Quantum Harmonic Oscillator Complete System

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The wave functions of a quantum harmonic oscillator form a complete and orthogonal system over the interval of negative infinity to positive infinity. Unlike the particle in a box, where the system is complete only within the boundaries of the well, the harmonic oscillator's wave functions extend infinitely. The Hermite polynomials are integral to this system, providing an orthogonal basis for the Hilbert space. This completeness and orthogonality are crucial for understanding quantum mechanics. Thus, the harmonic oscillator's wave functions are indeed complete and orthogonal across the entire real line.
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Over which interval do the wave functions of a harmonic oscillator form a complete and orthogonal system? Is it (-inf,+inf)? The case with particle in a box is rather clear(system is complete and orthogonal only for the interval of the well), however the harmonic oscillator is a bit less intuitive.
 
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Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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