Bargmann-Type Transforms and Modified Harmonic Oscillators
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We study some complete orthonormal systems on the real line. These systems are determined by Bargmann-type transforms, which are Fourier integral operators with complex-valued quadratic phase functions. Each system consists of eigenfunctions for a second-order elliptic differential operator like the Hamiltonian of the harmonic oscillator. We also study the commutative case of a certain class of systems of second-order differential operators called the non-commutative harmonic oscillators. By using the diagonalization technique, we compute the eigenvalues and eigenfunctions for the commutative case of the non-commutative harmonic oscillators. Finally, we study a family of functions associated with an ellipse in the phase plane. We show that the family is a complete orthogonal system on the real line.
KeywordsBargmann-type transforms Segal–Bargmann spaces Berezin–Toeplitz quantization Generalized Hermite functions Modified harmonic oscillators
Mathematics Subject ClassificationPrimary 47B35 Secondary 47B32 47G30
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