶 퐷훼훹 푥, 푡 ℏ2훼 퐸 퐸푘훼푥푘훼훼푥,훼 , 푥 ∈ (푎, ∞) Γ 1 − 훼 휕푡훼 0 푡 훼 훼 푠 휕훼 푠 퐶 훼 휕훼 퐷 퐷휕푡훼 훹 푥, 푡 , 푥 < 0 1 훹 푥, 푡 =൝ 푥 0 To obtain the discrete energy states, the following transcendental 퐷퐶퐷훼푓(푠)= න(푠 − 푧)−훼푓′ 푧 푑푧. 휕푥훼 퐷퐶퐷훼훹 푥, 푡 , 푥 ≥ 0 푎 푠 Γ(1 − 훼) 0 푥 equation can be used. 푎 훼 훼 훼 훼 훼 퐸훼,훼 푘 푎 휅 푐표푠훼,훼 휅 푎 Definition: Fractional sine and cosine functions 훼 훼 = − 훼 훼 훼 퐸훼 푘 푎 푘 푠푖푛훼 휅 푎 The fractional sine and cosine functions of parameter 훼 ∈ ℝ+ are ∞ ∞ (−1)푗푠2푗 (−1)푗푠2푗+1 푐표푠 (푠) = ෍ 푠푖푛 (푠) = ෍ 훼,훽 Γ(훼 2푗 + 훽) 훼,훽 Γ(훼 2푗 + 1 + 훽) Future work 푗=0 푗=0

푐표푠 (푠) 2 0.99,1 푠푖푛0.99,1(푠) Operators 푝ො and 퐸෠ are not self-adjoint in 퐿 ℝ . A suitable space of functions, in which these operators are self-adjoint, has to be constructed. 푠 푠 훼 훼 Study of scattering phenomena (finite potential well with ε > 푉0 ) 훼 훼 and quantum tunneling (finite potential barrier with ε < 푉0 ), physical implications of the singularity point, amongst others. Study of applications of the fractional quantum mechanical model to physical mathematics and quantum mechanics. Free particle with zero angular momentum (independent of 푡) ℏ2훼 휕훼 휕훼 The FSE with 푉 푥 = 0 reads 훹 푥, 푡 + ε훼훹 푥, 푡 = 0, Bibliography 2휇훼 휕푥훼 휕푥훼 훼 훼 훼 훼 [1] K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Heidelberg (2004). 퐴퐸훼 푖푘 (−푥) + 퐵퐸훼 −푖푘 (−푥) , 푥 < 0 [2] V. I. Kukulin, V. M. Krasnopolsky, J. Horacek, Theory of Resonances: Principles and Applications, Kluwer Academic Publishers, (1988). with solution 휓 푥 = ቊ 훼 훼 훼 훼 [3] E. Rebhan, Theoretische Physik: Quantenmechanik, Spektrum Akademischer Verlag, (2008). 퐴퐸 푖푘 푥 + 퐵퐸 −푖푘 푥 , 푠푥 ≥ 0, [4] H. P. Schieck, Nuclear Reactions, Springer-Verlag Berlin Heidelber, (2014). 훼 훼 [5] H.J. Haubold, A.M. Mathai, R.K. Saxena, Mittag-Leffler Functions and Their Applications, J. Appl. Math., 298-628, (2011).

[6] J. W. Hanneken, D. M. Vaught, B. N. Narahari Achar, Enumeration of the Real Zeros of the Mittag-Leffler Function E훼(z), 1 < 훼 < 2, Advances in 2휇휀 Fractional Calculus, pp 15-26, (2007). 2 [7] R. Goreno, J. Loutchko, Y. Luchko, Computation of the Mittag-Leffler function E and its derivative, Fract. Calc. Appl. Anal. 5, No 4, 491-518, (2002). wave number 푘 = and priviliged point/singularity at 0. 훼 ℏ2